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UiraCaiadoEx02.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>UiraCaiadoEx02</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
<style type="text/css">
/*!
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<h1 id="Derivativos-de-Ações-e-Commodities">Derivativos de Ações e Commodities<a class="anchor-link" href="#Derivativos-de-Ações-e-Commodities">¶</a></h1><h2 id="Precificação-com-o-Método-de-Diferenças-Finitas">Precificação com o Método de Diferenças Finitas<a class="anchor-link" href="#Precificação-com-o-Método-de-Diferenças-Finitas">¶</a></h2><p><sub>Uirá Caiado. 29 de Junho, 2016<sub></p>
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<h4 id="Resumo">Resumo<a class="anchor-link" href="#Resumo">¶</a></h4><p><em>Neste projeto vou implementar o método de diferenças finitas explícito para precificar diferentes derivativos. Como o preço de opções geralmente é descrito por equações diferenciais de difusão ou parabólicas, o método de diferenças finitas se mostra adequado, uma vez que é utilizado justamente para encontrar soluções numéricas para equações diferencias. Depois de implementar o modelo, vou compará-lo com os resultados obtidos pela solução analítica de cada instrumento.</em></p>
<h2 id="1.-Introdução">1. Introdução<a class="anchor-link" href="#1.-Introdução">¶</a></h2><p>Nesta seção dou uma breve descrição sobre o método de diferenças finitas e declaro o problema que será abordado.</p>
<h3 id="1.1.-O-Método-Utilizado">1.1. O Método Utilizado<a class="anchor-link" href="#1.1.-O-Método-Utilizado">¶</a></h3><p>Como colocado por Wilmott, encontrar soluções fechadas para precificação de muitos derivativos pode ser difícil, ou mesmo inviável. Como o preço destes instrumentos frequentemente são descritos por equações diferencias parciais, frequentemente utilizam-se métodos numéricos para encontrar uma solução, como <a href="https://en.wikipedia.org/wiki/Binomial_options_pricing_model">árvores binomiais</a>, simulação de <a href="https://en.wikipedia.org/wiki/Monte_Carlo_method">monte carlo</a> e diferenças finitas.</p>
<p>O método de diferenças finitas encontra o valor de um derivativo calculando-o em todo domínio (factivel) de preços do instrumento base, incluíndo a passagem de tempo até seu vencimento. Assim, são similares às árvores binomiais. Porém, ao invéz e discretizar os preços do ativo e a passagem do tempo em uma estrutura de árvore, discretiza em um grid.</p>
<p>Diferenças finitas também são muitos boas para lidar com problemas com poucas dimensões e equações diferenciais não lineares (o preço e tempo já são duas, e o Wilmott sugere até quatro). Para muitas dimensões, a implementação começa a ficar complicada e pouco eficiente, coisa que o Monte Carlo lida melhor. Porém, se há exercício antecipado e não linearidade, o método de diferenças finitas acaba se mostrando como solução mais viável.</p>
<h3 id="1.2.-O-Problema">1.2. O Problema<a class="anchor-link" href="#1.2.-O-Problema">¶</a></h3><p>Considerando um ativo-objeto cuja dinâmica do preço satisfaz a seguinte EDE:</p>
$$\frac{\mathrm{d} S_t}{S_t}=\mu\cdot \mathrm{d}t + \sigma\cdot \mathrm{d}W_{t}$$<p>Deseja-se calcular o preço justo de um derivativo um derivativo com característica europeia cujo payoff é descrito por uma função qualquer $V_T=V(T, S_T)$, onde $T$ é o vencimento do derivativo. $S_T$ é o preço do ativo-objeto em $T$ e é possível negociar qualquer quantidade dele em qualquer instante. Não há custo de transação (corretagem, emolumento, bid-ask spread, etc) e posições vendidas a descoberto no subjacente são permitidas, não havendo custos associados.</p>
