equation_gpt_train_financial.jsonl

{"completion": " The related materials of Asian Options is: 1.What Is an Asian Option? How They Work Vs. Standard Options:<url>https://www.investopedia.com/terms/a/asianoption.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Asian Options?"}
{"completion": " The related materials of Black-Derman-Toy BDT is: 1.A Binomial Interest Rate Model and the Black-Derman-Toy Model:<url>https://web.ma.utexas.edu/users/mcudina/Lecture24_4and5.pdf</url>;2.Black-Derman-Toy Model Developed by Team at Goldman Sachs:<url>https://www.goldmansachs.com/our-firm/history/moments/1986-black-derman-toy-model.html</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Black-Derman-Toy BDT?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Standard Brownian Motionis <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Standard Brownian Motion?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Put Option Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options is: put protection if the stock price should fall..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts is: Risk-neutral equations.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Calls and Puts with Different Strikesis <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Calls and Puts with Different Strikes?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Stock Prices as Geometric Brownian Motionis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Bonds and Interest Ratesis <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Bonds and Interest Rates?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity is: the present value of the strike price (x),.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Ito Lemmais <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Ito Lemma?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options is: put protection if the stock price should fall..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations is: The variance of stock price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>U</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates is: Non-continuous annualized rate.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{1}.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Ito Lemma is: <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. Here is the brief introduction of Ito Lemma:\n        Latex code for the Ito Lemma.\n        \n: Diffusion\n: Stochastic differential equation for X(t)\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Ito Lemma?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t)</equation> in Standard Brownian Motion is: Value of Z at time stamp t.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>G(T)</equation> in Asian Options is: Geometric average of option price at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>G(T)</equation> in Asian Options?"}
{"completion": " The equation for Geometric Brownian Motion is: <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. Here is the brief introduction of Geometric Brownian Motion:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Geometric Brownian Motion?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: Pay the strike earlier and therefore miss the interest on that money.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations is: The variance of stock price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts is: Risk-neutral equations.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio is: Sharpe ratio of any asset written on a GBM.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity is: the present value of the strike price (x),.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\sigma</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Bonds and Interest Ratesis <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Bonds and Interest Rates?"}
{"completion": " The equation for Call and Put Price Bounds is: <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. Here is the brief introduction of Call and Put Price Bounds:\n\n        Latex code for the Calls and Puts Arbitrage. The following equations give the bounds on the prices of European calls and puts. Note that the lower bounds are no less than zero. We can also compare the prices of European and American options using the following inequalities.\n        \n\n: European Call Option Price\n: European Put Option Price\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Call and Put Price Bounds?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage is: Strike price of option 3.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The Latex Code for equation Call and Put Price Boundsis <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Call and Put Price Bounds?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>X</equation> in Ito Lemma is: Diffusion.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>X</equation> in Ito Lemma?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The Latex Code for equation Cox-Ingersoll-Ross CIRis <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion is: Observed stock price S(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Monte-Carlo Simulationsis <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Monte-Carlo Simulations?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>g_{i}</equation> in Monte-Carlo Simulations is: The i-th simulated payoff.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>g_{i}</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Varying Times to Expirationis <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Varying Times to Expiration?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates is: Forward Bond Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Call and Put Price Bounds is: <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. Here is the brief introduction of Call and Put Price Bounds:\n\n        Latex code for the Calls and Puts Arbitrage. The following equations give the bounds on the prices of European calls and puts. Note that the lower bounds are no less than zero. We can also compare the prices of European and American options using the following inequalities.\n        \n\n: European Call Option Price\n: European Put Option Price\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Call and Put Price Bounds?"}
{"completion": " The equation for Early Exercise for American Options is: <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. Here is the brief introduction of Early Exercise for American Options:\n\n            So we exercise the call option if the pros are greater than the cons, specifically, we exercise if:\n        \n\n: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons. \n: Early Exercise getting the stock's dividend payments\n: Pay the strike earlier and therefore miss the interest on that money\n: put protection if the stock price should fall.\n: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise. \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Early Exercise for American Options?"}
{"completion": " The Latex Code for equation Black-Derman-Toy BDTis <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Black-Derman-Toy BDT?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of put option p.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Geometric Brownian Motionis <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Geometric Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs is: Volatility.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\sigma</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage is: Strike price of option 2.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>K_{2}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts is: Price of the power contract.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio is: Sharpe ratio.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Cox-Ingersoll-Ross CIRis <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Cox-Ingersoll-Ross CIR?"}
{"completion": " The Latex Code for equation Black-Derman-Toy BDTis <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Monte-Carlo Simulationsis <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Monte-Carlo Simulations?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is: <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. Here is the brief introduction of Cox-Ingersoll-Ross CIR:\n        Latex code for the Cox-Ingersoll-Ross model. \n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nCox-Ingersoll-Ross Model\nA THEORY OF THE TERM STRUCTURE OF INTEREST RATES\nCox-Ingersoll-Ross (CIR) model-Mathworks\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Calls and Puts with Different Strikes is: <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. Here is the brief introduction of Calls and Puts with Different Strikes:\n\n        Latex code for the Calls and Puts with Different Strikes. For European calls and puts, with strike prices K_{1} and K_{2}, where K_{1} < K_{2}, we know the following.\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\n            American options, For three different options with strike prices K1 < K2 < K3:\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Calls and Puts with Different Strikes?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\delta</equation> in Sharpe Ratio is: Compounded Dividend Rate.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\delta</equation> in Sharpe Ratio?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>U</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>U</equation> in Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Forwardsis <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Forwards?"}
{"completion": " The equation for Calls and Puts with Different Strikes is: <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. Here is the brief introduction of Calls and Puts with Different Strikes:\n\n        Latex code for the Calls and Puts with Different Strikes. For European calls and puts, with strike prices K_{1} and K_{2}, where K_{1} < K_{2}, we know the following.\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\n            American options, For three different options with strike prices K1 < K2 < K3:\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Calls and Puts with Different Strikes?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates is: Non-continuous annualized rate.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. Here is the brief introduction of Risk-Neutral Valuation and Power Contracts:\n        Latex code for Risk-Neutral Valuation and Power Contracts.\n        \n: Payoff a power contract at time T\n: Price of the power contract\n: Risk-neutral equations\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The related materials of Standard Brownian Motion is: 1.Investopedia Simple Interest:<url>https://www.investopedia.com/terms/s/simple_interest.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Standard Brownian Motion?"}
{"completion": " The equation for Standard Brownian Motion is: <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. Here is the brief introduction of Standard Brownian Motion:\n        Latex code for the Standard Brownian Motion. I will briefly introduce the notations in this formulation. {Z(t)} has independent increments, and {Z(t)} has stationary increments such that Z (t + s) ? Z (t) follows standard normal distribution\n        \n: Value of Z at time stamp t\n: Stationary increments of Standard Brownian Motion\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Standard Brownian Motion?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>g_{i}</equation> in Monte-Carlo Simulations is: The i-th simulated payoff.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>g_{i}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(T)</equation> in Monte-Carlo Simulations is: The stock price at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>S(T)</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{1}.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The Latex Code for equation Early Exercise for American Optionsis <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Early Exercise for American Options?"}
{"completion": " The equation for Ito Lemma is: <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. Here is the brief introduction of Ito Lemma:\n        Latex code for the Ito Lemma.\n        \n: Diffusion\n: Stochastic differential equation for X(t)\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Ito Lemma?"}
{"completion": " The Latex Code for equation Asian Optionsis <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Asian Options?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage is: Strike price of option 3.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Black-Derman-Toy BDT is: <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. Here is the brief introduction of Black-Derman-Toy BDT:\n        Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0.\n        The yield volatility for period-3 is .\n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nBlack-Derman-Toy Model Developed by Team at Goldman Sachs\nA Binomial Interest Rate Model and the Black-Derman-Toy Model\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Black-Derman-Toy BDT?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion SDEs is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>U</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity is: .", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is: <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. Here is the brief introduction of Cox-Ingersoll-Ross CIR:\n        Latex code for the Cox-Ingersoll-Ross model. \n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nCox-Ingersoll-Ross Model\nA THEORY OF THE TERM STRUCTURE OF INTEREST RATES\nCox-Ingersoll-Ross (CIR) model-Mathworks\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Cox-Ingersoll-Ross CIR?"}
