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": [
       "(-0.0006055366781411985, 1.0025231767688696)"
      ]
     },
     "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": [
      "[ 95.58846095 162.00533109 114.39066961 ... 126.40456905 114.30328996\n",
      " 104.58264143]\n"
     ]
    }
   ],
   "source": [
    "# Black-Scholes-Merton model (Geometric BM)\n",
    "s_bsm = spot * np.exp(sigma*np.sqrt(texp)*z - sigma**2*texp/2)\n",
    "print(s_bsm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(99.99838825660521, 20.260426506988097)"
      ]
     },
     "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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+fUmzJb0MnAP2RcSLwGBEnAFIjx9Jk98M/Lg0+6kUuzkNT46bmVlF2jo5KyLeBm6XNAA8K2nZFJM366ePKeJXvoA0QtENxODgIGNjY+2k2VUTExM9We5kjb5Gy3EDswamHN9rdc2vvF3rsp1bqXt+4By7ocr8OjojNyLOSxqj6Is/K2lBRJxJXTfn0mSngEWl2RYCp1N8YZN4s+VsA7YBDA0NxfDwcCdpdsXY2Bi9WO5kW3dsbTmu0ddg18VdFWbTmbrmt2f4Ly+4Vpft3Erd8wPn2A1V5tfO0TsfTi18JM0BVgOvAbuB9Wmy9cDl/+7dwDpJfZIWU/xgeyB1AV2QtDIdtXN/aR4zM6tAOy39BcD2dATOLGBnRHxL0vPATkkPAK8D9wJExBFJO4FXgUvAhtQ9BPAg8BQwB9ib/swqVb6eUaOv8RffpnzJZcvBtEU/In4AfLJJ/A3gzhbzbAY2N4kfBKb6PcDMzGaQz8g1M8uIi76ZWUZc9M3MMuKib2aWERd9M7OMuOibmWXERd/MLCMu+mZmGXHRNzPLiIu+mVlGXPTNzDLiom9mlhEXfTOzjHR0ExWbOeXL/VpvtNoGvuSyXU/c0jczy4iLvplZRlz0zcwy4qJvZpaRdm6MvkjSc5KOSjoi6aEUny9pn6Tj6XFeaZ5NksYlHZN0Vym+XNLhNO6RdIN0MzOrSDst/UvAv4mIvwGsBDZIuhXYCOyPiKXA/vScNG4dcBuwBngs3VQd4HFgBFia/tZ08b2Ymdk0pi36EXEmIl5KwxeAo8DNQAPYnibbDqxNww1gNCIuRsQJYBxYIWkBMDcino+IAJ4uzWNmZhVQUX/bnFi6BfgOsAx4PSIGSuPeioh5kh4FXoiIZ1L8CWAvcBLYEhGrU/wO4OGIuLvJckYovhEwODi4fHR09Kre3LWYmJigv7+/suWNvzne8TwDswY4/8757ifTJXXPD9rLccn8JdUk00TV++HVcI7XbibyW7Vq1aGIGJocb/vkLEn9wO8BX4yIn07RHd9sREwRvzIYsQ3YBjA0NBTDw8Ptptk1Y2NjVLncrTu2djxPo6/Brou7ZiCb7qh7ftBejnuGe3dyVtX74dVwjteuyvzaOnpH0vsoCv7XIuKbKXw2ddmQHs+l+ClgUWn2hcDpFF/YJG5mZhVp5+gdAU8ARyPiK6VRu4H1aXg9sKsUXyepT9Jiih9sD0TEGeCCpJXpNe8vzWNmZhVop3vn08CvAoclvZxi/xbYAuyU9ADwOnAvQEQckbQTeJXiyJ8NEfF2mu9B4ClgDkU//97uvA0zM2vHtEU/Iv43zfvjAe5sMc9mYHOT+EGKH4HNzKwHfEaumVlGXPTNzDLi6+mbTcPX2bfriVv6ZmYZcdE3M8uIi76ZWUZc9M3MMuIfcs2ukn/gtfcit/TNzDLiln6FWrUMzcyq4pa+mVlGXPTNzDLiom9mlhEXfTOzjLjom5llxEXfzCwjLvpmZhlp5x65T0o6J+mVUmy+pH2SjqfHeaVxmySNSzom6a5SfLmkw2ncI+k+uWZmVqF2Ts56CngUeLoU2wjsj4gtkjam5w9LuhVYB9wGfBT4Q0kfS/fIfRwYAV4Afh9Yg++Ra9chX57B6mzaln5EfAd4c1K4AWxPw9uBtaX4aERcjIgTwDiwQtICYG5EPB8RQfEBshYzM6uUiho8zUTSLcC3ImJZen4+IgZK49+KiHmSHgVeiIhnUvwJitb8SWBLRKxO8TuAhyPi7hbLG6H4VsDg4ODy0dHRq36DV2tiYoL+/v6uvub4m+Ndfb2BWQOcf+d8V1+zm+qeH1Sb45L5SzqeZyb2w25zjtduJvJbtWrVoYgYmhzv9rV3mvXTxxTxpiJiG7ANYGhoKIaHh7uSXCfGxsbo9nK37tja1ddr9DXYdXFXV1+zm+qeH1Sb457hzrt3ZmI/7DbneO2qzO9qj945m7psSI/nUvwUsKg03ULgdIovbBI3M7MKXW3R3w2sT8PrgV2l+DpJfZIWA0uBAxFxBrggaWU6auf+0jxmZlaRabt3JO0AhoGbJJ0CvgxsAXZKegB4HbgXICKOSNoJvApcAjakI3cAHqQ4EmgORT+/j9yxrPioHquDaYt+RNzXYtSdLabfDGxuEj8ILOsou/coXzffzOrKZ+