py/BlackScholes_MC.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Option pricing with MC Simulations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pyfeng as pf\n",
    "import scipy.stats as spst"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma = 0.2\n",
    "texp = 1\n",
    "spot = 100\n",
    "m_bs = pf.Bsm(sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generate random numbers\n",
    "z = np.random.normal(size=100000)\n",
    "#anti-thetic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.002654646236199765, 1.0006030308752767)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(z), np.std(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Black-Scholes model (Geomegric Brownian Motion)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 60.65931887  88.43979654 107.46450935 ... 110.19237137  97.11623848\n",
      "  73.42832176]\n"
     ]
    }
   ],
   "source": [
    "# Black-Scholes-Merton model (Geometric BM)\n",
    "s_bsm = spot * np.exp(-(sigma**2)*texp/2 + sigma*np.sqrt(texp)*z)\n",
    "print(s_bsm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(100.0562734226824, 20.23872983741818)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## Sanity check\n",
    "np.mean(s_bsm), np.std(s_bsm)\n",
    "#spst.describe(s_bsm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HgfuA1ZJWAJuBAxGxDDiQHiPpboq5dO8BVgNPSpqTnuspYAPFvLnL0nIzMytJKxOjR0SMpYfvSX8B1IBdKb4LeCjdrwGDEXE5Ik4CI8BySQuAuRHxUkQE8GzdOmZmVgIV9XeSRkVP/TCwFPi9iNgkaTQiuuraXIyIeZJ2AC9HxHMpvpNiAvRTwLaIWJXiK4FNEbGmwfY2UHwjoLu7u2dwcHB6r7JFY2NjdHZ2lrKt8UYujEy4vOu2LkbfGS0nmSmqam5L5y8F2vu+TqbKuUG186tybtD+/Pr7+w9HRO/4eEsHciPiKnCfpC7geUkTTW7eaJw+Jog32t4AMADQ29sbfX19raQ5bcPDw5S1rfG2794+4fJaR42hy0MlZTM1Vc1tX19x7Z12vq+TqXJuUO38qpwbVDe/KZ2yGRGjwDDFWPy5NGRDuj2fmp0GFtWtthA4k+ILG8TNzKwkrZy984HUw0fSHcAq4A1gL7AuNVsHXOvq7QXWSuqQtITigO3BiDgLXJK0Ip2182jdOmYz7sHdD/Lg7gcZuTDy0/s+NdZy18rwzgJgVxrXvw3YExHfkPQSsEfSeuAt4GGAiDgqaQ9wDLgCbEzDQwCPAc8Ad1CM8++fyRdjZmYTm7ToR8T3gY80iL8N3N9kna3A1gbxQ8BExwPMzGwW+TIMZmYZcdE3M8uIi76ZWUZc9M3MMuKib2aWERd9M7OMuOibmWXERd/MLCMu+mZmGXHRNzPLiIu+mVlGXPTNzDLiom9mlhEXfTOzjLQ0XaLZraTZRCr7HtlXciZm5XNP38wsIy76ZmYZaWWO3EWSXpR0XNJRSY+n+HxJL0h6M93Oq1tni6QRSSckPVAX75F0JC17Is2Va2ZmJWmlp38F+HcR8feAFcBGSXcDm4EDEbEMOJAek5atBe4BVgNPpvl1AZ4CNlBMlr4sLTczs5K0MkfuWeBsun9J0nHgg0AN6EvNdgHDwKYUH4yIy8BJSSPAckmngLkR8RKApGeBh8hocvRmBxDNzMqiiGi9sbQY+DbF5OZvRURX3bKLETFP0g7g5Yh4LsV3UhT2U8C2iFiV4iuBTRGxpsF2NlB8I6C7u7tncHDwhl7cVI2NjdHZ2Tlrzz9yYeSG1+26rYvRd0ZnLpkZVOXcoPX8ls5fOvvJjDPbn7npqnJ+Vc4N2p9ff3//4YjoHR9v+ZRNSZ3AHwGfi4gfTTAc32hBTBC/PhgxAAwA9Pb2Rl9fX6tpTsvw8DCzua3tu7ff8Lq1jhpDl4dmMJuZU+XcoPX89vWVf8rmbH/mpqvK+VU5N6hufi2dvSPpPRQF