{"completion": " The Latex Code for equation Standard Brownian Motionis <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Standard Brownian Motion?"}
{"completion": " The Latex Code for equation Sharpe Ratiois <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Sharpe Ratio?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. Here is the brief introduction of Stock Prices as Geometric Brownian Motion:\n        Latex code for Stock Prices as Geometric Brownian Motion.\n        \n: Observed stock price S(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{2}.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The Latex Code for equation Cox-Ingersoll-Ross CIRis <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity is: the present value of the strike price (x),.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Early Exercise for American Optionsis <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Early Exercise for American Options?"}
{"completion": " The Latex Code for equation Stock Prices as Geometric Brownian Motionis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Ito Lemmais <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Ito Lemma?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Monte-Carlo Simulationsis <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Monte-Carlo Simulations?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. Here is the brief introduction of Risk-Neutral Valuation and Power Contracts:\n        Latex code for Risk-Neutral Valuation and Power Contracts.\n        \n: Payoff a power contract at time T\n: Price of the power contract\n: Risk-neutral equations\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates is: Non-continuous annualized rate.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts is: Price of the power contract.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is: <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. Here is the brief introduction of Geometric Brownian Motion SDEs:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(0, S)</equation> in Bonds and Interest Rates is: Price of an s-year zero..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P(0, S)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Forwards is: <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. Here is the brief introduction of Forwards:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. A forward contract is an agreement in which the buyer agrees at time t to pay the seller at time T and receive the asset at time T.\n        \n: Forward Contract at strike price S\n: Interest Rate\n: Future Value\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Forwards?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: Early Exercise getting the stock's dividend payments.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The Latex Code for equation Asian Optionsis <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Asian Options?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion is: Stationary increments of Standard Brownian Motion.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\sigma</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts is: Payoff a power contract at time T.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>r</equation> in Forwards is: Interest Rate.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>r</equation> in Forwards?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\mu</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Calls and Puts with Different Strikes is: <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. Here is the brief introduction of Calls and Puts with Different Strikes:\n\n        Latex code for the Calls and Puts with Different Strikes. For European calls and puts, with strike prices K_{1} and K_{2}, where K_{1} < K_{2}, we know the following.\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\n            American options, For three different options with strike prices K1 < K2 < K3:\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>Y(t)</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{2}.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Forwards is: <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. Here is the brief introduction of Forwards:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. A forward contract is an agreement in which the buyer agrees at time t to pay the seller at time T and receive the asset at time T.\n        \n: Forward Contract at strike price S\n: Interest Rate\n: Future Value\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Forwards?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>F_{t,T}(S)</equation> in Forwards is: Forward Contract at strike price S.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>F_{t,T}(S)</equation> in Forwards?"}
{"completion": " The equation for Bonds and Interest Rates is: <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. Here is the brief introduction of Bonds and Interest Rates:\n        Latex code for the Bonds and Interest Rates. The price of an s-year zero is P(0, S). The forward bond price formula  is calculated as . And the \n        non-continuous annualized rate is .\n        \n: Price of an s-year zero.\n: Forward Bond Price\n: Non-continuous annualized rate\n\n\nRelated Documents\n\nBond and Interest Rate\nUnderstanding Interest Rates, Inflation, and Bonds\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Bonds and Interest Rates?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of call option c.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The equation for Geometric Brownian Motion is: <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. Here is the brief introduction of Geometric Brownian Motion:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Geometric Brownian Motion?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates is: Forward Bond Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations is: The variance of stock price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts is: Risk-neutral equations.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion SDEs is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>U</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The related materials of Geometric Brownian Motion is: 1.Geometric Brownian Motion:<url>https://www.quantstart.com/articles/Geometric-Brownian-Motion/</url>;2.Geometric Brownian motion:<url>http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Geometric Brownian Motion?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Monte-Carlo Simulations is: <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. Here is the brief introduction of Monte-Carlo Simulations:\n        Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as , convert to normal variable .  The variance of the Monte-Carlo estimate is calculated as . \n        \n: The stock price at time T\n: The stock price at time t, which is nearer to final stage stock price S(T)\n: The i-th simulated payoff\n: The variance of stock price\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Monte-Carlo Simulations?"}
{"completion": " The equation for Monte-Carlo Simulations is: <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. Here is the brief introduction of Monte-Carlo Simulations:\n        Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as , convert to normal variable .  The variance of the Monte-Carlo estimate is calculated as . \n        \n: The stock price at time T\n: The stock price at time t, which is nearer to final stage stock price S(T)\n: The i-th simulated payoff\n: The variance of stock price\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Monte-Carlo Simulations?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>F_{t,T}(S)</equation> in Forwards is: Forward Contract at strike price S.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>F_{t,T}(S)</equation> in Forwards?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio is: Sharpe ratio.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Calls and Puts Arbitrageis <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Calls and Puts Arbitrage?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>X</equation> in Ito Lemma is: Diffusion.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>X</equation> in Ito Lemma?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The Latex Code for equation Stock Prices as Geometric Brownian Motionis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Black-Derman-Toy BDTis <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Calls and Puts with Different Strikesis <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Calls and Puts with Different Strikes?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>A(T)</equation> in Asian Options is: Arithmetic average of option price at time T.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>A(T)</equation> in Asian Options?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>G(T)</equation> in Asian Options is: Geometric average of option price at time T.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>G(T)</equation> in Asian Options?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The related materials of Stock Prices as Geometric Brownian Motion is: 1.Geometric Brownian Motion:<url>https://www.quantstart.com/articles/Geometric-Brownian-Motion/</url>;2.Geometric Brownian motion:<url>http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Calls and Puts with Different Strikesis <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Calls and Puts with Different Strikes?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t)</equation> in Standard Brownian Motion is: Value of Z at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Varying Times to Expiration is: <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. Here is the brief introduction of Varying Times to Expiration:\n\n            For American options, when expiration T2 > T1, the above equations holds.\n        \n\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Varying Times to Expiration?"}
{"completion": " The Latex Code for equation Early Exercise for American Optionsis <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Early Exercise for American Options?"}
{"completion": " The equation for Monte-Carlo Simulations is: <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. Here is the brief introduction of Monte-Carlo Simulations:\n        Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as , convert to normal variable .  The variance of the Monte-Carlo estimate is calculated as . \n        \n: The stock price at time T\n: The stock price at time t, which is nearer to final stage stock price S(T)\n: The i-th simulated payoff\n: The variance of stock price\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Monte-Carlo Simulations?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>G(T)</equation> in Asian Options is: Geometric average of option price at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>G(T)</equation> in Asian Options?"}
{"completion": " The equation for Put-Call Parity is: <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. Here is the brief introduction of Put-Call Parity:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. Call options give the owner the right, but not the obligation, to buy an asset at some time in the future for a predetermined strike price. Put options give the owner the right to sell. The price of calls and puts is compared in the following put-call parity formula for European options.\n        \n: Price of call option c\n: Price of put option p\n: the present value of the strike price (x), \n: \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Put-Call Parity?"}
{"completion": " The equation for Calls and Puts Arbitrage is: <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. Here is the brief introduction of Calls and Puts Arbitrage:\n\n        Latex code for the Calls and Puts Arbitrage. Three different options have strike prices K1, K2, K3 and K1 < K2 < K3 holds.\n        An important formula for determining arbitrage opportunities comes from the following equations.\n        \n\n: Strike price of option 1\n: Strike price of option 2\n: Strike price of option 3\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Calls and Puts Arbitrage?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>g_{i}</equation> in Monte-Carlo Simulations is: The i-th simulated payoff.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>g_{i}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of put option p.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Cox-Ingersoll-Ross CIRis <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>Y(t)</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>FV</equation> in Forwards is: Future Value.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>FV</equation> in Forwards?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>FV</equation> in Forwards is: Future Value.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>FV</equation> in Forwards?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(t)</equation> in Monte-Carlo Simulations is: The stock price at time t, which is nearer to final stage stock price S(T).", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>S(t)</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Black-Derman-Toy BDTis <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Early Exercise for American Optionsis <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Early Exercise for American Options?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage is: Strike price of option 2.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The related materials of Geometric Brownian Motion SDEs is: 1.Geometric Brownian Motion:<url>https://www.quantstart.com/articles/Geometric-Brownian-Motion/</url>;2.Geometric Brownian motion:<url>http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Geometric Brownian Motion SDEs?"}