SamWXERd/MLCMu+mZmGXHRNzPLiIu+mVlGun1Grpl1yIdyWpXc0jczy4iLvplZRty9Y1ZT9+y4h0Zfo+mF+tz1Y1fLLX0zs4y4pX8NfLkFM3uvcUvfzCwjbumbvQf5ME+7Wm7pm5llxC19s+uIvwHYdNzSNzPLiFv6ZhnwNwC7zEW/DT40065X/jDIT+VFX9Ia4KvAbOC3I2JL1TmY2dT8YXD9qrToS5oN/FfgF4FTwHcl7Y6IV6vMo5Xyjt7q9HeznDX7MJjqf8UfEvVTdUt/BTAeET8CkDQKNIBaFH0z665udY22+vCY6vX9gdOcIqK6hUn/CFgTEf8sPf9V4G9GxBcmTTcCjKSnHweOVZbkX7oJ+JMeLLcTdc+x7vlB/XOse37gHLthJvL7+Yj48ORg1S19NYld8akTEduAbTOfTmuSDkbEUC9zmE7dc6x7flD/HOueHzjHbqgyv6qP0z8FLCo9XwicrjgHM7NsVV30vwsslbRY0vuBdcDuinMwM8tWpd07EXFJ0heA/0FxyOaTEXGkyhw60NPupTbVPce65wf1z7Hu+YFz7IbK8qv0h1wzM+stX3vHzCwjLvpmZhlx0QcknZR0WNLLkg6m2HxJ+yQdT4/zepjfx1Nul/9+KumLkn5D0k9K8c9VnNeTks5JeqUUa7neJG2SNC7pmKS7epTff5b0mqQfSHpW0kCK3yLpz0rr8rdmOr8pcmy5Xateh1Pk+Dul/E5KejnFK1+PkhZJek7SUUlHJD2U4rXYF6fIrzf7YkRk/wecBG6aFPtPwMY0vBH4zV7nmXKZDfwx8PPAbwC/3sNcPgN8CnhluvUG3Ap8H+gDFgM/BGb3IL9fAm5Iw79Zyu+W8nQ9XodNt2sv1mGrHCeN3wr8u16tR2AB8Kk0/CHg/6R1VYt9cYr8erIvuqXfWgPYnoa3A2t7l8q73An8MCL+qNeJRMR3gDcnhVuttwYwGhEXI+IEME5xWY5K84uIb0fEpfT0BYpzRXqmxTpspfJ1CFPnKEnAPwZ2zHQerUTEmYh4KQ1fAI4CN1OTfbFVfr3aF130CwF8W9KhdAkIgMGIOAPFRgM+0rPs3m0d7/4H+0L6evhkL7ugSlqtt5uBH5emO5VivfRPgb2l54slfU/S/5J0R6+SSppt1zquwzuAsxFxvBTr2XqUdAvwSeBFargvTsqvrLJ90UW/8OmI+BTwWWCDpM/0OqFmVJzQ9nngd1PoceCvAbcDZyi+ZtdVW5fgqIqkLwGXgK+l0Bngr0bEJ4F/DXxd0twepddqu9ZqHSb38e5GSM/Wo6R+4PeAL0bET6eatElsxtdjq/yq3hdd9IGIOJ0ezwHPUnzVOytpAUB6PNe7DP/CZ4GXIuIsQEScjYi3I+Id4L9RwVf9NrRab7W5BIek9cDdwC9H6kRNX/XfSMOHKPp5P9aL/KbYrrVZhwCSbgD+AfA7l2O9Wo+S3kdRUL8WEd9M4drsiy3y68m+mH3Rl3SjpA9dHqb4ceUVistDrE+TrQd29SbDd3lXq+ryDp38fYq8e63VetsNrJPUJ2kxsBQ4UHVyKm7i8zDw+Yj401L8wyru94CkX0j5/ajq/NLyW23XWqzDktXAaxFx6nKgF+sx/a7wBHA0Ir5SGlWLfbFVfj3bF2fqF+L3yh/wCxS/5H8fOAJ8KcV/DtgPHE+P83uc5weBN4C/Uor9d+Aw8AOKHXlBxTntoPgq+ucUracHplpvwJcoWi3HgM/2KL9xiv7cl9Pfb6Vp/2Ha/t8HXgLu6eE6bLldq16HrXJM8aeAfzFp2srXI/B3KLpnflDarp+ry744RX492Rd9GQYzs4xk371jZpYTF30zs4y46JuZZcRF38wsIy76ZmYZcdE3M8uIi76ZWUb+P6wKJ2NIRfYtAAAAAElFTkSuQmCC\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": [
      "4.310116158007301\n"
     ]
    }
   ],
   "source": [
    "### Price a call option\n",
    "\n",
    "strike = 110\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": [
       "4.292010941409885"
      ]
     },
     "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": [
      "6.94359770223014\n"
     ]
    }
   ],
   "source": [
    "### Can price different options without extra computation\n",
    "\n",
    "strike = 98\n",
    "cp = -1\n",
    "price = np.mean(np.fmax(cp*(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": [
      "7.029318044708391\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": [
      "7.039141746655447\n"
     ]
    }
   ],
   "source": [
    "strike = 98\n",
    "cp = -1\n",
    "price = np.mean(np.fmax(cp*(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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