{
 "metadata": {
  "name": "BSPDE_NonUniformGrid",
  "zanadu": {
   "authors": [
    "Antoine Jacquier"
   ],
   "category": "1",
   "clearance": "Internal",
   "group_name": "IC MathFin Num Methods",
   "md5": "22a2c97584188032f2300fe3706e0013",
   "notebook_id": "1982BD76-41AA-46F8-9FD0-2040F77FEA60",
   "python_name": null,
   "reviewer_id": "B81D4EC4-269D-42B3-9975-85BC42EE5712",
   "status": "Review Pending"
  }
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "Coordinate transformation for the Black-Scholes PDE\n======\n***"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "import numpy as np\nimport matplotlib.pyplot as plt",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": "",
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {
      "zanadu": {}
     },
     "source": "Coordinate mapping"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "The Python function $\\textrm{Smap}(\\cdot)$ below maps the $\\xi$ coordinates into $S$ coordinates:\n$$\nS(\\xi) := B + \\alpha\\mathrm{sinh}(c_{0}\\xi + c_{1}(1-\\xi)),\n$$\nwith\n$c_{0} := \\mathrm{arcsinh}\\left(\\frac{\\overline{S} - B}{\\alpha}\\right)$\nand\n$c_{1} := \\mathrm{arcsinh}\\left(\\frac{\\underline{S} - B}{\\alpha}\\right)$."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def SMap(xi, SMin, SMax, B, alpha):\n    c = arcsinh((SMax - B) / alpha), arcsinh((SMin - B) / alpha)\n    return (B + alpha*sinh(c[0]*xi + c[1]*(1.-xi)))",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {
      "zanadu": {}
     },
     "source": "Numerical example"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B, SMin, SMax = 50., 0., 100.",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "xixi = linspace(0,1,100)\nhorizLine = [1. for xi in xixi]\naalpha = [0.01, 10., 100.]\nfor alpha in aalpha:\n    SS = [SMap(xi, SMin, SMax, B, alpha) for xi in xixi] \n    plt.figure(figsize=(15,3))\n    plt.title('Non-uniform grid with alpha = %.2f' %alpha, fontsize=13, fontweight='bold')\n    plt.scatter(SS, horizLine, s=2)\nplt.show()",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": "",
       "is_hidden": true
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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CYQEAAADAzc1j2MvJyVFQUJACAwPl7e2thIQEZWVlVSqzdu1aJSYmSpKio6NV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\nAAAAAGBB/wd32PeGIusIpwAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x7f7e19e782d0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Kiopqbfvcc8/p9ddfb4IpAQAAAAA8hr3CwkL5+/u7b9vtdhUWFtap5tSpUzW2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\nDwAAAAAsiLAHAAAAABb0/wCGSwU+VKZXFQAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x7f7e1a6e2a50>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0x7f7e1a180450>"
      }
     ],
     "prompt_number": 11
    }
   ],
   "metadata": {}
  }
 ]
}