{"completion": " The related materials of Calls and Puts with Different Strikes is: 1.Investopedia Put Call Parity:<url>https://www.investopedia.com/terms/p/putcallparity.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Calls and Puts with Different Strikes?"}
{"completion": " The Latex Code for equation Put-Call Parityis <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Put-Call Parity?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of call option c.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>U</equation> in Geometric Brownian Motion?"}
{"completion": " The related materials of Early Exercise for American Options is: 1.Investopedia Put Call Parity:<url>https://www.investopedia.com/terms/p/putcallparity.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Early Exercise for American Options?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t)</equation> in Standard Brownian Motion is: Value of Z at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates is: Forward Bond Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates?"}
{"completion": " The Latex Code for equation Ito Lemmais <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Ito Lemma?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Call Option Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>Y(t)</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. Here is the brief introduction of Risk-Neutral Valuation and Power Contracts:\n        Latex code for Risk-Neutral Valuation and Power Contracts.\n        \n: Payoff a power contract at time T\n: Price of the power contract\n: Risk-neutral equations\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Sharpe Ratio is: <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. Here is the brief introduction of Sharpe Ratio:\n        Latex code for Sharpe Ratio.\n        \n: Compounded Dividend Rate\n: Sharpe ratio\n: Sharpe ratio of any asset written on a GBM\n\n\nRelated Documents\n\nSharpe Ratio Formula and Definition With Examples\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Sharpe Ratio?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: Pay the strike earlier and therefore miss the interest on that money.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Ito Lemma is: <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. Here is the brief introduction of Ito Lemma:\n        Latex code for the Ito Lemma.\n        \n: Diffusion\n: Stochastic differential equation for X(t)\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Ito Lemma?"}
{"completion": " The Latex Code for equation Asian Optionsis <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Asian Options?"}
{"completion": " The Latex Code for equation Risk-Neutral Valuation and Power Contractsis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Stock Prices as Geometric Brownian Motionis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Varying Times to Expirationis <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Varying Times to Expiration?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>X</equation> in Ito Lemma is: Diffusion.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>X</equation> in Ito Lemma?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations is: The variance of stock price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>F_{t,T}(S)</equation> in Forwards is: Forward Contract at strike price S.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>F_{t,T}(S)</equation> in Forwards?"}
{"completion": " The equation for Varying Times to Expiration is: <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. Here is the brief introduction of Varying Times to Expiration:\n\n            For American options, when expiration T2 > T1, the above equations holds.\n        \n\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Varying Times to Expiration?"}
{"completion": " The equation for Black-Derman-Toy BDT is: <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. Here is the brief introduction of Black-Derman-Toy BDT:\n        Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0.\n        The yield volatility for period-3 is .\n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nBlack-Derman-Toy Model Developed by Team at Goldman Sachs\nA Binomial Interest Rate Model and the Black-Derman-Toy Model\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Risk-Neutral Valuation and Power Contractsis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Put-Call Parity is: <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. Here is the brief introduction of Put-Call Parity:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. Call options give the owner the right, but not the obligation, to buy an asset at some time in the future for a predetermined strike price. Put options give the owner the right to sell. The price of calls and puts is compared in the following put-call parity formula for European options.\n        \n: Price of call option c\n: Price of put option p\n: the present value of the strike price (x), \n: \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Put-Call Parity?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion SDEs is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>U</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Bonds and Interest Rates is: <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. Here is the brief introduction of Bonds and Interest Rates:\n        Latex code for the Bonds and Interest Rates. The price of an s-year zero is P(0, S). The forward bond price formula  is calculated as . And the \n        non-continuous annualized rate is .\n        \n: Price of an s-year zero.\n: Forward Bond Price\n: Non-continuous annualized rate\n\n\nRelated Documents\n\nBond and Interest Rate\nUnderstanding Interest Rates, Inflation, and Bonds\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Bonds and Interest Rates?"}
{"completion": " The Latex Code for equation Call and Put Price Boundsis <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Call and Put Price Bounds?"}
{"completion": " The Latex Code for equation Geometric Brownian Motion SDEsis <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion is: Observed stock price S(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Standard Brownian Motion is: <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. Here is the brief introduction of Standard Brownian Motion:\n        Latex code for the Standard Brownian Motion. I will briefly introduce the notations in this formulation. {Z(t)} has independent increments, and {Z(t)} has stationary increments such that Z (t + s) ? Z (t) follows standard normal distribution\n        \n: Value of Z at time stamp t\n: Stationary increments of Standard Brownian Motion\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Standard Brownian Motion?"}
{"completion": " The Latex Code for equation Stock Prices as Geometric Brownian Motionis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{1}.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Call and Put Price Bounds is: <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. Here is the brief introduction of Call and Put Price Bounds:\n\n        Latex code for the Calls and Puts Arbitrage. The following equations give the bounds on the prices of European calls and puts. Note that the lower bounds are no less than zero. We can also compare the prices of European and American options using the following inequalities.\n        \n\n: European Call Option Price\n: European Put Option Price\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Call and Put Price Bounds?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>G(T)</equation> in Asian Options is: Geometric average of option price at time T.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>G(T)</equation> in Asian Options?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(T)</equation> in Monte-Carlo Simulations is: The stock price at time T.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>S(T)</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Calls and Puts Arbitrageis <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Calls and Puts Arbitrage?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: Early Exercise getting the stock's dividend payments.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Stock Prices as Geometric Brownian Motionis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage is: Strike price of option 3.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>K_{3}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>r</equation> in Forwards is: Interest Rate.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>r</equation> in Forwards?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Sharpe Ratiois <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Sharpe Ratio?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion is: Stationary increments of Standard Brownian Motion.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion is: Observed stock price S(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma is: Stochastic differential equation for X(t).", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\mathrm{d}X(t)</equation> in Ito Lemma?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\sigma</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{1}.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\mu</equation> in Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Calls and Puts with Different Strikesis <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Calls and Puts with Different Strikes?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>FV</equation> in Forwards is: Future Value.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>FV</equation> in Forwards?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>r</equation> in Forwards is: Interest Rate.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>r</equation> in Forwards?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. Here is the brief introduction of Risk-Neutral Valuation and Power Contracts:\n        Latex code for Risk-Neutral Valuation and Power Contracts.\n        \n: Payoff a power contract at time T\n: Price of the power contract\n: Risk-neutral equations\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Ito Lemmais <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Ito Lemma?"}
{"completion": " The Latex Code for equation Call and Put Price Boundsis <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Call and Put Price Bounds?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P_{1}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The related materials of Varying Times to Expiration is: 1.Investopedia Put Call Parity:<url>https://www.investopedia.com/terms/p/putcallparity.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Varying Times to Expiration?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>U</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Call Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>A(T)</equation> in Asian Options is: Arithmetic average of option price at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>A(T)</equation> in Asian Options?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio is: Sharpe ratio of any asset written on a GBM.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Bonds and Interest Ratesis <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Bonds and Interest Rates?"}
