python/European_Call.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Introduction**\n",
    "\n",
    "Suppose a European call option with strike price $K$ and an underlying asset whose spot price at maturity $S_T$ follows a given random distribution. The corresponding payoff function is defined as: \n",
    "\n",
    "$$\\max\\{S_T - K, 0\\}$$\n",
    "In the following, a quantum algorithm based on amplitude estimation is used to estimate the expected payoff, i.e., the fair price before discounting, for the option: \n",
    "\n",
    "$$\\mathbb{E}\\left[ \\max\\{S_T - K, 0\\} \\right]$$\n",
    "as well as the corresponding $\\Delta$, i.e., the derivative of the option price with respect to the spot price, defined as: \n",
    "\n",
    "$$\n",
    "\\Delta = \\mathbb{P}\\left[S_T \\geq K\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "\n",
    "from qiskit import BasicAer\n",
    "from qiskit.aqua.algorithms import AmplitudeEstimation\n",
    "from qiskit.aqua.components.uncertainty_models import LogNormalDistribution\n",
    "from qiskit.aqua.components.uncertainty_problems import UnivariateProblem\n",
    "from qiskit.aqua.components.uncertainty_problems import UnivariatePiecewiseLinearObjective as PwlObjective\n",
    "\n",
    "# number of qubits to represent the uncertainty\n",
    "num_uncertainty_qubits = 3\n",
    "\n",
    "# parameters for considered random distribution\n",
    "S = 2.0 # initial spot price\n",
    "vol = 0.4 # volatility of 40%\n",
    "r = 0.05 # annual interest rate of 4%\n",
    "T = 40 / 365 # 40 days to maturity\n",
    "\n",
    "# resulting parameters for log-normal distribution\n",
    "mu = ((r - 0.5 * vol**2) * T + np.log(S))\n",
    "sigma = vol * np.sqrt(T)\n",
    "mean = np.exp(mu + sigma**2/2)\n",
    "variance = (np.exp(sigma**2) - 1) * np.exp(2*mu + sigma**2)\n",
    "stddev = np.sqrt(variance)\n",
    "\n",
    "# lowest and highest value considered for the spot price; in between, an equidistant discretization is considered.\n",
    "low  = np.maximum(0, mean - 3*stddev)\n",
    "high = mean + 3*stddev\n",
    "\n",
    "# construct circuit factory for uncertainty model\n",
    "uncertainty_model = LogNormalDistribution(num_uncertainty_qubits, mu=mu, sigma=sigma, low=low, high=high)\n",
    "\n",
    "# plot probability distribution\n",
    "x = uncertainty_model.values\n",
    "y = uncertainty_model.probabilities\n",
    "plt.bar(x, y, width=0.2)\n",
    "plt.xticks(x, size=15, rotation=90)\n",
    "plt.yticks(size=15)\n",
    "plt.grid()\n",
