{
 "metadata": {
  "name": "FiniteDifference_BS",
  "zanadu": {
   "authors": [
    "Antoine Jacquier"
   ],
   "category": "7",
   "clearance": "Internal",
   "group_name": "IC MathFin Num Methods",
   "md5": "42C0C3D8A33394E97A12690E4CA982BD",
   "notebook_id": "AE43D0D2-BDB1-4987-B774-C9F583ECB346",
   "python_name": null,
   "reviewer_id": null,
   "status": "WIP"
  }
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "Finite-difference scheme for the Black-Scholes PDE\n======\n***"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from scipy import interpolate\nimport numpy as np",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {
      "zanadu": {}
     },
     "source": "Description of the scheme"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Boundary conditions"
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {
      "zanadu": {}
     },
     "source": "At maturity"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def bcTime0(x, args): ## Function specifying the boundary condition\n    [strike, r, sigma] = args\n    a = (r - 0.5*sigma*sigma) / (sigma*sigma)\n    b = 2.*r / (sigma*sigma) + a*a\n    return np.exp(a*x) * max(np.exp(x) - 1.0, 0.0)## From the Call option payoff",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": "",
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {
      "zanadu": {}
     },
     "source": "In the space domain"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def bcxXMin(x):\n    return 0.\n\ndef bcxXMax(x, tau, args):\n    [strike, r, sigma] = args\n    a = (r - 0.5*sigma*sigma) / (sigma*sigma)\n    b = 2.*r / (sigma*sigma) + a*a\n    return (np.exp(x) - np.exp(-2.*r*tau/(sigma*sigma))) * np.exp(a*x + b*tau)",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": "",
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Explicit Finite Difference Scheme"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def heatEq_FDExplicit(xMin, xMax, tauMax, nbStepTime, nbStepSpace, *extraArgs):\n## bc represents the function (of x) at the boundary tauMin\n## We follow the conventions in the notes:\n## there are (nbStepSpace+1) points on the x-axis\n## and (nbStepTime+1) points on the tau-axis\n\n    sigma = extraArgs[0][2]\n    print \"sigma: \", sigma\n    dx = (xMax - xMin) / (1.*nbStepSpace)       # Grid step in space\n    dt = tauMax / (1.*nbStepTime)    # Grid step in time\n    cfl = dt / (dx*dx)\n    if cfl > 1.:\n        print '***********************************************************'\n        print '*****WARNING: THE CFL CONDITION IS NOT SATISFIED!!!!!!*****'\n        print \"dt = \", dt\n        print \"dx2 = \", dx*dx\n        print '*********    clf ratio = %s    ***************************' % cfl\n        print '***********************************************************'\n    else:\n        print \"dt =\", dt\n        print \"dx2 = \", dx*dx\n        print \"CFL condition satisfied\"\n        \n    # Boundary conditions\n    xx = np.linspace(xMin,xMax,nbStepSpace+1) ## set the grid on the x-axis\n    uu = [[0.]*(nbStepSpace+1)]*(nbStepTime+1)\n    timeVector = np.linspace(0.,tauMax,nbStepTime+1)\n    uu[0] = np.array([bcTime0(x, extraArgs[0]) for x in xx])\n    \n    for i in range(0, nbStepTime):\n        uu[i+1][0] = bcxXMin(xx[0])\n        uu[i+1][-1] = bcxXMax(xx[-1], timeVector[i+1], extraArgs[0])\n        \n        for j in range(1, nbStepSpace):\n            uu[i+1][j] = cfl*(uu[i][j+1] - 2.0*uu[i][j] + uu[i][j-1]) + uu[i][j]\n    return [xx, timeVector, uu]",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": false
      }
     },
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Numerical example"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "xMin, xMax, nbStepTime, nbStepSpace = -0.8, 0.5, 200, 50 ##(try with 100)\nT, strike, r, sigma = 2., 1.2, 0., 0.2",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "extraArgs = strike, r, sigma\n[xx, timeVector, uu] = heatEq_FDExplicit(xMin, xMax, 0.5*sigma*sigma*T, nbStepTime, nbStepSpace, extraArgs)\n#plot(xx, uu[0], 'b', xx, uu[-1], 'r')\n#show()\n\ndef mappingXToS(xx, tautau, uu, T, r, sigma, strike):\n    ## xx is the x coordinate vector; tautau the time-coordinate vector\n    ## uu the value function on the grid\n    a = (r - 0.5*sigma*sigma) / (sigma*sigma)\n    b = 2.*r / (sigma*sigma) + a*a\n    tt = T - tautau/(0.5*sigma*sigma)\n    SS = strike*np.exp(xx)\n    output = [0.0]*len(tautau)\n    for n in arange(len(tautau)):\n#        for p in arange(len(xx)):\n        output[n] = strike*np.exp(-a*xx - b*tautau[n])*uu[n]\n    return [SS, tt, output]\n\n[SS, tt, CC] = mappingXToS(xx, timeVector, uu, T, r, sigma, strike)\nplot(SS, CC[0], 'b')\nplot(SS, CC[-1], 'ro')\nshow()\nm = 99\nprint \"Remaining maturity: \", tt[m]\nf = interpolate.interp1d(SS, CC[m], 'cubic')\nprint \"Call option value: \", float(f(1.2))",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "sigma:  0.2\ndt = 0.0002\ndx2 =  0.000676\nCFL condition satisfied\n"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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A5O5wOJCdne16rNPpcPTo0aDH9Pb2xiy5u2+ld0/A/r4fnpS7ARzyk5TDTdjB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\nEVECYnInIkpA/x9N12/DmyjdMAAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x7f4f18329f10>"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Remaining maturity:  1.01\nCall option value:  0.161562738184\n"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 6
    }
   ],
   "metadata": {}
  }
 ]
}