<p>Pede-se que se implemente o algoritmo de diferenças finitas explícito e se calcule o preço justo, o Delta e o Gamma de cada instrumento abaixo. Os valores encontrados devem ser comparados com o resultado de suas expressões analítica. A simulação deve ser feita para os payoffs abaixo. $K$ é o Strike da opção.</p>
<ul>
<li>$V(T, S_T)=ln(S_T)$</li>
<li>$V(T, S_T)=(ln(S_T))^2$</li>
<li>$V(T, S_T)=(S_T-K)^2$</li>
<li>$V(T, S_T)=\mathbf{1}_{S_T > K}$</li>
<li>$V(T, S_T)=max(S_T-K, 0)$</li>
</ul>
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<h2 id="2.-Diferenças-Finitas">2. Diferenças Finitas<a class="anchor-link" href="#2.-Diferenças-Finitas">¶</a></h2><p>Abaixo, apresentarei os elementos necessários para implementar o método diferenças finitas, sendo eles: a <a href="https://pt.wikipedia.org/wiki/Diferenciação_numérica">diferenciação</a> das derivadas parciais necessárias usando o Grid; a discretização da condição final para cada derivativo; e suas respectivas condições de contorno (nos limites do grid).</p>
<h3 id="2.1.-Diferenciação-no-Grid">2.1. Diferenciação no Grid<a class="anchor-link" href="#2.1.-Diferenciação-no-Grid">¶</a></h3><p>De acordo com Wilmott, o grid utilizado pelo método de diferenças finitas tem passos de tempo e de preço (ou log do preço) geralmente homogênios. Porém, <strong>não há</strong> restrição para a forma do Grid, desde que os ajustes necessários sejam feitos. Como exposto por</p>
<p>Considerando que podemos discretizar $t$ como $t=T-k\delta t$ e do $S$ como $S = i\delta S$, onde $i$ e $k$ são seus respectivos passos no grid, podemos escrever o valor da opção em cada ponto do grig como sendo:</p>
$$V^{k}_{i} = ( i \delta S, T-k \delta t )$$<p>Seguindo notas de aula, aplicando Taylor ao preço $V(T, S_T)$ de um derivativo genérico, posso escrever a seguinte equação diferencial parabólica:</p>
$$\frac{\partial V}{\partial t} + a(S, t) \cdot \frac{\partial V^2}{\partial S^2} + b(S, t) \cdot \frac{\partial V}{\partial S} + c(S, t)\cdot V = 0 $$<p>Sendo que $\frac{\partial V}{\partial t}$ também é chamado de theta ($\theta$), $\frac{\partial V}{\partial S}$ de delta ($\Delta$) e $\frac{\partial V^2}{\partial S^2}$ de gamma ($\Gamma$). Como demostrado por Wilmott, como a definição da primeira derivada $V$ em relação a $t$ é dado por</p>
$$\frac{\partial V}{\partial t} = \underset{h \, \rightarrow \, 0}{\lim}\, \frac{V(S, t + h) - V(S, t)}{h}$$<p>Então podemos aproximar $\theta$ como sendo $\frac{\partial V}{\partial t} (S, t) \approx \frac{V_{i}^{k} - V_{i}^{k+1} }{\delta t}$. Note que $i$, que é relacionado ao passo do ativo, ficou fixo. <em>Também estou ignorando os erros de aproximação, como ignorarei em todas as outras diferenciações</em>.</p>
<p>O mesmo raciocínio pode ser utilizado para aproximar o $\Delta$. Porém, Wilmott ainda sugere que se utilize a <a href="http://math.stackexchange.com/questions/888259/can-someone-explain-in-general-what-a-central-difference-formula-is-and-what-it">diferença centrada</a>, que é dada por $\frac{\partial V}{\partial S} (S, t) \approx \frac{V_{i+1}^{k} - V_{i-1}^{k} }{2 \delta S}$. A discretização anterior também poderia ser utilizada, mas esta oferece um erro de aproximação menor. O único problema desta abordagem é que preciso saber os valores $S + \delta S$ e $S - \delta S$ e nas fronteiras do Grid não terei esta informação. Por tanto, nestes casos utilizarei a discretrização utilizando apenas um dos lados.</p>