{"completion": " The equation for Sharpe Ratio is: <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. Here is the brief introduction of Sharpe Ratio:\n        Latex code for Sharpe Ratio.\n        \n: Compounded Dividend Rate\n: Sharpe ratio\n: Sharpe ratio of any asset written on a GBM\n\n\nRelated Documents\n\nSharpe Ratio Formula and Definition With Examples\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Sharpe Ratio?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>R_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of put option p.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage is: Strike price of option 2.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>K_{2}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t)</equation> in Standard Brownian Motion is: Value of Z at time stamp t.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion is: <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. Here is the brief introduction of Geometric Brownian Motion:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Geometric Brownian Motion?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>r</equation> in Forwards is: Interest Rate.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>r</equation> in Forwards?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity is: .", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts is: Price of the power contract.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Early Exercise for American Options is: <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. Here is the brief introduction of Early Exercise for American Options:\n\n            So we exercise the call option if the pros are greater than the cons, specifically, we exercise if:\n        \n\n: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons. \n: Early Exercise getting the stock's dividend payments\n: Pay the strike earlier and therefore miss the interest on that money\n: put protection if the stock price should fall.\n: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise. \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Early Exercise for American Options?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. Here is the brief introduction of Stock Prices as Geometric Brownian Motion:\n        Latex code for Stock Prices as Geometric Brownian Motion.\n        \n: Observed stock price S(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Varying Times to Expiration is: <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. Here is the brief introduction of Varying Times to Expiration:\n\n            For American options, when expiration T2 > T1, the above equations holds.\n        \n\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Varying Times to Expiration?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage is: Strike price of option 1.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Geometric Brownian Motion is: <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. Here is the brief introduction of Geometric Brownian Motion:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Geometric Brownian Motion?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t)</equation> in Standard Brownian Motion is: Value of Z at time stamp t.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\delta</equation> in Sharpe Ratio is: Compounded Dividend Rate.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\delta</equation> in Sharpe Ratio?"}
{"completion": " The equation for Bonds and Interest Rates is: <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. Here is the brief introduction of Bonds and Interest Rates:\n        Latex code for the Bonds and Interest Rates. The price of an s-year zero is P(0, S). The forward bond price formula  is calculated as . And the \n        non-continuous annualized rate is .\n        \n: Price of an s-year zero.\n: Forward Bond Price\n: Non-continuous annualized rate\n\n\nRelated Documents\n\nBond and Interest Rate\nUnderstanding Interest Rates, Inflation, and Bonds\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Bonds and Interest Rates?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates is: Non-continuous annualized rate.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Risk-Neutral Valuation and Power Contractsis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Call and Put Price Boundsis <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Call and Put Price Bounds?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>FV</equation> in Forwards is: Future Value.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>FV</equation> in Forwards?"}
{"completion": " The equation for Forwards is: <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. Here is the brief introduction of Forwards:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. A forward contract is an agreement in which the buyer agrees at time t to pay the seller at time T and receive the asset at time T.\n        \n: Forward Contract at strike price S\n: Interest Rate\n: Future Value\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Forwards?"}
{"completion": " The Latex Code for equation Put-Call Parityis <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Put-Call Parity?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{2}.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: Early Exercise getting the stock's dividend payments.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage is: Strike price of option 3.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>K_{3}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Monte-Carlo Simulationsis <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Monte-Carlo Simulations?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>Y(t)</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Calls and Puts Arbitrageis <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Calls and Puts Arbitrage?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Geometric Brownian Motionis <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Geometric Brownian Motion?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(t)</equation> in Monte-Carlo Simulations is: The stock price at time t, which is nearer to final stage stock price S(T).", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>S(t)</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Varying Times to Expiration is: <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. Here is the brief introduction of Varying Times to Expiration:\n\n            For American options, when expiration T2 > T1, the above equations holds.\n        \n\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Varying Times to Expiration?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: Pay the strike earlier and therefore miss the interest on that money.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>FV</equation> in Forwards is: Future Value.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>FV</equation> in Forwards?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>g_{i}</equation> in Monte-Carlo Simulations is: The i-th simulated payoff.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>g_{i}</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Forwardsis <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Forwards?"}
{"completion": " The related materials of Sharpe Ratio is: 1.Sharpe Ratio Formula and Definition With Examples:<url>https://www.investopedia.com/terms/s/sharperatio.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Sharpe Ratio?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: Pay the strike earlier and therefore miss the interest on that money.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Black-Derman-Toy BDT is: <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. Here is the brief introduction of Black-Derman-Toy BDT:\n        Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0.\n        The yield volatility for period-3 is .\n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nBlack-Derman-Toy Model Developed by Team at Goldman Sachs\nA Binomial Interest Rate Model and the Black-Derman-Toy Model\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Black-Derman-Toy BDT?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: Early Exercise getting the stock's dividend payments.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity is: the present value of the strike price (x),.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P_{1}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The Latex Code for equation Put-Call Parityis <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Put-Call Parity?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Monte-Carlo Simulations is: <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. Here is the brief introduction of Monte-Carlo Simulations:\n        Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as , convert to normal variable .  The variance of the Monte-Carlo estimate is calculated as . \n        \n: The stock price at time T\n: The stock price at time t, which is nearer to final stage stock price S(T)\n: The i-th simulated payoff\n: The variance of stock price\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Monte-Carlo Simulations?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion is: Stationary increments of Standard Brownian Motion.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Early Exercise for American Options is: <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. Here is the brief introduction of Early Exercise for American Options:\n\n            So we exercise the call option if the pros are greater than the cons, specifically, we exercise if:\n        \n\n: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons. \n: Early Exercise getting the stock's dividend payments\n: Pay the strike earlier and therefore miss the interest on that money\n: put protection if the stock price should fall.\n: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise. \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Early Exercise for American Options?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\sigma</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{1}.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>c(K_{1})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The Latex Code for equation Risk-Neutral Valuation and Power Contractsis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Calls and Puts Arbitrage is: <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. Here is the brief introduction of Calls and Puts Arbitrage:\n\n        Latex code for the Calls and Puts Arbitrage. Three different options have strike prices K1, K2, K3 and K1 < K2 < K3 holds.\n        An important formula for determining arbitrage opportunities comes from the following equations.\n        \n\n: Strike price of option 1\n: Strike price of option 2\n: Strike price of option 3\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Calls and Puts Arbitrage?"}
{"completion": " The Latex Code for equation Asian Optionsis <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Asian Options?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>g_{i}</equation> in Monte-Carlo Simulations is: The i-th simulated payoff.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>g_{i}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion SDEs is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>U</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts Arbitrage is: <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. Here is the brief introduction of Calls and Puts Arbitrage:\n\n        Latex code for the Calls and Puts Arbitrage. Three different options have strike prices K1, K2, K3 and K1 < K2 < K3 holds.\n        An important formula for determining arbitrage opportunities comes from the following equations.\n        \n\n: Strike price of option 1\n: Strike price of option 2\n: Strike price of option 3\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Calls and Puts Arbitrage?"}
{"completion": " The equation for Put-Call Parity is: <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. Here is the brief introduction of Put-Call Parity:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. Call options give the owner the right, but not the obligation, to buy an asset at some time in the future for a predetermined strike price. Put options give the owner the right to sell. The price of calls and puts is compared in the following put-call parity formula for European options.\n        \n: Price of call option c\n: Price of put option p\n: the present value of the strike price (x), \n: \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Put-Call Parity?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma is: Stochastic differential equation for X(t).", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio is: Sharpe ratio.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio?"}
{"completion": " The equation for Standard Brownian Motion is: <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. Here is the brief introduction of Standard Brownian Motion:\n        Latex code for the Standard Brownian Motion. I will briefly introduce the notations in this formulation. {Z(t)} has independent increments, and {Z(t)} has stationary increments such that Z (t + s) ? Z (t) follows standard normal distribution\n        \n: Value of Z at time stamp t\n: Stationary increments of Standard Brownian Motion\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Standard Brownian Motion?"}
{"completion": " The equation for Varying Times to Expiration is: <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. Here is the brief introduction of Varying Times to Expiration:\n\n            For American options, when expiration T2 > T1, the above equations holds.\n        \n\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Varying Times to Expiration?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts is: Price of the power contract.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Sharpe Ratiois <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Sharpe Ratio?"}
{"completion": " The equation for Put-Call Parity is: <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. Here is the brief introduction of Put-Call Parity:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. Call options give the owner the right, but not the obligation, to buy an asset at some time in the future for a predetermined strike price. Put options give the owner the right to sell. The price of calls and puts is compared in the following put-call parity formula for European options.\n        \n: Price of call option c\n: Price of put option p\n: the present value of the strike price (x), \n: \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Put-Call Parity?"}
