{
 "metadata": {
  "name": "CEV1",
  "zanadu": {
   "authors": [
    "Antoine Jacquier"
   ],
   "category": "7",
   "clearance": "Private",
   "group_name": "Imperial College",
   "md5": "393ae999e2e18e7935ac1036e3fdb93f",
   "notebook_id": "2B65A42E-0A35-4BA6-91E3-D2D8BA06232B",
   "python_name": null,
   "reviewer_id": null,
   "status": "WIP"
  }
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "Pricing European Call options under the CEV model with $\\beta=1$\n======\n***"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from scipy.stats import norm\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import quad\nimport pandas as pd",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": "",
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "We consider here the Constant Elasticity of Volatility model $\\mathrm{d}S_t = S_t^{1+\\beta} \\mathrm{d}W_t$, with $S_0 = S_0>0$,\nand where $W$ is a one-dimensional standard Brownian motion.\nWe are interested here in the particular case where $\\beta=1$.\nIn this case, however, the stock price process is a strictly positive strict local martingale.\nWe wish to study how this martingale defect impacts the price of European Calls and Puts."
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {
      "zanadu": {}
     },
     "source": "European option pricing"
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {
      "zanadu": {}
     },
     "source": "Pricing with closed-form formulae"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "In the case $\\beta=1$ considered here, European Calls and Puts, defined as expected values of the final payoff (following Cox-Hobson's Theorem), admit closed-form representations:\n\\begin{align*}\n\\mathbb{E}(S_T - K)_+ & = S_0\\Big(\\mathcal{N}(\\kappa - \\delta) - \\mathcal{N}(-\\delta) + \\mathcal{N}(\\delta) - \\mathcal{N}(\\kappa + \\delta)\\Big)\n  - K\\left(\\mathcal{N}(\\kappa + \\delta) - \\mathcal{N}(\\delta - \\kappa) + \\frac{n(\\kappa + \\delta) - n(\\kappa - \\delta)}{\\delta}\\right),\\\\\n\\mathbb{E}(K-S_T)_+ & = S_0K\\sqrt{T}\\left(\\zeta_{+}\\mathcal{N}(\\zeta_+) + n(\\zeta_+) - \\zeta_{-}\\mathcal{N}(\\zeta_-) - n(\\zeta_-)\\right),\n\\end{align*}\nwhere \n$$\n\\delta := \\frac{1}{S_0\\sqrt{T}},\\qquad\n\\kappa := \\frac{1}{K\\sqrt{T}},\\qquad\n\\zeta_{\\pm} := \\frac{1}{\\sqrt{T}}\\left(\\pm\\frac{1}{S_0} - \\frac{1}{K}\\right).\n$$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def n(x):\n    return np.exp(-x*x/2.)/np.sqrt(2.*pi)\n\ndef cev2Call_closed(S, K, T):\n    delta = 1. / (S*np.sqrt(T))\n    kappa = 1. / (K*np.sqrt(T))\n    tPlus = n(kappa+delta)\n    NMinus = norm.cdf(kappa-delta)\n    NPlus = norm.cdf(kappa+delta)\n    temp1 = NMinus - norm.cdf(-delta) + norm.cdf(delta) - NPlus\n    temp2 = NPlus - (1.-NMinus) + (n(kappa+delta) - n(kappa-delta))/delta\n    return S*temp1 - K*temp2\n    \ndef cev2Put_closed(S, K, T):\n    dp = (1./S - 1./K) / np.sqrt(T)\n    dm = -(1./S + 1./K) / np.sqrt(T)\n    return S*K*np.sqrt(T)*(dp*norm.cdf(dp) + n(dp) - dm*norm.cdf(dm) - n(dm))",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {
      "zanadu": {}