<p>Por fim, para achar o $\gamma$, Wilmott subtrai o delta forward do delta backward e divide o resultado por $\delta S$, chegando na aproximação $\frac{\partial V^2}{\partial S^2} (S, t) \approx \frac{V_{i+1}^{k} - 2 V_{i}^{k} + V_{i-1}^{k} }{\delta S^2}$. Como demonstrado em notas de aula, todos estes resultados podem ser checados aplicando a expansão de Taylor.</p>
<h3 id="2.2.-Condição-Terminal-e-Payoffs">2.2. Condição Terminal e Payoffs<a class="anchor-link" href="#2.2.-Condição-Terminal-e-Payoffs">¶</a></h3><p>O valor da opção no vencimento é simplemente seu payoff. Assim, não é necessário resolver nada para $T$, apenas para $S$. Usando a notação de diferenças finitas, os payoffs desejados ficam sendo da forma:</p>
<ul>
<li>$V_{i}^{0}=ln(i\delta S)$</li>
<li>$V_{i}^{0}=(ln(i\delta S))^2$</li>
<li>$V_{i}^{0}=(i\delta S-K)^2$</li>
<li>$V_{i}^{0}=\mathbf{1}_{i\delta S > K}$</li>
<li>$V_{i}^{0}=max(i\delta S-K, 0)$</li>
</ul>
<p>Como o Wilmott explica, o método de diferenças finitas começa de trás para frente, será destes valores que a iteração começará, como se estivéssemos calculando o preço de um derivativo por árvores binomiais, também de trás para frente.</p>
<h3 id="2.3.-Condições-de-Contorno">2.3. Condições de Contorno<a class="anchor-link" href="#2.3.-Condições-de-Contorno">¶</a></h3><p>Quando estivermos percorrendo o Grid, necessitaremos definir o preço do derivativo em seus extremos, quando $S=0$ e $S=I \delta S$, onde $I$ é o ponto mais alto do Grid. Esta condição depende do instrumento que está sendo precificado. Para <strong>todos</strong> dos contratos, utilizaremos a seguinte condição para quando $S=0$:</p>
$$V^{k}_{0} = (1 - r \delta t)V^{k-1}_{0}$$<p>Para a condição de contorno superior, utilizaremos para a <strong>maioria dos contratos</strong>:</p>
$$V^{k}_{I} = 2V^{k}_{I-1} - V^{k}_{I-2}$$<p>Esta condição é adequada pois, a medida que $S \rightarrow \infty$, o $\Gamma$ da maioria dos contratos tende a zero. Porém, isso não é verdade para o contrato cujo payoff é descrito por $V_{i}^{0}=(i\delta S-K)^2$. Diferenciando novamente o <a href="https://nbviewer.jupyter.org/github/ucaiado/Replicating_Strategy/blob/master/UiraCaiadoEx01.ipynb">delta</a> deste contrato, chegamos que o gamma será dado por $2 \cdot e^{(r + \sigma^2)(T-t)}$. Neste caso,
partindo da equação discretizada do $\Gamma$ e resolvendo para $V^{k}_{I}$, tenho que:</p>
\begin{equation}
\begin{aligned}
\Gamma &= \frac{V_{I+1}^{k} - 2 V_{I}^{k} + V_{I-1}^{k} }{\delta S^2}\\
2 \cdot e^{(r + \sigma^2)(\delta t)} &= \frac{V_{I+1}^{k} - 2 V_{I}^{k} + V_{I-1}^{k} }{\delta S^2}\\
V_{I}^{k} &= 2 \delta S^2 e^{(r + \sigma^2)(\delta t)} + 2V_{I-1}^{k} - V_{I-2}^{k}\\
V_{I}^{k} &\approx 2 \delta S^2 + 2V_{I-1}^{k} - V_{I-2}^{k}\\
\end{aligned}
\end{equation}
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<h2 id="3.-Implementando-o-Modelo">3. Implementando o Modelo<a class="anchor-link" href="#3.-Implementando-o-Modelo">¶</a></h2><p>Nesta seção vou decsrever o método de diferenças finitas explícito e implementar os códigos necessários.</p>
<h3 id="3.1.-Método-Explícito">3.1. Método Explícito<a class="anchor-link" href="#3.1.-Método-Explícito">¶</a></h3><p>Seguindo notas de aula, substituindo as diferenciações encontradas na equação parabólica mencionada anteriormente e reescrevendo os outros termos com a notação de diferenças finitas, temos que:</p>
\begin{equation}
\begin{aligned}
\frac{\partial V}{\partial t} + a(S, t) \cdot \frac{\partial V^2}{\partial S^2} + b(S, t) \cdot \frac{\partial V}{\partial S} + c(S, t)\cdot V &= 0 \\
\frac{V_{i}^{k} - V_{i}^{k+1} }{\delta t} + a_{i}^{k} \cdot \frac{V_{i+1}^{k} - 2 V_{i}^{k} + V_{i-1}^{k} }{\delta S^2} + \\b_{i}^{k} \cdot \frac{V_{i+1}^{k} - V_{i-1}^{k} }{2 \delta S} + c_{i}^{k}\cdot V_{i}^{k} &= 0
\end{aligned}