{"completion": " The equation for Asian Options is: <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. Here is the brief introduction of Asian Options:\n        Latex code for the Asian options are options. Asian options are options that are based on averages in place of either the price or the strike. The average can be either an arithmetic average or a geometric average. Then to price the option replace either the strike or the price with the appropriate path-dependent average, calculate the payoffs, and then discount them.\n        \n: Arithmetic average of option price at time T\n: Geometric average of option price at time T\n\n\nRelated Documents\n\nWhat Is an Asian Option? How They Work Vs. Standard Options\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Asian Options?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is: <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. Here is the brief introduction of Cox-Ingersoll-Ross CIR:\n        Latex code for the Cox-Ingersoll-Ross model. \n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nCox-Ingersoll-Ross Model\nA THEORY OF THE TERM STRUCTURE OF INTEREST RATES\nCox-Ingersoll-Ross (CIR) model-Mathworks\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Cox-Ingersoll-Ross CIR?"}
{"completion": " The Latex Code for equation Put-Call Parityis <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Put-Call Parity?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. Here is the brief introduction of Risk-Neutral Valuation and Power Contracts:\n        Latex code for Risk-Neutral Valuation and Power Contracts.\n        \n: Payoff a power contract at time T\n: Price of the power contract\n: Risk-neutral equations\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion is: Stationary increments of Standard Brownian Motion.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The equation for Early Exercise for American Options is: <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. Here is the brief introduction of Early Exercise for American Options:\n\n            So we exercise the call option if the pros are greater than the cons, specifically, we exercise if:\n        \n\n: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons. \n: Early Exercise getting the stock's dividend payments\n: Pay the strike earlier and therefore miss the interest on that money\n: put protection if the stock price should fall.\n: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise. \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Early Exercise for American Options?"}
{"completion": " The Latex Code for equation Standard Brownian Motionis <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Standard Brownian Motion?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of put option p.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Call and Put Price Boundsis <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Call and Put Price Bounds?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity is: .", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity?"}
{"completion": " The equation for Sharpe Ratio is: <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. Here is the brief introduction of Sharpe Ratio:\n        Latex code for Sharpe Ratio.\n        \n: Compounded Dividend Rate\n: Sharpe ratio\n: Sharpe ratio of any asset written on a GBM\n\n\nRelated Documents\n\nSharpe Ratio Formula and Definition With Examples\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Sharpe Ratio?"}
{"completion": " The Latex Code for equation Geometric Brownian Motion SDEsis <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Call and Put Price Bounds is: <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. Here is the brief introduction of Call and Put Price Bounds:\n\n        Latex code for the Calls and Puts Arbitrage. The following equations give the bounds on the prices of European calls and puts. Note that the lower bounds are no less than zero. We can also compare the prices of European and American options using the following inequalities.\n        \n\n: European Call Option Price\n: European Put Option Price\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Call and Put Price Bounds?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\mu</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>X</equation> in Ito Lemma is: Diffusion.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>X</equation> in Ito Lemma?"}
{"completion": " The Latex Code for equation Cox-Ingersoll-Ross CIRis <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The equation for Asian Options is: <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. Here is the brief introduction of Asian Options:\n        Latex code for the Asian options are options. Asian options are options that are based on averages in place of either the price or the strike. The average can be either an arithmetic average or a geometric average. Then to price the option replace either the strike or the price with the appropriate path-dependent average, calculate the payoffs, and then discount them.\n        \n: Arithmetic average of option price at time T\n: Geometric average of option price at time T\n\n\nRelated Documents\n\nWhat Is an Asian Option? How They Work Vs. Standard Options\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Asian Options?"}
{"completion": " The related materials of Cox-Ingersoll-Ross CIR is: 1.A THEORY OF THE TERM STRUCTURE OF INTEREST RATES:<url>https://pages.stern.nyu.edu/~dbackus/BCZ/discrete_time/CIR_Econometrica_85.pdf</url>;2.Cox-Ingersoll-Ross (CIR) model-Mathworks:<url>https://www.mathworks.com/help/finance/cir.html</url>;3.Cox-Ingersoll-Ross Model:<url>https://www.investopedia.com/terms/c/cox-ingersoll-ross-model.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Cox-Ingersoll-Ross CIR?"}
{"completion": " The Latex Code for equation Calls and Puts with Different Strikesis <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Calls and Puts with Different Strikes?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion is: Observed stock price S(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage is: Strike price of option 1.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>K_{1}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The Latex Code for equation Bonds and Interest Ratesis <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Bonds and Interest Rates?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(0, S)</equation> in Bonds and Interest Rates is: Price of an s-year zero..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P(0, S)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Bonds and Interest Rates is: <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. Here is the brief introduction of Bonds and Interest Rates:\n        Latex code for the Bonds and Interest Rates. The price of an s-year zero is P(0, S). The forward bond price formula  is calculated as . And the \n        non-continuous annualized rate is .\n        \n: Price of an s-year zero.\n: Forward Bond Price\n: Non-continuous annualized rate\n\n\nRelated Documents\n\nBond and Interest Rate\nUnderstanding Interest Rates, Inflation, and Bonds\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Bonds and Interest Rates?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Put Option Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Asian Options is: <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. Here is the brief introduction of Asian Options:\n        Latex code for the Asian options are options. Asian options are options that are based on averages in place of either the price or the strike. The average can be either an arithmetic average or a geometric average. Then to price the option replace either the strike or the price with the appropriate path-dependent average, calculate the payoffs, and then discount them.\n        \n: Arithmetic average of option price at time T\n: Geometric average of option price at time T\n\n\nRelated Documents\n\nWhat Is an Asian Option? How They Work Vs. Standard Options\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Asian Options?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P_{1}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts is: Payoff a power contract at time T.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Black-Derman-Toy BDT is: <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. Here is the brief introduction of Black-Derman-Toy BDT:\n        Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0.\n        The yield volatility for period-3 is .\n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nBlack-Derman-Toy Model Developed by Team at Goldman Sachs\nA Binomial Interest Rate Model and the Black-Derman-Toy Model\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Black-Derman-Toy BDT?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>A(T)</equation> in Asian Options is: Arithmetic average of option price at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>A(T)</equation> in Asian Options?"}
{"completion": " The Latex Code for equation Ito Lemmais <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Ito Lemma?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>F_{t,T}(S)</equation> in Forwards is: Forward Contract at strike price S.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>F_{t,T}(S)</equation> in Forwards?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The Latex Code for equation Early Exercise for American Optionsis <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Early Exercise for American Options?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: Pay the strike earlier and therefore miss the interest on that money.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>G(T)</equation> in Asian Options is: Geometric average of option price at time T.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>G(T)</equation> in Asian Options?"}
{"completion": " The equation for Ito Lemma is: <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. Here is the brief introduction of Ito Lemma:\n        Latex code for the Ito Lemma.\n        \n: Diffusion\n: Stochastic differential equation for X(t)\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Ito Lemma?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage is: Strike price of option 1.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>K_{1}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates is: Forward Bond Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates?"}
{"completion": " The Latex Code for equation Sharpe Ratiois <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Sharpe Ratio?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Put Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The Latex Code for equation Forwardsis <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Forwards?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The Latex Code for equation Put-Call Parityis <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Put-Call Parity?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\mu</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio is: Sharpe ratio.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\delta</equation> in Sharpe Ratio is: Compounded Dividend Rate.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\delta</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Geometric Brownian Motion SDEsis <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Geometric Brownian Motion SDEs?"}
{"completion": " The related materials of Put-Call Parity is: 1.Investopedia Put Call Parity:<url>https://www.investopedia.com/terms/p/putcallparity.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Put-Call Parity?"}