     },
     "source": "Pricing by Put-Call parity"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def cev2Call_pc(S, K, T): ##By PC Parity\n    return S - K + cev2Put_closed(S, K, T)\n\ndef expectS(S, T): ##Computes the expectation of the stock price at time zero\n    return S*(1. - 2.*norm.cdf(-1/(S*np.sqrt(T))))",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {
      "zanadu": {}
     },
     "source": "Pricing by integration"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "In the case $\\beta=1$ (as well as in the other cases for that matter), the density of the CEV is known in closed form:\n$$\n\\mathbb{P}(S_T \\in \\mathrm{d}z) = \\frac{S_0}{z^3}\\left\\{\\exp\\left(-\\frac{\\left(z^{-1} - S_0^{-1}\\right)^2}{2T}\\right) - \\exp\\left(-\\frac{\\left(z^{-1} + S_0^{-1}\\right)^2}{2T}\\right)\\right\\}\\frac{\\mathrm{d}z}{\\sqrt{2\\pi T}},\n\\qquad \\text{for all }z>0.\n$$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def cev2_density(z, s, T):\n    temp1 = -(1./z - 1./s)*(1./z - 1./s)/(2.*T)\n    temp2 = -(1./z + 1./s)*(1./z + 1./s)/(2.*T)\n    return S0*(np.exp(temp1) - np.exp(temp2))/(z*z*z*np.sqrt(2.*pi*T))\n\ndef cev2Put_int(s, K, T):\n    def toIntegrate(z, s, K, T):\n        return (K-z)*cev2_density(z, s, T)\n    return quad(toIntegrate, 1.E-10, K, args=(s, K, T), epsabs=1.49e-15)[0]\n\ndef cev2Call_int(s, K, T):\n    def toIntegrate(z, s, K, T):\n        return (z-K)*cev2_density(z, s, T)\n    return quad(toIntegrate, K, 40.*s, args=(s, K, T), epsabs=1.49e-15)[0]",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {
      "zanadu": {}
     },
     "source": "Numerical tests"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "strikes = arange(0., 2., 0.1)\nS0, T = 1., 1.\nlowerBound = array([max(S0-K, 0.) for K in strikes])\nupperBound = array([S0 for K in strikes])",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "print \"Expectation S_T= \", expectS(S0, T)",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Expectation S_T=  0.682689492137\n"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Call option"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Vcev2Call_pc = [cev2Call_pc(S0, K, T) for K in strikes]\nVcev2Call_int = [cev2Call_int(S0, K, T) for K in strikes]\nVcev2Call_closed = [cev2Call_closed(S0, K, T) for K in strikes]\n\nplt.figure(figsize=(8,4))\nplt.plot(strikes, lowerBound, 'r', linewidth=2)\nplt.plot(strikes, upperBound, 'r', linewidth=2)\nplt.plot(strikes, Vcev2Call_pc, 'ys', label='Parity')\nplt.plot(strikes, Vcev2Call_int, 'g-', label='Integration', linewidth=2)\nplt.plot(strikes, Vcev2Call_closed, 'bo', label='Closed-form')\nplt.fill_between(strikes, lowerBound, upperBound, where=upperBound>=lowerBound, facecolor='grey', interpolate=True)\n#lt.figtext(.95, .9, \"The shaded area corresponds to the no-arbitrage region, \\\n#nwhile the red plot is its boundary $(S_0 - K)_+$\", size=15)\nplt.title(\"Call option prices in the CEV model with \" r\"$\\beta=1$\")\nplt.legend()\nplt.xlabel(u'strike', fontsize=12)\nprint 'The shaded area corresponds to the no-arbitrage region, while the red plot is its boundary $(S_0 - K)_+$'\nplt.show()",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "The shaded area corresponds to the no-arbitrage region, while the red plot is its boundary $(S_0 - K)_+$\n"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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AgABYWlpi6NChOH/+fI1rqA8Gc10xnImoAVhaytVOl8kUeluHg4MDcnJyUFZW\nVu2yWVlZKtdIlkgk8PT0RGZmJvz8/LBs2TJER0fD2dkZI0eORHZ2NgAgNTUVQ4cOhb29Pezt7dGx\nY0dIpVLcunUL2dnZKqNMWltbw8HBQfnYxsYGtra2aN68udqWdFZWlsp1HgDA29sbWVlZKjVW5uzs\nrLxvZWWl8lgmkylb5g2NwVwfDGci0rGoqP7w9Z2jMs3XdzamTu2nt3WEhITA0tISu/432pJG7u7u\nSEtLUz4WQiA9PR3u7u4AgJEjR+LkyZNIS0uDRCLBzJkzAQBeXl6Ii4tDXl6e8l9hYSHc3Nzg6uqq\nMhx0YWEh7ty5o3xcUFCA/Px83L9/v8ow0QDg5uaG9PR0lWPhaWlpyppqylDnmLNXdn2xtzYR6VB5\nz+kVK+aiqMgcMpkCU6eG16pHdX3XYWdnh48//hjvvvsupFIp+vXrBwsLCxw+fBjx8fEqvbJHjBiB\nRYsW4ejRo+jZsye++OILyGQyPPfcc0hOTkZGRga6d+8OS0tLyGQyZdi9/fbbmD17Nr755ht4eXnh\n77//xunTpxEREYHhw4cjODgYp06dQteuXfHhhx/WqPVe7tlnn4W1tTWWLFmC6dOn49SpU9i3b5/y\neg3GPqgLg1kXGM5EpEODBvWq9elRul7H9OnT4eLigvnz5+O1116Dra0tgoKCMGfOHBw8eFB5TnH7\n9u2xefNmTJ06FZmZmQgMDMTevXshlUpRXFyMWbNm4dKlS7CwsED37t2xdu1aAMB7770HIQT69++P\nrKwstGrVCpGRkYiIiEDHjh3x5ZdfYtSoUXjw4AGmT5+udtdzRRXPcbawsMDevXvxzjvvYOHChfDw\n8MCmTZvQrl075bI1ORe74jI1fY4ucEhOXeLwnURUCxySs3HgkJzGjMeciYionhjMusZwJiKiemAw\nNwSGMxER1RE7fzWUSh3Czh/dgowcJzysNMZsUVELhIRUP7oOERE1DQzmhlQhnEvOxGDS7L+rLLJs\nmf7LIiIi48Vd2Q3tcTiXubkauhIiIjIBbDHrg0QCs7ZtAWQbuhIiMjKGuLYxGTe9BnNY2Ad1uq5o\n48AvHxGpKh+Jigzn5MmT6NKlC5YsWWLoUpT0GsyHDs1HSsqj8VubZjhXZVVYaOgSiIjIiOh9V/aj\na4LObXLBLJP54H/jwQuUXbkCs6xseNz8G20dkzhCGBERATDQMebaXFe0sVi8eIPqhIrDd3JsbSIi\neswgvbJ+uOldAAAgAElEQVRrc13RRouDkBARkRr6D2b7V9AmrETvmzVKDGciIqpEr8Hc4dkxwIBY\nrL6/BKvOrdLnpo0Xw5mIiCrQazBfOr0Ry6MWAADePfAuw7kcw5mIiB7T+67sqcFTsTx8OQCGswqG\nMxERwUCdvxjOGjCciYiaPIONlc1w1oDhTETUpBn0IhYMZw0YzkRETZbBry7FcNaA4UxE1CQZPJgB\nhrNGDGcioibHKIIZYDhrxHAmImpSjCaYAYazRgxnIqImwyAXsdBmavBUAEBUXBTePfAuAOCdru8Y\nsiTjUB7OAGQxMRj++MIXG3MvQia7W2XxoqIWCAkZqv86iYioXqoN5ri4OEybNg0KhQJvvvkmZs6c\nWWWZ+Ph4/OMf/0BpaSkcHR0RHx9fr6IYzhqoCee9Lzhi0rS/qyy6bJn+yyMiovrTGswKhQJTpkzB\n4cOH4e7ujq5duyIiIgL+/v7KZe7evYt3330XBw8ehIeHB3JycnRSGMNZg0rh7HS7aigTEZHp0nqM\n+ezZs/Dz84OPjw8sLCwQGRmJ3bt3qyzz3Xff4eWXX4aHhwcAwNHRUWfF8ZizBhWOOUsMXQsREemU\n1mDOzMyEp6en8rGHhwcyMzNVlrly5Qpyc3PRp08fBAUFYdOmTTotkOGsweNwLnNzNXQlRESkQ1p3\nZUsk1bfHSktL8dtvv+HIkSMoLCxESEgInn32WbRt21ZnRXK3tgYSCczatgWQbehKiIhIR7QGs7u7\nO9LT05WP09PTlbusy3l6esLR0RFWVlawsrJCr1698Pvvv6sN5ujoaOX90NBQhIaG1rhQhrN6Mllr\n7NoFlF25ArOsbAgAf7dyQpFZC0OXRkTUpMTHx9e78zMASIQQQtNMuVyO9u3b48iRI3Bzc0O3bt2w\nZcsWlc5fly9fxpQpU3Dw4EEUFxcjODgY27ZtQ8eOHVU3JJFAy6ZqbEXCCkTFRQHJNvBPGYpWsjaw\ntJQjKqo/Bg3qVe/1mywhUDR5MmQxMSiRSLAjMhJX2rc3dFVEREbt5MmT6NKlC5YsWaLzddc197S2\nmKVSKVauXImwsDAoFApMmDAB/v7+iImJAQBMmjQJHTp0QHh4ODp37gwzMzO89dZbVUJZl6YGT8Wf\nP6VhbWw6LuVtxKXH01NS5gBA0w1nDec5M5yJiEyL1hazTjekoxYzAISFfYBDh+armT4XcXGf6GQb\nJostZyKiGjPGFrNRDclZU8XF6hv6RUXmeq7ECHH4TiIik2aSwWxpKVc7/XbxdT1XYqQYzkREJssk\ngzkqqj98feeoTrR/BZfa7OR5zuUYzkREJsnoLmJRE+UdvFasmIuiInPIZAp49/PC2oICnkpVETuE\nERGZHJMMZuBROFfugd0pwZvnOVfGcCYiMikmuStbEw7fqQF3axMRmQyTbTFrwhHCNKhDy/n06V28\n1jMRkZ41umAGGM4a1TKcZbK7mDYtrcp0XuuZiKjhNKpd2RVxt