\end{equation}<p>Rearranjando equação acima para isolar $V_{i}^{k+1}$ e renomeando alguns termos, ficamos com:</p>
$$V_{i}^{k+1} = A_{i}^{k}V_{i-1}^{k} + \left( 1 + B_{i}^{k} \right)V_{i}^{k} + C_{i}^{k}V_{i+1}^{k}$$<p>Onde:</p>
\begin{equation}
\begin{aligned}
A_{i}^{k} &= \nu_1 a_{i}^{k} - 0.5\nu_2 b_{i}^{k} \\
B_{i}^{k} &= -2\nu_1 a_{i}^{k} - \delta t c_{i}^{k} \\
C_{i}^{k} &= \nu_1 a_{i}^{k} + 0.5\nu_2 b_{i}^{k}
\end{aligned}
\end{equation}<p>Sendo que $\nu_1=\frac{\delta t}{\delta S ^2}$ e $\nu_2=\frac{\delta t}{\delta S}$. A equação acima está definida apenas entre $i=1, ..., I-1$. OS pontos restantes necessários para discretização vem das condições de contorno. Como nós conhecemos o valor terminal em $V_{i}^{0}$, podemos calcular o valor de $V_{i}^{1}$ e assim por diante. Como o valor do instrumento em $k+1$ só depende dos valores dele em $k$, chamamos este método de <strong>método explícito</strong>.</p>
<h3 id="3.2.-Outros-Detalhes">3.2. Outros Detalhes<a class="anchor-link" href="#3.2.-Outros-Detalhes">¶</a></h3><h4 id="a.-Estabilidade">a. Estabilidade<a class="anchor-link" href="#a.-Estabilidade">¶</a></h4><p>Para que a solução deste método seja estável, é necessário que satisfaça $\delta t \, \leqslant \frac{\delta S^2}{2\sigma^2 S^2}$ e $\delta S \leqslant \frac{2a}{|b|}$. Vou utilizar a primeira delas explicitamente na implementação, porém vou fazer a seguinte transformação:</p>
$$\delta t \, \leqslant \frac{\delta S^2}{2\sigma^2 S^2} = \frac{1}{\sigma^2} \left(\frac{\delta S}{S}\right)^2 = \frac{1}{\sigma^2 I^2}$$<p>Onde $I = \delta S \times S$.</p>
<h4 id="b.-Interpolação-Bilinear">b. Interpolação Bilinear<a class="anchor-link" href="#b.-Interpolação-Bilinear">¶</a></h4><p>Devido a discretização do modelo, pode ser que seja necessário obter um valor que esteja entre os nós do Grid. Nestes casos, adotarei uma interpolação de duas dimensões (Preço e tempo) chamada <a href="https://en.wikipedia.org/wiki/Bilinear_interpolation">interpolação bilinear</a>. Ela consiste em realizar uma interpolação linear primeiro em uma direção e depois na outra. Considerando um quadrante específico do Grid, em que há quatro preços de opção, basicamente se divide o quadrante em quatro (onde os pontos de divisão são os valores desejados) e se utiliza a área de cada um deles como peso para os valores das opções. Assim, terminamos com a seguinte fórmula:</p>
$$\frac{\sum_{i=1}^{4} A_i V_i}{\sum_{i=1}^{4} A_i}$$<p>Onde $A$ é a área dos retângulos e $V$ o preço da opção em cada canto do quadrante principal. Cada $A_i$ é oposto ao $V_i$ correspondente (o $A_2$ é oposto ao $V_2$, por exemplo).</p>
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<h3 id="3.3.-Criando-o-Grid">3.3. Criando o Grid<a class="anchor-link" href="#3.3.-Criando-o-Grid">¶</a></h3><p>Primeiro, vou definir as classes para a criação do Grid. Como quero acessar os valores posteriormente, vou manter todos em uma estrutura, ainda que este provavelmente não seja o método mais adequado pensando em velocidade.</p>
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">finite_difference</span><span class="p">;</span> <span class="nb">reload</span><span class="p">(</span><span class="n">finite_difference</span><span class="p">);</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">finite_difference</span><span class="o">.</span><span class="n">Grid</span><span class="p">(</span><span class="n">f_vol</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<span class="n">f_value</span><span class="o">=</span><span class="mf">100.</span><span class="p">,</span>
<span class="n">f_time</span><span class="o">=</span><span class="mf">1.</span><span class="p">,</span>