{"completion": " The related materials of Monte-Carlo Simulations is: 1.Investopedia Simple Interest:<url>https://www.investopedia.com/terms/s/simple_interest.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Monte-Carlo Simulations?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Put Option Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Sharpe Ratio is: <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. Here is the brief introduction of Sharpe Ratio:\n        Latex code for Sharpe Ratio.\n        \n: Compounded Dividend Rate\n: Sharpe ratio\n: Sharpe ratio of any asset written on a GBM\n\n\nRelated Documents\n\nSharpe Ratio Formula and Definition With Examples\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Sharpe Ratio?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity is: the present value of the strike price (x),.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>F^{P}_{t,T}(S)</equation> in Put-Call Parity?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options is: Early Exercise getting the stock's dividend payments.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P_{V_{t},T}(dividends)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity is: .", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Varying Times to Expirationis <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Varying Times to Expiration?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage is: Strike price of option 1.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Asian Options is: <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. Here is the brief introduction of Asian Options:\n        Latex code for the Asian options are options. Asian options are options that are based on averages in place of either the price or the strike. The average can be either an arithmetic average or a geometric average. Then to price the option replace either the strike or the price with the appropriate path-dependent average, calculate the payoffs, and then discount them.\n        \n: Arithmetic average of option price at time T\n: Geometric average of option price at time T\n\n\nRelated Documents\n\nWhat Is an Asian Option? How They Work Vs. Standard Options\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Asian Options?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options is: put protection if the stock price should fall..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>p(S_{t}, K)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Put-Call Parity is: <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. Here is the brief introduction of Put-Call Parity:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. Call options give the owner the right, but not the obligation, to buy an asset at some time in the future for a predetermined strike price. Put options give the owner the right to sell. The price of calls and puts is compared in the following put-call parity formula for European options.\n        \n: Price of call option c\n: Price of put option p\n: the present value of the strike price (x), \n: \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Put-Call Parity?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(T)</equation> in Monte-Carlo Simulations is: The stock price at time T.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>S(T)</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Monte-Carlo Simulationsis <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Risk-Neutral Valuation and Power Contractsis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Early Exercise for American Optionsis <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Early Exercise for American Options?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Put Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>p(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Call Option Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Forwardsis <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Forwards?"}
{"completion": " The related materials of Risk-Neutral Valuation and Power Contracts is: 1.Geometric Brownian Motion:<url>https://www.quantstart.com/articles/Geometric-Brownian-Motion/</url>;2.Geometric Brownian motion:<url>http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Geometric Brownian Motion is: <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. Here is the brief introduction of Geometric Brownian Motion:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Geometric Brownian Motion?"}
{"completion": " The equation for Sharpe Ratio is: <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. Here is the brief introduction of Sharpe Ratio:\n        Latex code for Sharpe Ratio.\n        \n: Compounded Dividend Rate\n: Sharpe ratio\n: Sharpe ratio of any asset written on a GBM\n\n\nRelated Documents\n\nSharpe Ratio Formula and Definition With Examples\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Sharpe Ratio?"}
{"completion": " The equation for Ito Lemma is: <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. Here is the brief introduction of Ito Lemma:\n        Latex code for the Ito Lemma.\n        \n: Diffusion\n: Stochastic differential equation for X(t)\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Ito Lemma?"}
{"completion": " The equation for Call and Put Price Bounds is: <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. Here is the brief introduction of Call and Put Price Bounds:\n\n        Latex code for the Calls and Puts Arbitrage. The following equations give the bounds on the prices of European calls and puts. Note that the lower bounds are no less than zero. We can also compare the prices of European and American options using the following inequalities.\n        \n\n: European Call Option Price\n: European Put Option Price\n: American Call Option Price\n: American Put Option Price\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Call and Put Price Bounds?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>R_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts is: Payoff a power contract at time T.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Call Option Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The related materials of Calls and Puts Arbitrage is: 1.Investopedia Put Call Parity:<url>https://www.investopedia.com/terms/p/putcallparity.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Calls and Puts Arbitrage?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(0, S)</equation> in Bonds and Interest Rates is: Price of an s-year zero..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P(0, S)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts is: Price of the power contract.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>F^{p}_{t, T}(S^{a})</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Cox-Ingersoll-Ross CIRis <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion is: Observed stock price S(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>S(t)</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates is: Forward Bond Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>F_{t,T}[P(T, T+s)]</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts is: Payoff a power contract at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>X</equation> in Ito Lemma is: Diffusion.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>X</equation> in Ito Lemma?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>R_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts is: Payoff a power contract at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>S^{a}(T)</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma is: Stochastic differential equation for X(t).", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\mathrm{d}X(t)</equation> in Ito Lemma?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is: <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. Here is the brief introduction of Geometric Brownian Motion SDEs:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Geometric Brownian Motionis <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Geometric Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is: <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. Here is the brief introduction of Geometric Brownian Motion SDEs:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage is: Strike price of option 2.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The Latex Code for equation Put-Call Parityis <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Put-Call Parity?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of call option c.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The Latex Code for equation Sharpe Ratiois <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Sharpe Ratio?"}
{"completion": " The Latex Code for equation Standard Brownian Motionis <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Standard Brownian Motion?"}
{"completion": " The equation for Standard Brownian Motion is <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. The meaning of symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion is: Stationary increments of Standard Brownian Motion.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>Z(t+s) - Z(t)</equation> in Standard Brownian Motion?"}
{"completion": " The Latex Code for equation Geometric Brownian Motionis <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Geometric Brownian Motion?"}
{"completion": " The equation for Black-Derman-Toy BDT is: <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. Here is the brief introduction of Black-Derman-Toy BDT:\n        Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0.\n        The yield volatility for period-3 is .\n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nBlack-Derman-Toy Model Developed by Team at Goldman Sachs\nA Binomial Interest Rate Model and the Black-Derman-Toy Model\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Black-Derman-Toy BDT?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR is: 1-year interest rate..", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>R_{0}</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is: <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. Here is the brief introduction of Geometric Brownian Motion SDEs:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>Y(t)</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of put option p.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>p(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\mu</equation> in Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Calls and Puts with Different Strikesis <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Calls and Puts with Different Strikes?"}
{"completion": " The Latex Code for equation Geometric Brownian Motion SDEsis <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs is: Volatility.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. The meaning of symbol <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR is: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>P(1, 2, r_{u}), P(1, 2, r_{d})</equation> in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Early Exercise for American Options is: <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. Here is the brief introduction of Early Exercise for American Options:\n\n            So we exercise the call option if the pros are greater than the cons, specifically, we exercise if:\n        \n\n: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons. \n: Early Exercise getting the stock's dividend payments\n: Pay the strike earlier and therefore miss the interest on that money\n: put protection if the stock price should fall.\n: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise. \n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Early Exercise for American Options?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity is: .", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>Ke^{-r(T-t)}</equation> in Put-Call Parity?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(T)</equation> in Monte-Carlo Simulations is: The stock price at time T.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>S(T)</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Black-Derman-Toy BDTis <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Black-Derman-Toy BDT?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts is: Risk-neutral equations.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The Latex Code for equation Calls and Puts Arbitrageis <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Calls and Puts Arbitrage?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(0, S)</equation> in Bonds and Interest Rates is: Price of an s-year zero..", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P(0, S)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options is: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)})</equation> in Early Exercise for American Options?"}
{"completion": " The Latex Code for equation Sharpe Ratiois <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Sharpe Ratio?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\mu</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT is: Observed year-1 price of a 1-year bond.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P_{1}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: European Call Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>c(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The Latex Code for equation Forwardsis <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Forwards?"}
{"completion": " The Latex Code for equation Asian Optionsis <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Asian Options?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>A(T)</equation> in Asian Options is: Arithmetic average of option price at time T.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>A(T)</equation> in Asian Options?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio is: Sharpe ratio of any asset written on a GBM.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Varying Times to Expirationis <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Varying Times to Expiration?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options is: put protection if the stock price should fall..", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>p(S_{t}, K)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Asian Options is: <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. Here is the brief introduction of Asian Options:\n        Latex code for the Asian options are options. Asian options are options that are based on averages in place of either the price or the strike. The average can be either an arithmetic average or a geometric average. Then to price the option replace either the strike or the price with the appropriate path-dependent average, calculate the payoffs, and then discount them.\n        \n: Arithmetic average of option price at time T\n: Geometric average of option price at time T\n\n\nRelated Documents\n\nWhat Is an Asian Option? How They Work Vs. Standard Options\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Asian Options?"}