    "plt.xlabel('Spot Price at Maturity $S_T$ (\\$)', size=15)\n",
    "plt.ylabel('Probability (%)', size=15)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Payoff Function**\n",
    "\n",
    "The payoff function equals zero as long as the spot price at maturity $S_T$ is less than the strike price $K$ and then increases linearly. The implementation uses a comparator, that flips an ancilla qubit from $\\big|0\\rangle$ to $\\big|1\\rangle$ if $S_T \\geq K$, and this ancilla is used to control the linear part of the payoff function.\n",
    "\n",
    "\n",
    "The linear part itself is then approximated as follows. We exploit the fact that $\\sin^2(y + \\pi/4) \\approx y + 1/2$ for small $|y|$. Thus, for a given approximation scaling factor $c_{approx} \\in [0, 1]$ and $x \\in [0, 1]$ we consider $$ \\sin^2( \\pi/2 * c_{approx} * ( x - 1/2 ) + \\pi/4) \\approx \\pi/2 * c_{approx} * ( x - 1/2 ) + 1/2 $$ for small $c_{approx}$.\n",
    "\n",
    "\n",
    "We can easily construct an operator that acts as $$\\big|x\\rangle \\big|0\\rangle \\mapsto \\big|x\\rangle \\left( \\cos(a*x+b) \\big|0\\rangle + \\sin(a*x+b) \\big|1\\rangle \\right),$$ using controlled Y-rotations.\n",
    "Eventually, we are interested in the probability of measuring $\\big|1\\rangle$ in the last qubit, which corresponds to $\\sin^2(a*x+b)$. Together with the approximation above, this allows to approximate the values of interest. The smaller we choose $c_{approx}$, the better the approximation. However, since we are then estimating a property scaled by $c_{approx}$, the number of evaluation qubits $m$ needs to be adjusted accordingly.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# set the strike price (should be within the low and the high value of the uncertainty)\n",
    "strike_price = 1.896\n",
    "\n",
    "# set the approximation scaling for the payoff function\n",
    "c_approx = 0.25\n",
    "\n",
    "# setup piecewise linear objective fcuntion\n",
    "breakpoints = [uncertainty_model.low, strike_price]\n",
    "slopes = [0, 1]\n",
    "offsets = [0, 0]\n",
    "f_min = 0\n",
    "f_max = uncertainty_model.high - strike_price\n",
    "european_call_objective = PwlObjective(\n",
    "    uncertainty_model.num_target_qubits, \n",
    "    uncertainty_model.low, \n",
    "    uncertainty_model.high,\n",
    "    breakpoints,\n",
    "    slopes,\n",
    "    offsets,\n",
    "    f_min,\n",
    "    f_max,\n",
    "    c_approx\n",
    ")\n",
    "\n",
    "# construct circuit factory for payoff function\n",
    "european_call = UnivariateProblem(\n",