<span class="n">i_nas</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">i_nts</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="k">print</span> <span class="n">x</span>
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<pre> 0 1 2 3 4 5 6 7 8 9
0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9
1 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9
2 2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9
3 3,0 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9
4 4,0 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 4,9
5 5,0 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 5,9
6 6,0 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 6,9
7 7,0 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 7,9
8 8,0 8,1 8,2 8,3 8,4 8,5 8,6 8,7 8,8 8,9
9 9,0 9,1 9,2 9,3 9,4 9,5 9,6 9,7 9,8 9,9
10 10,0 10,1 10,2 10,3 10,4 10,5 10,6 10,7 10,8 10,9
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<div class=" highlight hl-ipython2"><pre><span class="o">%</span><span class="k">timeit</span> x = Grid(f_vol=0.2, f_value=100., f_time=1., i_nas=100)
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<pre>1000 loops, best of 3: 344 µs per loop
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<p>Como comentei, não é uma estrtutura eficiente. Demorou quase 5 segundos para criar uma estrtutura de 10.000 nós. Porém acredito que isso me ajudará adiante.</p>
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<h3 id="3.3.-Testando-Método">3.3. Testando Método<a class="anchor-link" href="#3.3.-Testando-Método">¶</a></h3><p>Seguindo implementação do livro, também há outra opção para se encontrar $V_{i}^{k+1}$. Substituindo a notação da fórmula de precificação e assumindo que possuimos os valores para $\Delta$, $\Gamma$ e $V_{i}^{k}$, podemos isolar o $\theta$, ficando com</p>
\begin{equation}
\begin{aligned}
\theta + a_{i}^{k} \cdot \Gamma + b_{i}^{k} \cdot \Delta + c_{i}^{k}\cdot V_{i}^{k} &= 0\\
-a_{i}^{k} \cdot \Gamma - b_{i}^{k} \cdot \Delta - c_{i}^{k}\cdot V_{i}^{k} &= \theta
\end{aligned}
\end{equation}<p>Assim, para uma opção que não paga dividendos, que o ativo base segue um movimento brawniano geométrico, e esta sendo avaliado no mundo neutro a risco, temos que o parâmetro $c=-r$, $b=rS$ e $a=0.5 \cdot \sigma^2 S^2$. Assim, fico com o theta sendo da forma:</p>
$$\theta = r V_{i}^{k} - rS_{i}^{k}\Delta - 0.5\sigma^2 S^2\Gamma$$<p>Então, o preço do derivativo em $k+1$ pode ser dado por:</p>
$$V_{i}^{k+1} = V_{i}^{k} - \delta t \theta$$<p>Para comparar os resultados obtidos, também serão calculados os preços e as gregas de cada opção analiticamente, seguindo as equações de preficição demostradas em <a href="https://nbviewer.jupyter.org/github/ucaiado/Replicating_Strategy/blob/master/UiraCaiadoEx01.ipynb">trabalho</a> anterior. Abaixo vou imprimir os resultados obtidos para uma Call Européia para comparação com exemplos do Wilmott.</p>
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">finite_difference</span><span class="p">;</span> <span class="nb">reload</span><span class="p">(</span><span class="n">finite_difference</span><span class="p">);</span>
<span class="n">d_param</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"f_St"</span><span class="p">:</span> <span class="mf">100.</span><span class="p">,</span> <span class="c1"># preco do ativo</span>
<span class="s2">"f_sigma"</span><span class="p">:</span> <span class="mf">0.2</span><span class="p">,</span> <span class="c1"># desvio padra do ativo objeto</span>
<span class="s2">"f_time"</span><span class="p">:</span> <span class="mf">1.</span><span class="p">,</span> <span class="c1"># tempo para vencimento em anos</span>