{"completion": " The Latex Code for equation Forwardsis <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Forwards?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage is: Strike price of option 1.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>K_{1}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(t)</equation> in Monte-Carlo Simulations is: The stock price at time t, which is nearer to final stage stock price S(T).", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>S(t)</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Calls and Puts Arbitrage is: <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. Here is the brief introduction of Calls and Puts Arbitrage:\n\n        Latex code for the Calls and Puts Arbitrage. Three different options have strike prices K1, K2, K3 and K1 < K2 < K3 holds.\n        An important formula for determining arbitrage opportunities comes from the following equations.\n        \n\n: Strike price of option 1\n: Strike price of option 2\n: Strike price of option 3\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Calls and Puts Arbitrage?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>\\mu</equation> in Geometric Brownian Motion is: Drift coefficient.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\mu</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs is: Volatility.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Black-Derman-Toy BDTis <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Varying Times to Expirationis <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Varying Times to Expiration?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The Latex Code for equation Standard Brownian Motionis <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Standard Brownian Motion?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{2}.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The equation for Calls and Puts with Different Strikes is <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. The meaning of symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes is: Call option of strike price K_{2}.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>c(K_{2})</equation> in Calls and Puts with Different Strikes?"}
{"completion": " The related materials of Call and Put Price Bounds is: 1.Investopedia Put Call Parity:<url>https://www.investopedia.com/terms/p/putcallparity.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Call and Put Price Bounds?"}
{"completion": " The equation for Calls and Puts with Different Strikes is: <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. Here is the brief introduction of Calls and Puts with Different Strikes:\n\n        Latex code for the Calls and Puts with Different Strikes. For European calls and puts, with strike prices K_{1} and K_{2}, where K_{1} < K_{2}, we know the following.\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\n            American options, For three different options with strike prices K1 < K2 < K3:\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Calls and Puts with Different Strikes?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Geometric Brownian Motion is <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>. The meaning of symbol <equation>Y(t)</equation> in Geometric Brownian Motion is: Observed value Y(t) at time stamp t.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>Y(t)</equation> in Geometric Brownian Motion?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(T)</equation> in Monte-Carlo Simulations is: The stock price at time T.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>S(T)</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Monte-Carlo Simulationsis <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Monte-Carlo Simulations?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. Here is the brief introduction of Stock Prices as Geometric Brownian Motion:\n        Latex code for Stock Prices as Geometric Brownian Motion.\n        \n: Observed stock price S(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Geometric Brownian Motion SDEsis <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Call and Put Price Boundsis <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Call and Put Price Bounds?"}
{"completion": " The Latex Code for equation Bonds and Interest Ratesis <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Bonds and Interest Rates?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(t)</equation> in Monte-Carlo Simulations is: The stock price at time t, which is nearer to final stage stock price S(T).", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>S(t)</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio is: Sharpe ratio.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\phi = \\frac{m + \\delta -r }{s}</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Geometric Brownian Motion SDEsis <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>", "tag": "Financial,Economics", "prompt": "Please give me some hints on equation Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Calls and Puts Arbitrageis <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Calls and Puts Arbitrage?"}
{"completion": " The related materials of Forwards is: 1.Investopedia Simple Interest:<url>https://www.investopedia.com/terms/s/simple_interest.asp</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Forwards?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma is: Stochastic differential equation for X(t).", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of call option c.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The equation for Asian Options is <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>. The meaning of symbol <equation>A(T)</equation> in Asian Options is: Arithmetic average of option price at time T.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>A(T)</equation> in Asian Options?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>C(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The Latex Code for equation Risk-Neutral Valuation and Power Contractsis <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>U</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The Latex Code for equation Varying Times to Expirationis <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Varying Times to Expiration?"}
{"completion": " The equation for Forwards is: <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. Here is the brief introduction of Forwards:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. A forward contract is an agreement in which the buyer agrees at time t to pay the seller at time T and receive the asset at time T.\n        \n: Forward Contract at strike price S\n: Interest Rate\n: Future Value\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Forwards?"}
{"completion": " The equation for Early Exercise for American Options is <equation>P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 ? e^{?r(T ?t)}) \\\\\\\\\\ K(1 ? e^{?r(T ?t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)</equation>. The meaning of symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options is: put protection if the stock price should fall..", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>p(S_{t}, K)</equation> in Early Exercise for American Options?"}
{"completion": " The equation for Call and Put Price Bounds is <equation>(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \\\\le c(S_{t},K,t,T) \\\\le F^{P}_{t,T}(S) \\\\\\\\\\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \\\\le p(S_{t},K,t,T) \\\\le Ke^{-r(T-t)} \\\\\\\\ c(S_{t},K,t,T) \\\\le C(S_{t},K,t,T) \\\\le S_{t} \\\\\\\\\\ p(S_{t},K,t,T) \\\\le P(S_{t},K,t,T) \\\\le K</equation>. The meaning of symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds is: American Call Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>C(S_{t},K,t,T)</equation> in Call and Put Price Bounds?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>U</equation> in Stock Prices as Geometric Brownian Motion is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>U</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage is: Strike price of option 3.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>K_{3}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The Latex Code for equation Bonds and Interest Ratesis <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>", "tag": "Financial,Economics", "prompt": "What is the equation of Bonds and Interest Rates?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. The meaning of symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion is: Volatility.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>\\sigma</equation> in Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>r</equation> in Forwards is: Interest Rate.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>r</equation> in Forwards?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\delta</equation> in Sharpe Ratio is: Compounded Dividend Rate.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\delta</equation> in Sharpe Ratio?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is: <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. Here is the brief introduction of Cox-Ingersoll-Ross CIR:\n        Latex code for the Cox-Ingersoll-Ross model. \n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nCox-Ingersoll-Ross Model\nA THEORY OF THE TERM STRUCTURE OF INTEREST RATES\nCox-Ingersoll-Ross (CIR) model-Mathworks\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates is: Non-continuous annualized rate.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)</equation> in Bonds and Interest Rates?"}
{"completion": " The related materials of Ito Lemma is: 1.Geometric Brownian Motion:<url>https://www.quantstart.com/articles/Geometric-Brownian-Motion/</url>;2.Geometric Brownian motion:<url>http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Ito Lemma?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs is: Volatility.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Geometric Brownian Motionis <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>", "tag": "Financial,Economics", "prompt": "Show me the latex code of equation Geometric Brownian Motion?"}
{"completion": " The equation for Forwards is <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. The meaning of symbol <equation>F_{t,T}(S)</equation> in Forwards is: Forward Contract at strike price S.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>F_{t,T}(S)</equation> in Forwards?"}
{"completion": " The related materials of Bonds and Interest Rates is: 1.Understanding Interest Rates, Inflation, and Bonds:<url>https://www.investopedia.com/articles/bonds/09/bond-market-interest-rates.asp</url>;2.Bond and Interest Rate:<url>https://www.wsj.com/market-data/bonds</url>;", "tag": "Financial,Economics", "prompt": "Can you help me find related materials of Bonds and Interest Rates?"}
{"completion": " The equation for Monte-Carlo Simulations is: <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. Here is the brief introduction of Monte-Carlo Simulations:\n        Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as , convert to normal variable .  The variance of the Monte-Carlo estimate is calculated as . \n        \n: The stock price at time T\n: The stock price at time t, which is nearer to final stage stock price S(T)\n: The i-th simulated payoff\n: The variance of stock price\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Monte-Carlo Simulations?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is: <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. Here is the brief introduction of Geometric Brownian Motion SDEs:\n        Latex code for the Geometric Brownian Motion. \n        \n: Observed value Y(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Standard Brownian Motion is: <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. Here is the brief introduction of Standard Brownian Motion:\n        Latex code for the Standard Brownian Motion. I will briefly introduce the notations in this formulation. {Z(t)} has independent increments, and {Z(t)} has stationary increments such that Z (t + s) ? Z (t) follows standard normal distribution\n        \n: Value of Z at time stamp t\n: Stationary increments of Standard Brownian Motion\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Standard Brownian Motion?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>U</equation> in Geometric Brownian Motion SDEs is: Any normal random variable.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>U</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations is: The variance of stock price.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>e^{-2rT} \\times \\frac{s^{2}}{n}</equation> in Monte-Carlo Simulations?"}