    "    uncertainty_model,\n",
    "    european_call_objective\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot exact payoff function (evaluated on the grid of the uncertainty model)\n",
    "x = uncertainty_model.values\n",
    "y = np.maximum(0, x - strike_price)\n",
    "plt.plot(x, y, 'ro-')\n",
    "plt.grid()\n",
    "plt.title('Payoff Function', size=15)\n",
    "plt.xlabel('Spot Price', size=15)\n",
    "plt.ylabel('Payoff', size=15)\n",
    "plt.xticks(x, size=15, rotation=90)\n",
    "plt.yticks(size=15)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "exact expected value:\t0.1623\n",
      "exact delta value:   \t0.8098\n"
     ]
    }
   ],
   "source": [
    "# evaluate exact expected value (normalized to the [0, 1] interval)\n",
    "exact_value = np.dot(uncertainty_model.probabilities, y)\n",
    "exact_delta = sum(uncertainty_model.probabilities[x >= strike_price])\n",
    "print('exact expected value:\\t%.4f' % exact_value)\n",
    "print('exact delta value:   \\t%.4f' % exact_delta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Evaluate Expected Payoff**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# set number of evaluation qubits (=log(samples))\n",
    "m = 6\n",
    "\n",
    "# construct amplitude estimation \n",
    "ae = AmplitudeEstimation(m, european_call)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# result = ae.run(quantum_instance=BasicAer.get_backend('qasm_simulator'), shots=100)\n",
    "result = ae.run(quantum_instance=BasicAer.get_backend('statevector_simulator'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exact value:    \t0.1623\n",
      "Estimated value:\t0.1196\n",
      "Probability:    \t0.4973\n"
     ]
    }
   ],
   "source": [
    "print('Exact value:    \\t%.4f' % exact_value)\n",
    "print('Estimated value:\\t%.4f' % result['estimation'])\n",
    "print('Probability:    \\t%.4f' % result['max_probability'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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g/4hYWEoFzMysZR2f7DIibgPePEiesQ3SX9tkmyeBw4ZVONu4TZiQHhe5RdWsDJ5N2axm8eJul8Csr/iGY2ZmVgkHGDMzq4QDjJmZVcIBxszMKuEAY2ZmlfBVZGY106Z1uwRmfcUBxqymdstkMyuFm8jMzKwSbQUYSfWmbzHrD4sWeRS/WYnabSK7T9K5wJyIuL2KApl1zcSJ6dEzKpuVot0msrOAI4A/Svq9pOmStq2gXGZm1uPaCjARMTMixgFvAZYCpwMPSPqJpIOqKKCZmfWmIXXyR8TVEfEBYBfg48ArgXmSlkuaKelFZRbSzMx6z3CvIpsIvIl0G+SVwLXAscAySe8f5r7NzKyHtR1gJO0u6UuS7gSuAnYFPgS8KCL+DdgdmAV8o9SSmplZT2nrKjJJ84F/Be4D5pCuJrsrnycinpb0U+ATpZXSzMx6TruXKT8ETAauiGh6LefNwEuHXCqzbljoO2ublandAPNdYHG94CJpa2DviLgmIp4C7tpga7ORrHbLZDMrRbt9MPOBPRuse2W23szMrO0AoybrtgbWDKMsZt01fXpazKwUgzaRSXoTMCmXdKykgwvZtgQOBW4tr2hmHXb22enRsyqblaKVPpg3kAZTAgRwJLC+kOcfwBLgM+UVzczMetmgASYivkE2pkXSX4F3RcTNVRfMzMx6W1tXkUWELz02M7OWtNIHMxm4LiJWZX83FRGXllIyMzPraa2cwfwKeCNwQ/Z30PhqsgB8UzIzM2spwLwUeCD3t1l/2nvvbpfArK+00sl/V72/zfqOb5dsVqpW+mC2ameHEeHBlmZm1lIT2WpS30qr3AdjZmYtBZgP0V6AMetNyq5daTpRuJm1qpU+mHM6UA4zM+szw71lspmZWV2tdPLfABwdEbdJupFBmssiYp+yCmdmZr2rlT6YPwFrc3+7gdrMzAbVSh/MB3N/H11paczMrG8MuQ9GyU6Smt2EzMzMNlJtzaYMz05+eSIwIdt+vaRFwFcj4pKSy2fWObNmdbsEZn2lrQAjaQbwPeAq4BPAQ8DOwGHA/0j6SET4v9R6k2+XbFaqds9gPgfMioiPFNLPknQW8HnAAcbMzNrug9kB+GWDdRcA2w+2A0l7SrpK0hpJ90s6WVLT6WUkjZUUdZa5dfJOkXSrpCcl3SZpaks1M5s9Oy1mVop2z2DmA/sDV9RZtz9wTbONJY0GrgRuA6YALwNOIwW6E1t4/U8Dv8k9X1HY/36kQPc94DhgMnCepJURcXkL+7eN2YwZ6dFNZWalaGWg5Z65p98Cvi9pB+AinuuDeRdwCHDsILv7MDAKOCwiVgFXSNoWmCnp1CytmaURcX2T9V8AromI47Ln8yXtBXwRcIAxM+ugVs5g/sjzB1cKmJEtxbtb/prmsykfAswrBJK5wCmkM6CLWyhPXZK2AA4gnbnkzQXmSNouIh4b6v7NzKw9rQSYA0p8vfHA1fmEiLhb0pps3WABZo6k7UlnTucBn4+I2iwDLwM2B5YUtrmd1AS3B3Dj8IpvZmatamUk/4ISX2808Gid9JXZukbWAd8lNXOtAiYBx5OCypTcvqmz/5WF9c8jaTowHWDMmDEMDAw0K/+Isnr16p4qb9nKrv+k7LFXjunG/v6Dj8FIr3/bAy1rJG0CbFlMr+KOlhHxAPCxXNKApAeB70n6l4i4ZRj7ng3MBpg4cWJMmjRpWGXtpIGBAXqpvGUbSv3HnvD8scDLv37oBnl65Zhu7O8/+BiM9Pq3dZlyNj3M8ZKWAU8Bj9dZmlkJbFcnfTTPnWm06vzscUJu39TZ/+jCejMz64B2x8EcB5wA/IDUuf9V4GTgDmA5WVNTE0tIfS3PkrQbsBUb9p0MJgqPd5KC3vhCvvHAM1kZzRqL8N0szUrUboCZBnwJODV7flFEnATsRQoQrxhk+8uAt0naJpc2lXQ7gHb7eo7IHhcBRMQ60jidIwv5pgK/8xVkZmad1W4fzEuBmyPiaUlPAS8EiIhnJH0P+D7pDKeRs0hnQRdKOgUYB8wETs9fupw1wS2IiGOy5zOBbUiDLFcBbwI+A1wYEX/I7f/LpP6ZM0njdCZny8Ft1tPMzIap3TOYR4Cts7/vBl6XWzeaNIiyoYhYCRxIGitzMXAScAbprChvM54/nmYJaZzMHOBS4L3AN7LH/P6vI53ZHATMA94BvNej+K0lEyakxcxK0e4ZzG+A15O+5H9KGoG/PfAP4KOkWZabiojbgDcPkmds4flc0oDJQUXERaSzF7P2LF7c7RKY9ZV2A8xM4MXZ318jNZEdTTpzuQL4eFkFMzOz3tZWgImIpcDS7O91pHvCfKKCcpmZWY8bzkDLfwZ2Be6PiPvKK5KZmfWDdjv5kfTvku4B7gJ+D9wt6V5JxZuQmZnZRqzdkfxfBL5DGs9yKDAxe7wM+Fa23szMrO0mso8CX4uILxTSf53NDfZR0sh+s94zbVq3S2DWV9oNMKNofNfKBfgqMutlvl2yWana7YO5CDiswbrDgV8NrzhmZtYvWrll8uTc08uAUyWNZcNbJu8F/Gf5RTTrkEWL0qNH85uVopUmsl+x4a2RXwy8rU7eH5PuNGnWeyZOTI+eUdmsFK0EmJdWXgozM+s7rdwy+a5OFMTMzPpL2yP5JW1G6tDfD9ge+DtwLWnq/PXlFs/MzHpVWwFG0s7A5cBrSHewfBDYlzT+5RZJb42Ih8supJmZ9Z52L1M+HdgBeGNEjIuIfSNiHPCGLP30sgtoZma9qd0AMxk4PiJuyCdGxI3AZ0nTxpiZmbXdB7MF8HiDdY8D/zS84ph10cKF3S6BWV9pN8BcDxwv6eqIeKKWKOkFwPHZerPe5AGWZqVqN8B8CpgP3CPpclIn/86kQZcCJpVaOjMz61lt9cFExM3AK4DZwE7AW0gB5izgFRFxS+klNOuU6dPTYmalaPkMRtLmwD7AXyPihOqKZNYlZ5+dHj2rslkp2jmDeRq4GhhfUVnMzKyPtBxgIuIZ4M/ALtUVx8zM+kW742A+D3xR0qurKIyZmfWPdq8iO5E0Yv9mSfeRriJ73tzmEbFPSWUzM7Me1