<span class="s2">"f_r"</span><span class="p">:</span> <span class="mf">0.05</span><span class="p">,</span> <span class="c1"># taxa de juros anual</span>
<span class="s2">"i_nas"</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="c1"># passos que o ativo sera discretizado</span>
<span class="s2">"i_nts"</span><span class="p">:</span> <span class="mi">10</span><span class="p">,</span> <span class="c1"># passos que o tempo sera discretizado (opcional)</span>
<span class="s2">"f_K"</span><span class="p">:</span> <span class="mf">100.</span> <span class="c1"># strike da opcao</span>
<span class="p">}</span>
<span class="o">%</span><span class="k">time</span> self = finite_difference.EuropianCall(**d_param)
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<pre>Wall time: 582 ms
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<div class=" highlight hl-ipython2"><pre><span class="c1"># imprime apenas a parte da matriz demonstrada no exemplo do livro, pag 1216, tabela 77.1</span>
<span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">df_opt_prices</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">df</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"{:.3f}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">df</span><span class="o">.</span><span class="n">columns</span><span class="p">]</span>
<span class="n">df</span><span class="o">.</span><span class="n">tail</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span><span class="o">.</span><span class="n">head</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
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<thead>
<tr style="text-align: right;">
<th></th>
<th>0.000</th>
<th>0.111</th>
<th>0.222</th>
<th>0.333</th>
<th>0.444</th>
<th>0.556</th>
<th>0.667</th>
<th>0.778</th>
<th>0.889</th>
<th>1.000</th>
</tr>
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<tr>
<th>60.0</th>
<td>0.0</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.001046</td>
<td>0.004769</td>
<td>0.012912</td>
<td>0.027040</td>
<td>0.048408</td>
</tr>
<tr>
<th>70.0</th>
<td>0.0</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.010816</td>
<td>0.040306</td>
<td>0.092362</td>
<td>0.168004</td>
<td>0.266547</td>
<td>0.386414</td>
</tr>
<tr>
<th>80.0</th>
<td>0.0</td>
<td>0.000000</td>
<td>0.000000</td>
<td>0.084278</td>
<td>0.248619</td>
<td>0.475041</td>
<td>0.746832</td>
<td>1.051237</td>
<td>1.379044</td>
<td>1.723707</td>
</tr>
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<th>90.0</th>
<td>0.0</td>
<td>0.000000</td>
<td>0.512500</td>
<td>1.148000</td>
<td>1.807437</td>
<td>2.461115</td>
<td>3.100759</td>
<td>3.724696</td>
<td>4.333380</td>
<td>4.927851</td>
</tr>
<tr>
<th>100.0</th>
<td>0.0</td>
<td>2.500000</td>
<td>4.013889</td>
<td>5.200154</td>
<td>6.223712</td>
<td>7.149693</td>
<td>8.008182</td>
<td>8.816223</td>
<td>9.584443</td>
<td>10.320129</td>
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<th>110.0</th>
<td>10.0</td>
<td>10.555556</td>
<td>11.571451</td>
<td>12.561620</td>
<td>13.500820</td>
<td>14.389205</td>
<td>15.235188</td>
<td>16.045095</td>
<td>16.824334</td>
<td>17.576942</td>
</tr>
<tr>
<th>120.0</th>
<td>20.0</td>
<td>20.555556</td>
<td>21.108025</td>
<td>21.790273</td>
<td>22.510062</td>
<td>23.243401</td>
<td>23.974649</td>
<td>24.698653</td>
<td>25.412183</td>
<td>26.114374</td>
</tr>
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<th>130.0</th>
<td>30.0</td>
<td>30.555556</td>
<td>31.108025</td>
<td>31.657425</td>
<td>32.248867</td>
<td>32.862026</td>
<td>33.491183</td>
<td>34.127231</td>
<td>34.766578</td>
