Convexity Adjustment.ipynb

{
 "metadata": {
  "name": "Convexity Adjustment"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Convexity Adjustment"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "import numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def generate_path(n_steps, start, sigma):\n    path = [start]\n    account = 0.0\n    discount = 1.0\n    for n in xrange(n_steps):\n        account = account * (1 + path[-1]/(100*365))\n        discount = discount / (1 + path[-1]/(100*365))\n        path.append(path[-1] + np.random.normal(0.0, sigma/np.sqrt(365)))\n        account = account - (path[-1] - path[-2])\n    return path, account, discount",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for n in xrange(500):\n    path, account, discount = generate_path(1000, 10, 1)\n    plt.plot((path[0]-path[-1])*discount, account*discount, 'ro', alpha=0.2)\nplt.plot([-3,5],[-3,5],'--')\nplt.xlabel('FRA value')\nplt.ylabel('ED value')\nplt.show()",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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EFizAlpWF7SPdbAPJ5NwfqBBn0HHD/6WXXgLgwQcfpKmpCb/fz3XXXXfaX/xx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\nFhJNJBgdGmKvyUTpkiXY8/P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LAbDZbFRXV+PxeNJcFVgsFgBisRjJZJKcnJw0VwR9fX28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\nrAAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0xc91c6d8>"
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Distribution of FRA values"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fra = []\nfor n in xrange(10000):\n    path, account, discount = generate_path(2000, 20, 3)\n    fra.append((path[0]-path[-1])*discount)\nplt.hist(fra, bins=100)\nplt.show()",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0xc357550>"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "print np.mean(fra)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "0.4608872083\n"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Distribution of ED values"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "ed = []\nfor n in xrange(10000):\n    path, account, discount = generate_path(2000, 20, 3)\n    ed.append(account*discount)\nplt.hist(ed, bins=100)\nplt.show()\nprint np.mean(ed)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0xc35c048>"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "-0.00673841296853\n"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "np.std(ed)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": "4.5468196054401258"
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "ends = []\nfor n in xrange(10000):\n    path, account, discount = generate_path(2000, 20, 3)\n    ends.append(path[-1]-path[0])\nplt.hist(ends, bins=100)\nplt.show()\nprint np.mean(ends)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0xc647978>"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "0.0754619067243\n"
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "print np.std(ends)\nprint 3*np.sqrt(2000.0/365.0)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "7.12032707421\n7.02246883177\n"
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}