7YG3K1NRGTUGm0wAwxnjRjORERG\nq1EHM8Bw1ojhTERklBp9MAMMZ40YzkRERqdRdv5Shx3CNNDSIayoqIXajl5FRbzWMxFRQ2kywQww\nnDXSEM7gKVFERHrXJHZlV8Td2hpwtzYRkVFocsEMMJw1YjgTERlckwxmgOGsEcOZiMigmtQx5sp4\nzFmDBr7wBYf6JCLSrEkHM8Bw1qgBw5lDfRIRadbkgxmoGs5//pSGlDgLFBdLYWkpR1RU/yqXmGwS\neMlIIiK9YzA/pgzn5bOxOjYVyNumnJeSMgcAGM4MZyKiBtdkO3+pMzV4KjqkDFEJZQBISfkUK1b8\naKCqjAA7hBER6Q2DuRJnma/a6UVF5nquxMgwnImI9IK7siuxtJSrnS6TKfRciRHS0W5tDvVJRKQZ\ng7mSqKj+SEmZg5SUT/830f4VeLzgbriijIkOwpmnRBERacZgrqS8g9eKFXNRVGSOv0tScbF1LL5+\nUIAOP7tjxnMzDFyhEWCHMCKiBsNgVmPQoF4qPbD/+2t3TNw3Ef/68V9QlCkws8dMA1ZnJIwwnDlw\nCRE1BgzmGnjrmbdgJjHDW3vfwvtH3keZKMOsnrMMXZbhGVk4c+ASImoM2Cu7hiZ0mYCvI76GBBLM\nPjob80/MN3RJxoG9tYmIdIrBXAvjA8dj/UvrIYEEc4/NxcfHPzZ0ScaB4UxEpDMM5loa+/RYbBy6\nEWYSM3wU/xGi46MhhDB0WYbHcCYi0gkGcx2M7jwam4ZugpnEDPOOz8NH8R8xnAGGMxGRDrDzVx2N\nemoUzCRmGL1zND458QkUZQrM7zsfEonE0KUZlgE7hDXkwCXs8U1E+sJgrofITpEwk5hh1P+NwoKf\nFqBMlGHB8wsYzgYK54YMSPb4JiJ94a7senrlyVewdfhWSM2kWHRqEWYensnd2gB3axMR1RGDWQeG\ndxyO7cO3Q2omxWc/f4YZh2YwnAGGMxFRHVQbzHFxcejQoQPatm2LxYsXa1zu3LlzkEql2Llzp04L\nNBVD/Ydix4gdsDCzwNIzSzH94HSGM8Bwrsbp07tw/vz6Kv9On95l6NKIyEC0HmNWKBSYMmUKDh8+\nDHd3d3Tt2hURERHw9/evstzMmTMRHh7epMPopQ4v4f9e+T+8vP1lLEtYhjJRhmXhy3jM2chGCDMm\nPHZNRJVpDeazZ8/Cz88PPj4+AIDIyEjs3r27SjCvWLECw4cPx7lz5xqsUFMxuP1g7Hx1J17e/jKW\nn10OhVBgxYAVDGcTD2deqpKI9EVrMGdmZsLT01P52MPDAwkJCVWW2b17N44ePYpz584xgAC82O5F\n7Hp1F4ZtG4Yvz32JMlGGlQNXwkzSxA/pm3A485QoItIXrcFck5CdNm0aFi1aBIlEAiFEk96VXdHA\ntgPxQ+QPGLJ1CFb/shplogyrBq1iOJtwOJsSnndNZLq0BrO7uzvS09OVj9PT0+Hh4aGyzK+//orI\nyEgAQE5ODmJjY2FhYYGIiIgq64uOjlbeDw0NRWhoaD1KN37hfuHYM3IPXtr6EmJ+jcH1hBzgbHsU\nF1vA0lKOqKj+KpeXbDIYzg2Ox66J9C8+Ph7x8fH1Xo/WYA4KCsKVK1eQmpoKNzc3bNu2DVu2bFFZ\n5tq1a8r748ePx+DBg9WGMqAazE1Ff9/+2DtyLwZFv4pD+8yBvE+V81JS5gAAw7kJh7OpHbtmS5xI\ns8oNznnz5tVpPVqDWSqVYuXKlQgLC4NCocCECRPg7++PmJgYAMCkSZPqtNGm5oU2L6Bz6iv4JW+1\nyvSUlE+xYsXcphnMAMMZpnfsmi1xooZX7ZCcAwYMwIABA1SmaQrk9evX66aqRugJM2e104uKzPVc\niZFhONNjbI0TPcKxsvXE0lKudrqZRaGeKzFCDGcCW+NE5RjMehIV1R8pKXOQkvK/Y8ywfwWX2lxA\nSu4k+Lb0NVxxxoDhrFOmduy6IbElTqaGwawn5ceRV6yYi6Iic0gsCnGjfTKuOV1ByNchOPDaAQS5\nBRm4SgNjOOsMA+d/GqolzsCnhsJg1qNBg3qpdPTKL56L4d8Px6GUQwjdEIodr+xAuF+4ASs0Agxn\no8aW+P805K53hn7TxmA2IFtLW+wduRdv7nkTmxI3YfCWwfhq8FcY+/RYQ5dmWAxno8VQ0A+28ps2\nBrOBNTNvhm+GfAN3W3csOrUI43aPQ2Z+Jmb1mNW0hzdlODc5bI03PAa+aWAwGwGJRIKFLyyEe3N3\nRMVGYc7ROci8n4nlA5bD3KwJn07FcG5S+AvcdJnibv3y9SoU93DpUhLee+8sAEAm88HixRvqvF5d\nYDAbkSndpsDFxgWjd47Gql9WIbsgG98O+xZWFlaGLs1wGM5UT2yJm7aGCn3V9d4FkAEA2GUEl0Jn\nMBuZ4R2Ho9UTrfDS1pew6/Iu9NvUD3tG7kFLq5aGLs1wGM5UDw3VEmfgU0NhMBuhXt698NP4nxD+\nbThOpZ9Cj3U9EDc6Dl52XoYuzXAYzmRkGnLXO0O/aWMwG6knWz2J0xNOY8C3A/Dn7T8R8nUIYl+L\nRWfnzoYuzXAYztREsJXftDGYjZhHcw+cHH8SL219CSfSTqDn+p744dUf0Kd1H0OXZjgMZ6I6Y+Cb\nBokQQuhlQxIJ9LSpRqdIXoQxu8bg+4vfo5l5M2wcshGvdnrV0GUZlhAomjwZspgYlEgkDGeiRqqh\ne2Xfu3cPNjY2aNPm0bDIuuyVXdfcYzCbiDJRhn/E/QPLzy4HACztvxT/CPmHgasyMIYzEdXTyZMn\n0aVLFyxZskTn665r7pnpvBJqEGYSMywLX4YlLzz64Zl+aDpmHJqBMlFm4MoMqHy39qRJaCYEhm/d\nirZJSYauioioXhjMJkQikeBf3f+FTUM3QWomxeenP8fonaNRLC82dGmGw3AmokaGnb9M0OjOo+H8\nhDOGbR+GLX9uwV+nbsDxj+5QlFrB0lKOqKj+KhfLaPTYIYyIGhEGs4nq59sPJ8adQJ85EUj8wR3I\nW6ycl5IyBwAYzgxnIjJB3JVtwgJdA9E59RUgb5vK9JSUT7FixY8GqsqAuFubiBoBBrOJMyuzVTu9\n8GETvTIVw5mITByD2cRZWsrVTk+88zOu3Lmi52qMBMOZiEwYg9nERUX1h6/vHJVp5g6jcC8gAU/H\nPI21v65tmuePM5yJyESx85eJK+/gtWLFXBQVmUMmU2D8pNHYIyT47o/vMGnfJOxL3oevIr5Cqyda\nGbhaPWOHMCIyQRz5qxH77o/v8M7+d3Cv+B5aPdEK6yLWYVC7QYYuS/84QhgRacCRv0ivRj01ComT\nExHqE4rbD27jxS0v4p3976CwtNDQpekXd2sTkQlhMDdyXnZeODLmCD7r9xkszCyw+pfVCIwJxC9Z\nvxi6NP1iOBORiWAwNwFmEjPMeG4Gzr11Dk86PYnkO8kI+ToEn574FIoyhaHL0x+GMxGZAAZzExLg\nEoBzb53De8HvQV4mxwfHPkDvDb1xPe+6oUvTH4YzERk5BnMTY2VhhWXhy3Bo9CG42brhVPopBKwJ\nwDcXvmk6nfMYzkRkxBjMTVQ/335IfDsRwzsOR35JPsbtHodXdryCO4V3DF2afjCcichIMZibMAdr\nB2wfvh3fDPkGts1ssePiDjy1+ikcSjlk6NL0g+FMREaIwdzESSQSjAkYg9/f/h3dPbsjuyAbYZvD\nMC1uGh6WPjR0eQ2P4UxERobBTACA1vatcXzccXza91NIzaT4IuELdP1vV6zYtAFhYR8gNDQaYWEf\nYP/+E4YuVfcYzkRkRDgkJymZm5ljds/Z6O/bH6N3jsZfp9IQFRurclnJRnutZw7fSURGgi1mqiLI\nLQi/TfoNnpf6Na1rPbPlTERGgMFMallbWKNN885q52Xm3m28p1YxnInIwGoUzHFxcejQoQPatm2L\nxYsXV5n/7bffIiAgAJ07d0b37t2RmJio80JJ/zRd6/nP3DPovq47jl4/queK9IThTEQGVG0wKxQK\nTJkyBXFxcbh48SK2bNmCS5cuqSzTpk0bnDhxAomJiZg7dy4mTpzYYAWT/qi71rOD+zuw7ZmF0xmn\n8fzG59Hnmz44deOUgSpsQAxnIjKQaoP57Nmz8PPzg4+PDywsLBAZGYndu3erLBMSEgI7OzsAQHBw\nMDIyMhqmWtKrQYN64YsvwhAWNhe9e0cjLGwuvomJRFZMEj7t+ylayFogPjUePdb3wIBvBzS+C2Mw\nnInIAKoN5szMTHh6eiofe3h4IDMzU+PyX3/9NQYOHKib6sjgBg3qhbi4TxAfH424uE8waFAv2DSz\nweyes3H9vev4sNeHsGlmg7ircej6364Yum0o/rj1h6HL1h014ezHcCaiBlTt6VISiaTGKzt27BjW\nrVuHU6fU79qMjo5W3g8NDUVoaGiN103Gp4WsBeb1mYepwVPx2anPsOLsCvxw+Qfsvrwbr3Z6FdG9\no9HesRGcblTpVKoRW7fi+8hIXOWpVERUQXx8POLj4+u9HomopnvtmTNnEB0djbi4OADAwoULYWZm\nhpkzZ6osl5iYiGHDhiEuLg5+fn5VNySRNN6evAQAuFlwEwtPLsSaX9egRFECM4kZxgSMwYe9PkRr\n+9aGLq/+hEDR229DtnYtSiQShjNRI3Dy5El06dIFS5Ys0fm665p71e7KDgoKwpUrV5CamoqSkhJs\n27YNERERKsvcuHEDw4YNw+bNm9WGMjUNLjYu+GLAF7g69SomdpkIM4kZNlzYgHYr22HyvsnIuG/i\nfQ8kEsjWrEHRxIloJgRGcLc2ETWAaoNZKpVi5cqVCAsLQ8eOHfHqq6/C398fMTExiImJAQB8/PHH\nyMvLw+TJkxEYGIhu3bo1eOFkvDztPBEzOAZJU5IwNmAsykQZ1vy6Bn7L/TAtbhpuFdzC/v0nTHOo\nT4YzETWwandl62xD3JXdZF3OuYzo+Ghs++vRKGLNUhxgfXg47mavUS7j6zsHX3wRZjpDfXK3NlGj\nYJK7sonqq4NjB2wdvhW/v/07Xmr/Ekp+bq0SyoAJDvXJljMRNRAGM+lNZ+fO+CHyBwQ69VQ7P+X2\nTdwpvKPnquqB4UxEDYDBTHrn1Nxa7fSr9y/A9XNXvLz9ZexJ2oNSRameK6sDhjMR6RiDmfRO3VCf\nrTynosuQJ6AQCuy8tBMvbX0J7kvdMS1uGi7cvGCgSmuI4UxEOsTOX2QQ+/efwIoVP6KoyBwymQJT\np/bDoEG9kJWfhW8Tv8WG3zfg4t8Xlct3du6MsQFj8dpTr8HZxtmAlWvBDmFEJscYO38xmMkoCSHw\na/av+ObCN/juz++Q+zAXAGAuMUe4XzjGBozF4PaDIZPKDFxpJQxnIpPCYGYwUx2UKEqwP3k/Nvy+\nAQeuHIC87NHlKO1l9ojsFImxAWPRzb1brYaPbVAMZyKTwWBmMFM93X5wG1v+2IJvfv8G52+eV05v\n79AeYwPGwjW7HbZ8fR7FxVJYWsoRFdXfMOdGM5yJTAKDmcFMOpR4KxHfXPgG3/7xLW49uAUk2wCx\nA4G8bcplDDpwCcOZyOgZYzCzVzaZrM7OnfF52OfImJ6BfSP3wfnP3iqhDDwauGTKRytwKOUQHpY+\n1G+B7K1NRHXAYCaTJzWTYlC7QehgH6R2fmrO3wjbHAb7xfbot6kflpxagvPZ51Emyhq+OIYzEdUS\ng5kaDUtLudrprZ1aoYtrFxQrinH42mHMPDwTXdZ2gcu/XTDq/0Zh/fn1DXvlK4YzEdUCjzFTo7F/\n/wm8995BpKR8qpzm6zsbX3wRjkGDeuHvB3/jyPUj+DHlR/x47Uek309Xeb6/oz/6temHfr79EOoT\nCptmNrotkMeciYyOMR5jZjBTo6Jp4JLKhBBIupOkDOljqcdQUFKgnG9hZoEQzxD0a9MP/X374xnX\nZxAXewrLlx+qX49vhjORUWEwM5jJSJUqSnEm4wx+vPYjDqUcwrmscyrHoJ+47gLEDcCDW+uU0+rc\n45vhTGQ0GMwMZjIReQ/zcCz1GA6lHMKP137EteUtgZRzVZbz7PIy5n31Ip5xewYdnTpCaiat2QYY\nzkRGwRiDuYa/RYiaFnsrewzzH4Zh/sMAAMHbZuJsStXl0u/cwRt73gAAWEmtEOASgGdcn0GQWxCe\ncX0G/k7+6sO6vEMYANnatRixdSvDmYgAMJiJaqSFjYXa6e2c3RHQcQR+zf4V1/Ku4UzGGZzJOKOc\nbyW1wtMuT/8vrN2eQQfHDo/C+nE470rLxZqDqXi4JQt3XfLxVF83tGvnpq+XRkRGhsFMVANRUf2R\nkjKnSo/vpR9OUh5jzn2Yi9+yf8MvWb/g1+xf8WvWr7h+9zpOZ5zG6YzTyudZSa0Q6BqIZ1yfgfSq\nPXYk+SAd3z+aeRO4tWcCEJHFcCZqoniMmaiGatrju6I7hXdUwzr7V6TeTf3fApuC1B679vLoh4Fv\n+qElWqIZmun4lRBROR5jJjJhgwb1qnUPbAdrB/TzfXRudLmcwhxlWC/d9hvuqHneDfNSrMEaAEBz\nNIcjHOEAB5X/m6M5zLSMEZScnIWEhHuQy2WQSosQHGzHVjiRCWAwE+mZo7Uj+vv2R3/f/jju+QEO\nXaq6jFPxA9jnACkOEtyX3Md93Mc1XFNZRgopWqJlldB2gAPSk/MQG2uGvLytyuVzcycD4C5yImPH\nYCYyIPXHrmfhMz8fDF35Cx6YC6x+bRB+a9Mcd3AHOchR/v8AD3D78a0ys4RglOWdUZmWl7capxJG\nwK+di9aWNhEZFoOZyIDKd42vWDG3wrHrARg0sCeK3n4bT6xdi6hNBx6fSvWcynOLUFQlrO88vsnl\nMrXbS5P/jU/wCZqjOey03GRQ/3yAu8iJGhqDmcjANB27ru48ZxlkcH98q6gMZdgg/Qs31GzLTFqE\nMgjce3zTxBKWagP7frLAuVhH3OMucqIGw2AmMlZ1HITEDGboEeyA2NzJyMtbrZxub/82BgQHoDVe\nwP3Ht3sabsUoVr+bPCEIyNupMikvbzX2JjyPgHYXYfP49gSeUN6XQQYJJDV6yWyNEzGYiYxbHcP5\nUZhlISFhJORyS0ilxQgObq4MufJOYuoICDzEQ7WBfUXeEiVqnpMvV+An/KR2feYwVwnqivcrPr6Z\nXIAjsTJ2WKMmj8FMZOzqEc51CTQJJLB+fHOFq8q8TdJLUDMyKZykT6AT+qDg8e0BHijvl6BE2ULX\nKiEIyFM9pzsvbzX2JbyAwHZJsHp8s4a1yn1LWFbbmY0tcTIlDGYiU2AkY2sHB9shV80u8n7BHmgH\n9UFXghKVoK54v+LjPLn661/fl8txHMc11iSBBDLIqgR2+f38ZOBSbGs8qNASv5M7CQpkwL+dRx3f\niUcY+NQQGMxEpsIIwrm6XeTqNHt8s4e91nVrao07SK3QCb1RiEI8fHyreL8Yxcr7aiUEAXlfq0y6\nmxeDbQldIW23ATLIYAlLlf/VTav8f0byPRyLtWqwXe8M/aaLwUxkSowknBsiIDS1xsOCvTS2xgFA\nAYUymCuHdiEKcUHeAgXqnih/AnLIlS33WtOw631nQk94tjum/IOk4s0SltVOl0KKK8nZDTZADAPf\n+DGYiUyNEYRzQ6hLaxx41LmsvCOZOtnSS2pj11fqilcxG0UoQjGKUfT4Vn6/uv/vyW2hbhTkIrk5\nruBKrV9/OQkkQEJXiLwElel5eauxOyEUvu3OwELDrRmaaZxnAQukJefix1gLkwv8pvbHBIOZyBQ1\n4nDW9S9cTS3x4ODmylZqXWja9e4mbYHeGIkSNbdiFFc7TQEFILdSu80HciARiXWqF4DGVv73Cd3R\nqt1+SCvcLGBR48e3kh/g11gXFFQI/JzciShECtq1c4U5zCGFFGYwq/Gpc+WSk7MadO/B+fNSJCVl\n4/ffP0BUVP9aj4ffEBjMRKaqkYazrtW1JV4dTYEfGuysddd7deSQY7M0Calq5jlLbRGCISitwa0E\nJVWmPdTQwa5UboFMZNa55keBv09l0r28tfghoSvQbpNymgQSZUiX/6/pfvn/aQlOyM87qrLuvLzV\nOJgQjjvt0mBeh5sUUqQm38HRWBnu5sUhNxfIyABSUuYAgMHDmcFMZMoYzjXSEC3xhgp8KaR4Ltge\n99SE/vPBbvUKfU2tfE9pS4ThTZSiFPIKN22PK95PlTuiSM16zeTN0QwyKKCAHHIICOVzakzeW+3k\nO/IiHNTSW79aavYepKR8ihUr5hp/MMfFxWHatGlQKBR48803MXPmzCrLREVFITY2FtbW1tiwYQMC\nAwMbpFgiUoPhbDAN1RFO3638nsFO8GiAwG8tbYXX8b7yseLxTQ658v/q7p+UPsTfatbdQmqBDnhW\nuc6K663JrUBTH4Ei8zq/D7qiNZgVCgWmTJmCw4cPw93dHV27dkVERAT8/f2Vyxw4cABXr17FlStX\nkJCQgMmTJ+PMmTNa1kqmKD4+HqGhoYYugzTREs7Xr19H69atDV0h1VJ56D/6/DrobJ36DPzg4OYq\ny5XvSq7NcX1ZcJaG4WXbNMjeA5lMUed16orWYD579iz8/Pzg4+MDAIiMjMTu3btVgnnPnj0YO3Ys\nACA4OBh3797FrVu34Ozs3HBVk94xmE2AhnA+lpXFYDZhqampOv38TGm3fkOuW90fE76+szF1ang9\nK64/rcGcmZkJT09P5WMPDw8kJCRUu0xGRgaDmcgQ1ITznvbt4ZqVZejKqI5s8/NN4vNztQF6P1/e\nQrZ89J+O6m6IdbvaAPbdcrDn9AtoJnOAX2BbTJ0abvDjy0A1wSyR1KxbuxCqe+pr+jwiagDl4SyR\nQBYTA//LlzHx8mVDV0V1lAVg4m+/GbqMRmslANwH8MNDQKb5OuT6pDWY3d3dkZ6ernycnp4ODw8P\nrctkZGTA3V31+rDlGNimbd68eYYugeqIn5xp4+enB1bqzx03BK3BHBQUhCtXriA1NRVubm7Ytm0b\ntmzZorJMREQEVq5cicjISJw5cwYtWrRQuxu7cquaiIiIqtIazFKpFCtXrkRYWBgUCgUmTJgAf39/\nxMTEAAAmTZqEgQMH4sCBA/Dz88MTTzyB9evX66VwIiKixkgi2JQlIiIyGtqvLl4HcXFx6NChA9q2\nbYvFixerXSYqKgpt27ZFQEAAzp8/r+sSqI6q++zi4+NhZ2eHwMBABAYGYv78+QaoktR544034Ozs\njKeeekrjMvzeGa/qPj9+94xXeno6+vTpgyeffBKdOnXC8uXL1S5Xq++f0CG5XC58fX3F9evXRUlJ\niQgICBAXL15UWWb//v1iwIABQgghzpw5I4KDg3VZAtVRTT67Y8eOicGDBxuoQtLmxIkT4rfffhOd\nOnVSO5/fO+NW3efH757xys7OFufPnxdCCJGfny/atWtX79zTaYu54oAkFhYWygFJKtI0IAkZVk0+\nO4Cd+IxVz549YW9vr3E+v3fGrbrPD+B3z1i5uLjg6aefBgDY2NjA398fWZXOsa7t90+nwaxusJHM\nzG9G5nAAAASKSURBVMxql8nIyNBlGVQHNfnsJBIJfv75ZwQEBGDgwIG4ePGivsukOuL3zrTxu2ca\nUlNTcf78eQQHB6tMr+33T6dXl+KAJKarJp9Bly5dkJ6eDmtra8TGxmLIkCFITk7WQ3WkC/zemS5+\n94xfQUEBhg8fji+++AI2NlUvr1mb759OW8y6HpCE9Kcmn52trS2sra0BAAMGDEBpaSlyc3P1WifV\nDb93po3fPeNWWlqKl19+GaNHj8aQIUOqzK/t90+nwVxxQJKSkhJs27YNERERKstERERg48aNAKB1\nQBLSr5p8drdu3VL+1Xf27FkIIdCyZUtDlEu1xO+daeN3z3gJITBhwgR07NgR06ZNU7tMbb9/Ot2V\nzQFJTFdNPrsdO3Zg9erVkEqlsLa2xtatWw1cNZUbOXIkjh8/jpycHHh6emLevHkoLS0FwO+dKaju\n8+N3z3idOnUKmzdvRufOnREYGAgAWLBgAW7cuAGgbt8/DjBCRERkRHQ+wAgRERHVHYOZiIjIiDCY\niYiIjAiDmYiIyIgwmImIiIwIg5mIiMiIMJiJGrmBAwdi06ZNAIANGzagZ8+eBq6IiLTR6QAjRKQ/\n0dHRSElJUYauJgcOHNBTRUSkC2wxEzVSQgheKpDIBDGYiUzA4sWL4eHhgebNm6NDhw44cOAAFi5c\niG3btsHW1lY5FGBoaCg++OADdO/eHTY2Nrh27RpCQ0Px9ddfq13vv/71L/Ts2RP5+fm4d+8eJkyY\nADc3N3h4eGDu3LkoKyvT58skInBXNpHRS0pKwpdffolffvkFLi4uuHHjBuRyOWbPno2UlBTl4Pjl\nNm/ejNjYWLRv3x5lZWWQSCRVLjEnhMDEiRORkZGBH3/8ETKZDEOHDoWLiwtSUlJQUFCAF198EZ6e\nnpg4caI+Xy5Rk8dgJjJy5ubmKC4uxl9//QUHBwd4eXkBUL+rWiKRYNy4cfD39wcAmJlV3SlWWlqK\nyMhIlJWVYe/evZBKpbh16xZiY2Nx9+5dyGQyWFlZYdq0afjvf//LYCbSMwYzkZHz8/PDsmXLEB0d\njb/++gthYWFYunSpxuU9PT21ru/q1atITExEQkICpNJHvwLS0tJQWloKV1dX5XJlZWXKPwKISH94\njJnIBIwcORInT55EWloaJBIJZs6cqbY1DKDKbuvK/P39sW7dOgwYMADJyckAHoW5paUl7ty5g7y8\nPOTl5eHevXv4448/dP5aiEg7BjORkUtOTsbRo0dRXFwMS0tLyGQymJubw9nZGampqVV2Z9ekJ3Zk\nZCQWLFiAF154AdeuXYOrqyv69++P6dOnIz8/H2VlZUhJScGJEyca6mURkQYMZiIjV1xcjFmzZsHJ\nyQmurq7IycnBwoULMWLECACAg4MDgoKClMtrazFX7Ag2ZswYfPjhh+jbty9u3LiBjRs3oqSkBB07\ndkTLli0xYsQI3Lx5s2FfHBFVIRE80ZGIiMhosMVMRERkRBjMRERERoTBTEREZEQYzEREREaEwUxE\nRGREGMxERERGhMFMRERkRBjMRERERoTBTEREZET+PxUecCW16/K0AAAAAElFTkSuQmCC\n",
       "text": "<matplotlib.figure.Figure at 0x7f4780d8e550>"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Put options"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "lowerBound = array([max(K-S0, 0.0) for K in strikes])\nupperBound = array([K for K in strikes])\nVcev2Put_closed = [cev2Put_closed(S0, K, T) for K in strikes]\nVcev2Put_int = [cev2Put_int(S0, K, T) for K in strikes]\n\nplt.figure(figsize=(8,4))\nplt.plot(strikes, lowerBound, 'r', linewidth=2)\nplt.plot(strikes, upperBound, 'r', linewidth=2)\nplt.plot(strikes, Vcev2Put_closed, 'bo', label='Closed-form')\nplt.plot(strikes, Vcev2Put_int, 'g-', label='Integration', linewidth=2)\nplt.fill_between(strikes, lowerBound, upperBound, where=upperBound>=lowerBound, facecolor='grey', interpolate=True)\nplt.title(\"Put option prices in the CEV model with \" r\"$\\beta=1$\")\nplt.legend(loc='upper left')\nplt.xlabel(u'strike', fontsize=12)\nprint 'The shaded area corresponds to the no-arbitrage region, while the red plot is its boundary $(K-S_0)_+$'\nplt.show()",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": "",
       "is_hidden": true