/8sR8fUUPidpQVq+ADif4qeBRXWrLwTOpPjCBnEzMytJK2fvCNgJHI+IL9Yt2gusS/fXAUN18bWSOiQtoThgezAdG7gkaUV6zkfr1jEzsxK0MrzzMeDXgSOSXkuxfw9sA/ZIWg+8BTwMEBFHJe0BjlGc+bMxIq6m9R4DngHuoBjnz+YgrplZFbRy9s7/pvF4PMD9TdbZCmxtED9EcRDYrHJ8eQbLgX+Ra2aWERd9M7OMuOibmWXERd/MLCMu+mZmGXHRNzPLiIu+mVlGXPTNzDLiom9mlhEXfTOzjLR8aWWzXPnyDHYrcU/fzCwjLvpmZhlx0Tczy4jH9GeBJ0A3s6pyT9/MLCOtTJf4tKTzkl6vi82X9IKkN9PtvLplWySNSDoh6YG6eI+kI2nZE5pgZnUzM5sdrfT0nwFWj4ttBg5ExDLgQHqMpLuBtcA9aZ0nJc1J6zwFbKCYM3dZg+c0M7NZ1sp0id+WtHhcuAb0pfu7gGFgU4oPRsRl4KSkEWC5pFPA3Ih4CUDSs8BDeI5cu4n5/H27GSkiJm9UFP1vRMS96fFoRHTVLb8YEfMk7QBejojnUnwnRWE/BWyLiFUpvhLYFBFrmmxvA8W3Arq7u3sGBwdv+AVOxdjYGJ2dndN+npELIzOQzbt13dbF6DujM/68M6HKuUH5+S2dv7TltjP1mZstVc6vyrlB+/Pr7+8/HBG94+MzffZOo3H6mCDeUEQMAAMAvb290dfXNyPJTWZ4eJiZ2Nb23dunn8w4tY4aQ5eHZvx5Z0KVc4Py89vX13pPf6Y+c7OlyvlVOTeobn43evbOOUkLANLt+RQ/DSyqa7cQOJPiCxvEzcysRDda9PcC69L9dcBQXXytpA5JSygO2B6MiLPAJUkr0lk7j9atY2ZmJZl0eEfSboqDtndJOg18AdgG7JG0HngLeBggIo5K2gMcA64AGyPianqqxyjOBLqDYpzfB3HNzErWytk7jzRZdH+T9luBrQ3ih4B7p5SdmZnNKP8i18wsI772jtkM8/n7VmXu6ZuZZcQ9/Wnw1TTN7Gbjom9WkkadhFpHjb6fXtHEbPZ5eMfMLCMu+mZmGXHRNzPLiIu+mVlGfCDXrM18Xr+VyT19M7OMuKdvVlET/Q7E3wLsRrmnb2aWEff0W+Bf3lrV+DiA3Sj39M3MMuKevtktxN8AbDIu+mYZ8H8Gdk3pRV/SauBLwBzgDyJiW9k5NDNyYYTtu7e3Ow0zs1lTatGXNAf4PeCXgdPAdyTtjYhjZeZhZoXJTlKoddTe1RHyN4ObX9k9/eXASET8EEDSIFCjmEh9xk31rJtaR2020jC7ZbTrTDb/ZzNzFBHlbUz6F8DqiPjN9PjXgb8fEZ8Z124DsCE9/BBwoqQU7wL+sqRtTZVzu3FVzq/KuUG186tybtD+/H4xIj4wPlh2T18NYtf9rxMRA8DA7KfzbpIORURv2dtthXO7cVXOr8q5QbXzq3JuUN38yj5P/zSwqO7xQuBMyTmYmWWr7KL/HWCZpCWS3gusBfaWnIOZWbZKHd6JiCuSPgN8k+KUzacj4miZOUyi9CGlKXBuN67K+VU5N6h2flXODSqaX6kHcs3MrL187R0zs4y46JuZZSTboi/plKQjkl6TdCjF5kt6QdKb6XZeG/L6UMrp2t+PJH1O0u9I+vO6+CdLzOlpSeclvV4Xa7qvJG2RNCLphKQH2pDbf5H0hqTvS3peUleKL5b0V3X78PdnM7cJ8mv6XlZg3321Lq9Tkl5L8VL3naRFkl6UdFzSUUmPp3hVPnfN8qvMZ6+piMjyDzgF3DUu9p+Bzen+ZuB325zjHOAvgF8Efgf47Tbl8XHgo8Drk+0r4G7ge0AHsAT4ATCn5Nx+Bbg93f/dutwW17dr475r+F5WYd+NW74d+A/t2HfAAuCj6f77gf+T9k9VPnfN8qvMZ6/ZX7Y9/SZqwK50fxfwUPtSAeB+4AcR8aftTCIivg1cGBdutq9qwGBEXI6Ik8AIxeU3SsstIr4VEVfSw5cpfg/SFk32XTNt33fXSBLwaWD3bG1/IhFxNiJeTfcvAceBD1Kdz13D/Kr02Wsm56IfwLckHU6XfQDojoizULypwC+0LbvCWt79j+4z6Wvj0+0Yehqn2b76IPBnde1Op1i7/Etgf93jJZK+K+l/SVrZrqRo/F5Wad+tBM5FxJt1sbbsO0mLgY8Ar1DBz924/OpV8rOXc9H/WER8FPgEsFHSx9udUD0VP177FPCHKfQU8LeB+4CzFF+9q6ilS22UQdLngSvAl1PoLPC3IuIjwL8FviJpbhtSa/ZeVmbfAY/w7g5HW/adpE7gj4DPRcSPJmraIDbr+65ZfhX+7OVb9CPiTLo9DzxP8VXwnKQFAOn2fPsy5BPAqxFxDiAizkXE1Yh4B/jvzOJX1xY121eVuNSGpHXAGuBXIw2qpq/+b6f7hynGff9O2blN8F5WZd/dDvwz4KvXYu3Yd5LeQ1FQvxwRX0/hynzumuRX6c8eZFr0Jd0p6f3X7lMcfHmd4pIQ61KzdcBQezIExvW0rn3Qk39KkW87NdtXe4G1kjokLQGWAQfLTEzFRD2bgE9FxE/q4h9QMacDkn4p5fbDMnNL2272XrZ93yWrgDci4vS1QNn7Lh1T2Akcj4gv1i2qxOeuWX5V/+wBeZ69A/wSxZH+7wFHgc+n+M8DB4A30+38NuX3PuBt4G/Uxf4HcAT4PsUHfEGJ+eym+Hr61xQ9qvUT7Svg8xQ9mRPAJ9qQ2wjF+O5r6e/3U9t/nt7v7wGvAg+2ad81fS/bve9S/BngX49rW+q+A/4RxfDM9+vex09W6HPXLL/KfPaa/fkyDGZmGclyeMfMLFcu+mZmGXHRNzPLiIu+mVlGXPTNzDLiom9mlhEXfTOzjPx/2yg5e2rqCMAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the histogram of the data\n",