m6A+WO2mJmZNdVSgJE0ijRNzB+BvwFXRsSDVRbMrOP23rvbJTDrK63cMnkccCUwNpe8StK7I+Lyqgpm1nG1WyabWSla6eQ/FXgG+FdgK2Av4CZgVoXlMjOzHtdKgNkXODEifhMRT0bE7cAM4CWSdq22eGZm1qtaCTC7An8ppN1JmnvMY2Ksf0hpMbNStHoVWQyexay/jT3hkuc9X/513/7IrJlWA8w8SevrpF9VTI+InYdfLDMz63WtBJiTKi+FmZn1nUEDTEQ4wJiZWdvanYvMzMysJQ4wZmZWiXbnIjPrX7M8dtisTA4wZjW+XbJZqdxEZmZmlXCAMauZPTstZlaKjgcYSXtKukrSGkn3SzpZ0qaDbPN6SXMkLcu2WyrpS5K2LOSbKSnqLAdXWyvrCzNmpMXMStHRPhhJo0lT/98GTAFeBpxGCnQnNtl0apb3FODPwGuAL2ePhxfyPgYUA8rtwy27mZm1p9Od/B8GRgGHRcQq4ApJ2wIzJZ2apdXz9YhYkXs+IOlJYJak3SPirty69RFxfTXFNzOzVnW6iewQYF4hkMwlBZ39G21UCC41N2WPLyqveGZmVpZOB5jxwJJ8QkTcDazJ1rVjX9KN0O4spL9Q0gpJT0m6SdJhQy6tmZkNmSI6NxO/pKeAz0TEmYX0e4FzI+JzLe5nF+APwKURcXQu/f3AzqSzm21IN0abDBweERc22Nd0YDrAmDFjJsydO7fdanXN6tWr2XrrrbtdjK4ZSv1vve+x5z1/9Yu3e/bvSQccAMDA/Pltb9sNG/v7Dz4GI6H+BxxwwKKImFhvXc8FGEn/RLpQ4J+BCRGxskleAb8FRkXEawfb98SJE2PhwoWDZRsxBgYGmDRpUreL0TVDqX/Te7rUbjbW4H9ipN0PZmN//8HHYCTUX1LDANPpJrKVQL2ffaOzdU1lAeNcYC9gcrPgAhApel4IvGawS6HNiGgYXMysfZ2+imwJhb4WSbsBW1Hom2ngTNLlzW+JiFbyQ7obp781zMw6rNNnMJcBb5O0TS5tKrAWWNBsQ0mfBT4GvD8irmvlxbIznsOBWyLi6aEV2czMhqLTZzBnAccBF0o6BRgHzAROz1+6LGkZsCAijsmevxf4GnAOcJ+kN+b2eWdEPJzlWwBcQDobegEwDXgD8M5qq2V9YcKE9LhoUXfLYdYnOhpgImKlpAOB7wAXA48CZ5CCTLFc+T6Tt2aPR2dL3gdJgQdgGfBJYFfSJcyLgUMj4rIyym99bvHibpfArK90fLr+iLgNePMgecYWnh/NhoGl3nbHDKNoZmZWIs+mbGZmlXCAMTOzSjjAmJlZJRxgzMysEh3v5DcbsaZN63YJzPqKA4xZjW+XbFYqN5GZmVklHGDMahYt8ih+sxK5icysZmI247hnVDYrhc9gzMysEg4wZmZWCQcYMzOrhAOMmZlVwgHGzMwq4QBjZmaV8GXKZjULF3a7BGZ9xQHGrKZ2y2QzK4WbyMzMrBIOMGY106enxcxK4QBjVnP22Wkxs1I4wJiZWSUcYMzMrBIOMGZmVgkHGDMzq4QDjJmZVcIDLc1q9t672yUw6ysOMGY1vl2yWancRGZmZpVwgDEzs0o4wJjVSGkxs1I4wJiZWSUcYMzMrBIOMGZmVgkHGDMzq4QDjJmZVcIBxszMKuGR/Nazbr3vMY4+4RIAln/90OHvcNas4e/DzJ7lAGNW49slm5XKAcasYmOzs6yaUs62zHqA+2DMambPTouZlcJnMGY1M2akRzeVmZXCZzBmZlaJjgcYSXtKukrSGkn3SzpZ0qYtbLedpDmSVkp6TNJPJO1QJ98USbdKelLSbZKmVlMTMzNrpqNNZJJGA1cCtwFTgJcBp5EC3YmDbP5zYA/gWOAZ4BTgIuBfc/vfD7gA+B5wHDAZOE/Syoi4vNTKWGncCd6Yj431sk73wXwYGAUcFhGrgCskbQvMlHRqlrYBSfsCbwX2j4hrsrT7gN9LOigirsyyfgG4JiKOy57Pl7QX8EXAAcbMrIM6HWAOAeYVAslc0tnI/sDFTbZ7sBZcACLiBkl/zdZdKWkL4ADSmUveXGCOpO0i4rGS6mFN5H91+xd39/jsx7qt0wFmPHB1PiEi7pa0JlvXKMCMB5bUSb89WwepuW3zOvluJzXB7QHcOLRiD129f/Jm//jNvpyL251z8Ataer1Wy9WIv6g2LsN5v4e67VB+lIw94RI+9er1Lc/m0G7Z/LkfPkVE515Megr4TEScWUi/Fzg3Ij7XYLsrgCci4p2F9B8D4yLif0v6P8B1wOsi4uZcnpcDfwbeVq8fRtJ0oHZd6iuBpUOuYOftCKzodiG6yPXfuOsPPgYjof67R8RO9VZs9ONgImI20JOj6yQtjIiJ3S5Ht7j+G3f9wcdgpNe/05cprwS2q5M+Ols3nO1qj8V8owvrzcysAzodYJbwXJ8JAJJ2A7aifh9Lw+0y+b6ZO4Gn6uQbT7qs+Y4hlNfMzIao0wHmMuBtkrbJpU0F1gILBtlul2ycCwCSJgLjsnVExDpgPnBkYdupwO/69AqynmzaK5Hrbxv7