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "The shaded area corresponds to the no-arbitrage region, while the red plot is its boundary $(K-S_0)_+$\n"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6pKRMNm9Oxte3D717+7J48Y3cfPMkq62/ubmnFrOIOBa1kMUKkpIy2fKpwcV+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\nrkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x7f477df76fd0>"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "At-the money comparison: $K = S_0$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "myK = 1.\nmyCol = ['Value']\nmyTable = pd.DataFrame.from_items([\n('Put by closed form', [cev2Put_closed(S0, myK, T)]),\n('Put by integration', [cev2Put_int(S0, myK, T)]),\n('Call by integration', [cev2Call_int(S0, myK, T)]),\n('Call by parity', [cev2Call_pc(S0, myK, T)]),\n('Call by closed form', [cev2Call_closed(S0, myK, T)]),\n('Expectation', [expectS(S0, T)])\n],orient='index', columns=myCol)\nprint myTable\nprint \"\\nNote that by Put-Call Parity, \\\nthe Call (evaluated by PC parity) and the Put have the same price\"",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "                        Value\nPut by closed form   0.390452\nPut by integration   0.390452\nCall by integration  0.072992\nCall by parity       0.390452\nCall by closed form  0.073141\nExpectation          0.682689\n\nNote that by Put-Call Parity, the Call (evaluated by PC parity) and the Put have the same price\n"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Low-strike comparison"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "###################################################################\n###################################################################\nmyK = 0.001\nprint \"************************************\"\nprint \"************************************\"\nprint \"Strike = \", myK\nmyCol = ['Value']\nmyTable = pd.DataFrame.from_items([\n('Put by closed form', [cev2Put_closed(S0, myK, T)]),\n('Put by integration', [cev2Put_int(S0, myK, T)]),\n('Call by integration', [cev2Call_int(S0, myK, T)]),\n('Call by parity', [cev2Call_pc(S0, myK, T)]),\n('Call by closed form', [cev2Call_closed(S0, myK, T)]),\n('E(S_T)', [expectS(S0, T)])\n],orient='index', columns=myCol)\nprint myTable\n",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "************************************\n************************************\nStrike =  0.001\n                        Value\nPut by closed form   0.000000\nPut by integration   0.000000\nCall by integration  0.681538\nCall by parity       0.999000\nCall by closed form  0.681689\nE(S_T)               0.682689\n"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "\\begin{bibliography}\n@article{Andersen,\n  author = {Leif Andersen},\n  title = {Option pricing with quadratic volatility: a revisit},\n  journal = {Finance and Stochastics},\n  year = {2011},\n  volume = {15},\n  issue = {2},\n  pages = {191-219}\n}\n@article{CoxHobson,\n  author = {Alex Cox and David Hobson},\n  title = {Local martingales, bubbles and option prices},\n  journal = {Finance and Stochastics},\n  year = {2005},\n  volume = {9},\n  pages = {477-49}\n}\n\\end{bibliography}"
    }
   ],
   "metadata": {}
  }
 ]
}