    "n, bins, patches = plt.hist(s_bsm, 50, facecolor='green', alpha=0.7)\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.109603503208742\n"
     ]
    }
   ],
   "source": [
    "### Price a call option\n",
    "\n",
    "strike = 102\n",
    "#cp_sign = 1\n",
    "price = np.mean(np.fmax(s_bsm - strike, 0))\n",
    "print( price )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.084494247829895"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Exact BSM price\n",
    "m_bs.price(strike, spot, texp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.898461998052294\n"
     ]
    }
   ],
   "source": [
    "### Can price different options without extra computation\n",
    "\n",
    "strike = 98\n",
    "cp_sign = -1\n",
    "price = np.mean(np.fmax(cp_sign*(s_bsm - strike), 0))\n",
    "print( price )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.925974960378305"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Exact price\n",
    "m_bs.price(strike, spot, texp, cp=-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bachelier Model (Arithmetic Brownian Motion)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma = 20\n",
    "texp = 1\n",
    "spot = 100\n",
    "m_norm = pf.Norm(sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(100000,)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_norm = spot + sigma * np.sqrt(texp) * z\n",
    "s_norm.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.0389173231624\n"
     ]
    }
   ],
   "source": [
    "strike = 102\n",
    "cp_sign = 1\n",
    "price = np.mean(np.fmax(s_norm - strike, 0))\n",
    "print( price )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.018706624094294"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Exact price\n",
    "m_norm.price(strike, spot, texp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.988602057400852\n"
     ]
    }
   ],
   "source": [
    "strike = 98\n",
    "cp_sign = -1\n",
    "price = np.mean(np.fmax(cp_sign*(s_norm - strike), 0))\n",
    "print( price )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.018706624094294"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Exact price\n",
    "m_norm.price(strike, spot, texp, cp=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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