MRjR9e90J/9o0iDLP5IuTR4HnA6cGREn5vItAxZExDG5tHnAK4BP89xAy4ciojjQcgD4DmkQ5uQs/8EeaGlm1lkdPYOJiJXAgcCmpEuSTwLOAL5UyLpZlidvKuks54fAucAi4F2F/V8HHAEcBMwD3gG818HFzKzzOnoGY2ZmGw/PptxDJE2T9OdsIs9Fkg5sYZuZkqLOcnAnyjwUVU+IOtINpf6SxjZ4n+d2qtxlkfRySbMk/UHS05IGWtyuL95/GNoxGImfgY1+HEyvkPQe4CxgJmlA6QeBX0l6fUT8cZDNHwOKAeX20gtZgqonRB3phll/SH2Ov8k97/YgvKHYi9R/ej1pdo5W9fz7nzPUYwAj6TMQEV56YCHNMPDD3PNNgFuBHw+y3UxgRbfL30Y9P0sas7RtLu0/gTX5tDrb7QsE8KZc2j5Z2kHdrlcH6j82q+vbu12HEo7BJrm/zwcGWtimL97/YR6DEfcZcBNZD+KLLd4AAALkSURBVJA0jvTL7Oe1tIh4BvgFabLPftJoQtRRpAlRm223wYSoQG1C1F4x1Pr3jeyz3a5+ef+BIR+DEccBpjfUBo/Wm8hze0l15wHKeaGkFZKeknSTpMPKL2JpNpjYNCLuJv2CrzfYtuF2mfyEqL1gqPWvmZO12T8g6XRJo6oo5AjUL+9/GUbMZ8B9ML2hNt3No4X0lbn1DzfYdhmpieUmYBtgBnCBpMMj4sKyC1qC0WxYT0h1HV0nvZXtxpVQrk4Zav3XAd8l3fdoFTAJOJ7UhzOl3CKOSP3y/g/HiPsMOMB0iaTtgF0HyxcRzabQGVRE/LjwuhcDvyXdhG0kBhgbgoh4APhYLmlA0oPA9yT9S0Tc0qWiWYeMxM+Am8i650jS6ftgC5Q4kWek3sALgde0culvF1Q5IWovKLMe52ePE4ZVot7QL+9/2br6GXCA6ZKI+H5EaLAly147i6k3keffI6JR81jDl8+WkajKCVF7wVDrX08UHvtZv7z/ZevqZ8ABpgdExF9Is0E/O5GnpE2y55e1sy9JAg4HbomIp8ssZ0kqmxC1Rwy1/vUckT0uKqNgI1y/vP9l6+5noNvXSXtpbQHeAzxNGmx3AHAO6UvnVbk8+wPrgf1zaQuA44C3kuZuu5Q0CO0d3a5Tg3qOBh4AriDNKTcdWA18pZBvGfCDQto84C/AYcA7SWOHru12nTpRf9J4p9Oyuh8EnJx9Pi7odp2GcAy2In0xHgH8DvhT7vlW/fz+D+cYjMTPQNcPpJc23iyYln2o1gGLgQML6yeRToUn5dJ+kP3TrQWeAK4FDul2XQap557A1VmZHwC+DGxayLMcOKeQ9kJgDulqolXAT4Edu12fTtQfOApYSJq14R/Z5+RkYItu12cI9R/Lc824xWVsv7//Qz0GI/Ez4MkuzcysEu6DMTOzSjjAmJlZJRxgzMysEg4wZmZWCQcYMzOrhAOMmZlVwgHGzMwq4QBjZmaV+P+5FBvFP+zXxgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot estimated values for \"a\"\n",
    "plt.bar(result['values'], result['probabilities'], width=0.5/len(result['probabilities']))\n",
    "plt.xticks([0, 0.25, 0.5, 0.75, 1], size=15)\n",
    "plt.yticks([0, 0.25, 0.5, 0.75, 1], size=15)\n",
    "plt.title('\"a\" Value', size=15)\n",
    "plt.ylabel('Probability', size=15)\n",
    "plt.ylim((0,1))\n",
    "plt.grid()\n",
    "plt.show()\n",
    "\n",
    "# plot estimated values for option price (after re-scaling and reversing the c_approx-transformation)\n",
    "plt.bar(result['mapped_values'], result['probabilities'], width=1/len(result['probabilities']))\n",
    "plt.plot([exact_value, exact_value], [0,1], 'r--', linewidth=2)\n",
    "plt.xticks(size=15)\n",
    "plt.yticks([0, 0.25, 0.5, 0.75, 1], size=15)\n",
    "plt.title('Estimated Option Price', size=15)\n",
    "plt.ylabel('Probability', size=15)\n",
    "plt.ylim((0,1))\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Evaluate Delta**\n",
    "\n",