{"completion": " The equation for Monte-Carlo Simulations is <equation>S(T) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})T + \\\\sigma \\\\sqrt{T} z} \\\\\\\\\\             S(T) = S(t) e^{(a - \\\\delta - \\\\frac{\\\\sigma^2}{2})(T-t) + \\\\sigma (Z(T) - Z(t))} \\\\\\\\\\             \\\\text{Variance} \\\\\\\\\\             e^{-2rT} \\\\times \\\\frac{s^{2}}{n} \\\\\\\\\\             s^{2} = \\\\frac{1}{n-1} \\\\sum [(g(S_{i}) - \\\\bar{g})]^{2}</equation>. The meaning of symbol <equation>S(t)</equation> in Monte-Carlo Simulations is: The stock price at time t, which is nearer to final stage stock price S(T).", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>S(t)</equation> in Monte-Carlo Simulations?"}
{"completion": " The Latex Code for equation Ito Lemmais <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>", "tag": "Financial,Economics", "prompt": "How to write equation Ito Lemma?"}
{"completion": " The equation for Put-Call Parity is <equation>c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}</equation>. The meaning of symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity is: Price of call option c.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>c(S_{t}, K, t, T)</equation> in Put-Call Parity?"}
{"completion": " The equation for Calls and Puts Arbitrage is: <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. Here is the brief introduction of Calls and Puts Arbitrage:\n\n        Latex code for the Calls and Puts Arbitrage. Three different options have strike prices K1, K2, K3 and K1 < K2 < K3 holds.\n        An important formula for determining arbitrage opportunities comes from the following equations.\n        \n\n: Strike price of option 1\n: Strike price of option 2\n: Strike price of option 3\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Calls and Puts Arbitrage?"}
{"completion": " The equation for Forwards is: <equation>F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\\\\text{Dividends}) = S_{t}e^{(r-\\\\delta)(T-t)}</equation>. Here is the brief introduction of Forwards:\n        Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. A forward contract is an agreement in which the buyer agrees at time t to pay the seller at time T and receive the asset at time T.\n        \n: Forward Contract at strike price S\n: Interest Rate\n: Future Value\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you give me a brief introduction to Forwards?"}
{"completion": " The equation for Calls and Puts Arbitrage is <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>. The meaning of symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage is: Strike price of option 2.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>K_{2}</equation> in Calls and Puts Arbitrage?"}
{"completion": " The equation for Bonds and Interest Rates is <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. The meaning of symbol <equation>P(0, S)</equation> in Bonds and Interest Rates is: Price of an s-year zero..", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P(0, S)</equation> in Bonds and Interest Rates?"}
{"completion": " The equation for Varying Times to Expiration is <equation>T_{2} \\\\ge T_{1} \\\\\\\\\\ C(S_{t},K,t,T_{2}) \\\\ge C(S_{t},K,t,T_{1}) \\\\le S_{t} \\\\\\\\\\ P(S_{t},K,t,T_{2}) \\\\ge P(S_{t},K,t,T_{1}) \\\\le S_{t}</equation>. The meaning of symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration is: American Put Option Price.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>P(S_{t},K,t,T)</equation> in Varying Times to Expiration?"}
{"completion": " The Latex Code for equation Asian Optionsis <equation>A(T) = \\\\frac{1}{n} \\\\sum S(ih) \\\\\\\\\\ G(T) = [\\\\prod S(ih)]^{\\\\frac{1}{n}}</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Asian Options?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio is: Sharpe ratio of any asset written on a GBM.", "tag": "Financial,Economics", "prompt": "What is the meaning of the symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio?"}
{"completion": " The equation for Geometric Brownian Motion SDEs is <equation>\\\\mathrm{d}Y(t) = \\\\mu Y(t)dt + \\\\sigma Y(t) \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\mathrm{d}[\\\\ln Y(t)] = (\\\\mu - \\\\frac{\\\\sigma^2}{2}) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             Y(t) = T(0) e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma Z(t)}</equation>. The meaning of symbol <equation>\\sigma</equation> in Geometric Brownian Motion SDEs is: Volatility.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\sigma</equation> in Geometric Brownian Motion SDEs?"}
{"completion": " The Latex Code for equation Calls and Puts Arbitrageis <equation>K_{1} < K_{2} < K_{3} \\\\\\\\\\ K_{2} = \\\\lambda K_{1} + (1 - \\\\lambda) K_{3} \\\\\\\\\\ \\\\lambda = \\\\frac{K_{3} - K_{2}}{K_{3} - K_{1}}</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Calls and Puts Arbitrage?"}
{"completion": " The equation for Black-Derman-Toy BDT is <equation>\\\\text{First Node: 1-year bond price} \\\\\\\\\\             P_{0} = \\\\frac{1}{1 + R_{0}} \\\\\\\\\\             \\\\text{Second Node} \\\\\\\\\\             P_{1} = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2} P(1,2,r_{u}) + \\\\frac{1}{2} P(1,2,r_{d})] \\\\\\\\\\             = \\\\frac{1}{1+R_{0}} [\\\\frac{1}{2(1 + R_{1}e^{2\\\\sigma_{1}})} + \\\\frac{1}{2(1 + R_{1})}] \\\\\\\\\\             R_{0} = \\\\frac{1}{2} \\\\ln (\\\\frac{R_{1} e^{2\\\\sigma_{1}} }{R_{1}})</equation>. The meaning of symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT is: 1-year bond price.", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>P_{0}</equation> in Black-Derman-Toy BDT?"}
{"completion": " The Latex Code for equation Geometric Brownian Motionis <equation>Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\\\\mu t + \\\\sigma Z(t)]} \\\\\\\\\\             E(e^{kU}) = e^{kE(U) + \\\\frac{1}{2}k^{2}\\\\text{Var}(U)} \\\\\\\\\\             E[Y^{k}(t)] = Y^{k}(0) e^{(k\\\\mu + \\\\frac{1}{2}k^{2}\\\\sigma^{2})t} \\\\\\\\\\             \\\\ln Y(t) \\\\sim N(\\\\ln Y(0) + \\\\mu t, \\\\sigma^{2} t)</equation>", "tag": "Financial,Economics", "prompt": "What is the latex code of equation Geometric Brownian Motion?"}
{"completion": " The equation for Standard Brownian Motion is: <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>. Here is the brief introduction of Standard Brownian Motion:\n        Latex code for the Standard Brownian Motion. I will briefly introduce the notations in this formulation. {Z(t)} has independent increments, and {Z(t)} has stationary increments such that Z (t + s) ? Z (t) follows standard normal distribution\n        \n: Value of Z at time stamp t\n: Stationary increments of Standard Brownian Motion\n\n\nRelated Documents\n\nInvestopedia Simple Interest\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Standard Brownian Motion?"}
{"completion": " The equation for Calls and Puts with Different Strikes is: <equation>K_{1} < K_{2} \\\\\\\\\\             0 \\\\le c(K_{1}) - c(K_{2}) \\\\le (K_{2} - K_{1})e^{-rT} \\\\\\\\\\             0 \\\\le p(K_{2}) - p(K_{1}) \\\\le (K_{2}) - K_{1})e^{-rT} \\\\\\\\\\             \\\\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \\\\ge \\\\frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\\\\\\\\\             \\\\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \\\\le \\\\frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}</equation>. Here is the brief introduction of Calls and Puts with Different Strikes:\n\n        Latex code for the Calls and Puts with Different Strikes. For European calls and puts, with strike prices K_{1} and K_{2}, where K_{1} < K_{2}, we know the following.\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\n            American options, For three different options with strike prices K1 < K2 < K3:\n        \n\n: Call option of strike price K_{1}\n: Call option of strike price K_{2}\n\n\nRelated Documents\n\nInvestopedia Put Call Parity\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "What's the meaning of Calls and Puts with Different Strikes?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio is: Sharpe ratio of any asset written on a GBM.", "tag": "Financial,Economics", "prompt": "What is the meaning of <equation>\\phi = \\frac{a - r}{\\sigma}</equation> in Sharpe Ratio?"}
{"completion": " The Latex Code for equation Standard Brownian Motionis <equation>Z(t) \\\\sim N(0, t) \\\\\\\\\\ Z(t+s) - Z(t) \\\\sim N(0, s) \\\\\\\\\\ Z(t+s) \\\\sim N(Z(t), s)</equation>", "tag": "Financial,Economics", "prompt": "Can you write equation Standard Brownian Motion?"}
{"completion": " The equation for Bonds and Interest Rates is: <equation>P(0, S) = \\\\frac{1}{[1 + r(0, s)]^{s}} \\\\text{or} e^{-r(0,s)s} \\\\\\\\\\ \\\\text{Forward bond price} \\\\\\\\\\ F_{t,T}[P(T, T+s)] = \\\\frac{P(t, T+s)}{P(t, T)} \\\\\\\\\\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)</equation>. Here is the brief introduction of Bonds and Interest Rates:\n        Latex code for the Bonds and Interest Rates. The price of an s-year zero is P(0, S). The forward bond price formula  is calculated as . And the \n        non-continuous annualized rate is .\n        \n: Price of an s-year zero.\n: Forward Bond Price\n: Non-continuous annualized rate\n\n\nRelated Documents\n\nBond and Interest Rate\nUnderstanding Interest Rates, Inflation, and Bonds\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Bonds and Interest Rates?"}
{"completion": " The equation for Risk-Neutral Valuation and Power Contracts is <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (r - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d} \\\\tilt{Z}(t) \\\\\\\\\\             \\\\tilt{Z}(t) = Z(t) + \\\\phi t \\\\\\\\\\             V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\\\\\\\\\             F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\\\\delta) + \\\\frac{1}{2} a(a-1)\\\\sigma^{2})(T-t)}</equation>. The meaning of symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts is: Risk-neutral equations.", "tag": "Financial,Economics", "prompt": "Help me find the meaning of symbol <equation>V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]</equation> in Risk-Neutral Valuation and Power Contracts?"}
{"completion": " The equation for Sharpe Ratio is <equation>\\\\frac{\\\\mathrm{d}X(t)}{X(t)} = m \\\\mathrm{d}t + s \\\\mathrm{d}Z(t) \\\\\\\\\\             \\\\phi = \\\\frac{m + \\\\delta -r }{s} \\\\\\\\\\             \\\\phi = \\\\frac{a - r}{\\\\sigma}</equation>. The meaning of symbol <equation>\\delta</equation> in Sharpe Ratio is: Compounded Dividend Rate.", "tag": "Financial,Economics", "prompt": "What does the symbol mean <equation>\\delta</equation> in Sharpe Ratio?"}
{"completion": " The equation for Cox-Ingersoll-Ross CIR is: <equation>\\\\mathrm{d} r(t) = a[b - r(t)]  \\\\mathrm{d} t + \\\\sigma \\\\sqrt{r(t)} \\\\mathrm{d} Z(t) \\\\\\\\\\             P(r, t, T) = A(T-t)e^{-rB(T-t)} \\\\\\\\\\             \\\\gamma = \\\\sqrt{(a-\\\\bar{\\\\phi})^{2} + 2 \\\\sigma^{2}} \\\\\\\\\\             q(r, t, T) = \\\\sigma \\\\sqrt{r} B(T-t) \\\\\\\\\\             \\\\text{yield to maturity} \\\\\\\\\\             \\\\frac{2ab}{ a - \\\\bar{\\\\phi} + \\\\gamma}</equation>. Here is the brief introduction of Cox-Ingersoll-Ross CIR:\n        Latex code for the Cox-Ingersoll-Ross model. \n        \n: 1-year interest rate.\n: 1-year bond price\n: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.\n: Observed year-1 price of a 1-year bond\n\n\nRelated Documents\n\nCox-Ingersoll-Ross Model\nA THEORY OF THE TERM STRUCTURE OF INTEREST RATES\nCox-Ingersoll-Ross (CIR) model-Mathworks\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "I am interested in Cox-Ingersoll-Ross CIR?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. Here is the brief introduction of Stock Prices as Geometric Brownian Motion:\n        Latex code for Stock Prices as Geometric Brownian Motion.\n        \n: Observed stock price S(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Please give me detailed tutorial of Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Stock Prices as Geometric Brownian Motion is: <equation>\\\\frac{\\\\mathrm{d}S(t)}{S(t)} = (a - \\\\delta) \\\\mathrm{d}t + \\\\sigma \\\\mathrm{d}Z(t) \\\\\\\\\\             S(t) = S(0) e^{(a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2})t + \\\\sigma Z(t)} \\\\\\\\\\             \\\\mathrm{d}[\\\\ln S(t)] = (a - \\\\delta - \\\\frac{\\\\sigma^{2}}{2}) \\\\mathrm{d}t + \\\\sigma \\\\sigma \\\\mathrm{d} Z(t) \\\\\\\\\\             S(t) \\\\sim \\\\ln( \\\\ln S(0) + (a - \\\\delta - \\\\frac{\\\\sigma^2}{2})t, \\\\sigma^{2}t)</equation>. Here is the brief introduction of Stock Prices as Geometric Brownian Motion:\n        Latex code for Stock Prices as Geometric Brownian Motion.\n        \n: Observed stock price S(t) at time stamp t\n: Any normal random variable\n: Drift coefficient\n: Volatility\n\n\nRelated Documents\n\nGeometric Brownian Motion\nGeometric Brownian motion\n\nRelated Videos\n\n\n", "tag": "Financial,Economics", "prompt": "Can you explain Stock Prices as Geometric Brownian Motion?"}
{"completion": " The equation for Ito Lemma is <equation>\\\\mathrm{d}X(t) = a(t, X(t)) \\\\mathrm{d}t + b(t, X(t))\\\\mathrm{d} Z(t) \\\\\\\\\\             Y(t) = f(t, X(t)) \\\\mathrm{d}t \\\\\\\\\\             \\\\mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\\\\mathrm{d} X(t) + \\\\frac{1}{2} f_{xx}(t, X(t))[\\\\mathrm{d}X(t)]^{2} \\\\\\\\\\             [\\\\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\\\\mathrm{d} t</equation>. The meaning of symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma is: Stochastic differential equation for X(t).", "tag": "Financial,Economics", "prompt": "Can you explain the symbol <equation>\\mathrm{d}X(t)</equation> in Ito Lemma?"}