<td>35.405282</td>
</tr>
<tr>
<th>140.0</th>
<td>40.0</td>
<td>40.555556</td>
<td>41.108025</td>
<td>41.657425</td>
<td>42.203772</td>
<td>42.764972</td>
<td>43.335178</td>
<td>43.915298</td>
<td>44.500544</td>
<td>45.090213</td>
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<p>O título das colunas se refere ao tempo, começando no vencimento (tempo zero) até $t$ (o dia em que a opção está sendo avaliada). As linhas se referem ao preço do ativo objeto. <em>Note que começa a aparecer uma diferença em relação ao livro quando o ativo está valendo $R\$ 110.00$ ou menos. Como o método utiliza os três nós subjacentes anteriores para calcular o valor do nó posterior, o nó $(\,90.0 \,; 0.111)$ não deveria apresentar valor algum de fato. Porém, no exemplo do Wilmott, apresentou.</em></p>
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<h2 id="4.-Precificando-Instrumentos">4. Precificando Instrumentos<a class="anchor-link" href="#4.-Precificando-Instrumentos">¶</a></h2><p>Nesta seção vou precificar alguns instrumentos e comparar com suas soluções analíticas.</p>
<h3 id="4.1.-Call-Européia">4.1. Call Européia<a class="anchor-link" href="#4.1.-Call-Européia">¶</a></h3><p>Partindo de implementação anterior, apenas resta imeplementar o gamma analítico da equação, que é dado abaixo:
$$\frac{\partial V^2}{\partial S^2} = \frac{\Delta}{\partial S} = \frac{N'(d1)}{S \sigma \sqrt{t}} $$</p>
<p>Abaixo vou comparar os resultados analíticos com o que obtive do método de diferenças finitas.</p>
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">finite_difference</span><span class="p">;</span> <span class="nb">reload</span><span class="p">(</span><span class="n">finite_difference</span><span class="p">);</span>
<span class="n">d_param</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"f_St"</span><span class="p">:</span> <span class="mf">100.</span><span class="p">,</span> <span class="c1"># preco do ativo</span>
<span class="s2">"f_sigma"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">,</span> <span class="c1"># desvio padra do ativo objeto</span>
<span class="s2">"f_time"</span><span class="p">:</span> <span class="o">.</span><span class="mi">5</span><span class="p">,</span> <span class="c1"># tempo para vencimento em anos</span>
<span class="s2">"f_r"</span><span class="p">:</span> <span class="mf">0.10</span><span class="p">,</span> <span class="c1"># taxa de juros anual</span>
<span class="s2">"i_nas"</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="c1"># passos que o ativo sera discretizado</span>
<span class="s2">"f_K"</span><span class="p">:</span> <span class="mf">100.</span> <span class="c1"># strike da opcao</span>
<span class="p">}</span>
<span class="o">%</span><span class="k">time</span> my_option = finite_difference.EuropianCall(**d_param)
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<pre>CPU times: user 2.24 s, sys: 23.1 ms, total: 2.26 s
Wall time: 2.28 s
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<div class=" highlight hl-ipython2"><pre><span class="c1"># plotando graficos do inicio do periodo</span>
<span class="n">l_prices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">50.</span><span class="p">,</span> <span class="mf">151.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">)</span>
<span class="n">my_option</span><span class="o">.</span><span class="n">compare_to_analytical_solutions</span><span class="p">(</span><span class="n">l_prices</span><span class="p">,</span> <span class="n">d_param</span><span class="p">[</span><span class="s1">'f_time'</span><span class="p">])</span>
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