    "The Delta is a bit simplier to evaluate than the expected payoff. Similarly to the expected payoff, we use a comparator circuit and an ancilla qubit to identify the cases where $S_T \\gt K$. However, since we are only interested in the probability of this condition being true, we can directly use this ancilla qubit as the objective qubit in amplitude estimation without any futher approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# setup piecewise linear objective fcuntion\n",
    "breakpoints = [uncertainty_model.low, strike_price]\n",
    "slopes = [0, 0]\n",
    "offsets = [0, 1]\n",
    "f_min = 0\n",
    "f_max = 1\n",
    "c_approx = 1  # no approximation necessary\n",
    "european_delta_objective = PwlObjective(\n",
    "    uncertainty_model.num_target_qubits, \n",
    "    uncertainty_model.low, \n",
    "    uncertainty_model.high,\n",
    "    breakpoints,\n",
    "    slopes,\n",
    "    offsets,\n",
    "    f_min,\n",
    "    f_max,\n",
    "    c_approx\n",
    ")\n",
    "\n",
    "# construct circuit factory for payoff function\n",
    "european_call_delta = UnivariateProblem(\n",
    "    uncertainty_model,\n",
    "    european_delta_objective\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# set number of evaluation qubits (=log(samples))\n",
    "m = 6\n",
    "\n",
    "# construct amplitude estimation \n",
    "ae_delta = AmplitudeEstimation(m, european_call_delta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# result_delta = ae_delta.run(quantum_instance=BasicAer.get_backend('qasm_simulator'), shots=100)\n",
    "result_delta = ae_delta.run(quantum_instance=BasicAer.get_backend('statevector_simulator'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exact delta:   \t0.8098\n",
      "Esimated value:\t0.8172\n",
      "Probability:   \t0.8829\n"
     ]
    }
   ],
   "source": [
    "print('Exact delta:   \\t%.4f' % exact_delta)\n",
    "print('Esimated value:\\t%.4f' % result_delta['estimation'])\n",
    "print('Probability:   \\t%.4f' % result_delta['max_probability'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot estimated values for delta\n",
    "plt.bar(result_delta['values'], result_delta['probabilities'], width=0.5/len(result_delta['probabilities']))\n",
    "plt.plot([exact_delta, exact_delta], [0,1], 'r--', linewidth=2)\n",
    "plt.xticks(size=15)\n",
    "plt.yticks([0, 0.25, 0.5, 0.75, 1], size=15)\n",
    "plt.title('Estimated Option Delta', size=15)\n",
    "plt.ylabel('Probability', size=15)\n",
    "plt.ylim((0,1))\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}