Add a pure-python version of the Black-Scholes function for comparison with the Cython equivalent

examples/Builtin Extensions/Cython Magics.ipynb

@@ -1,278 +1,366 b''
 {
  "metadata": {
-  "name": "Cython Magics"
+  "name": "",
+  "signature": "sha256:c357b93e9480d6347c6677862bf43750745cef4b30129c5bc53cb879a19d4074"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
  "worksheets": [
   {
    "cells": [
     {
      "cell_type": "heading",
      "level": 1,
      "metadata": {},
      "source": [
       "Cython Magic Functions"
      ]
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
       "Loading the extension"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
       "IPython has a `cythonmagic` extension that contains a number of magic functions for working with Cython code. This extension can be loaded using the `%load_ext` magic as follows:"
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "%load_ext cythonmagic"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
      "prompt_number": 1
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
       "The %cython_inline magic"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
       "The `%%cython_inline` magic uses `Cython.inline` to compile a Cython expression. This allows you to enter and run a function body with Cython code. Use a bare `return` statement to return values. "
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "a = 10\n",
       "b = 20"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
      "prompt_number": 2
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "%%cython_inline\n",
       "return a+b"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
+       "metadata": {},
        "output_type": "pyout",
        "prompt_number": 3,
        "text": [
         "30"
        ]
       }
      ],
      "prompt_number": 3
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
       "The %cython_pyximport magic"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
       "The `%%cython_pyximport` magic allows you to enter arbitrary Cython code into a cell. That Cython code is written as a `.pyx` file in the current working directory and then imported using `pyximport`.  You have the specify the name of the module that the Code will appear in. All symbols from the module are imported automatically by the magic function."
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "%%cython_pyximport foo\n",
       "def f(x):\n",
       "    return 4.0*x"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
      "prompt_number": 4
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "f(10)"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
+       "metadata": {},
        "output_type": "pyout",
        "prompt_number": 5,
        "text": [
         "40.0"
        ]
       }
      ],
      "prompt_number": 5
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
       "The %cython magic"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
       "Probably the most important magic is the `%cython` magic.  This is similar to the `%%cython_pyximport` magic, but doesn't require you to specify a module name. Instead, the `%%cython` magic uses manages everything using temporary files in the `~/.cython/magic` directory.  All of the symbols in the Cython module are imported automatically by the magic.\n",
       "\n",
       "Here is a simple example of a Black-Scholes options pricing algorithm written in Cython. Please note that this example might not compile on non-POSIX systems (e.g., Windows) because of a missing `erf` symbol."
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "%%cython\n",
       "cimport cython\n",
       "from libc.math cimport exp, sqrt, pow, log, erf\n",
       "\n",
       "@cython.cdivision(True)\n",
-      "cdef double std_norm_cdf(double x) nogil:\n",
+      "cdef double std_norm_cdf_cy(double x) nogil:\n",
       "    return 0.5*(1+erf(x/sqrt(2.0)))\n",
       "\n",
       "@cython.cdivision(True)\n",
-      "def black_scholes(double s, double k, double t, double v,\n",
-      "                 double rf, double div, double cp):\n",
+      "def black_scholes_cy(double s, double k, double t, double v,\n",
+      "                     double rf, double div, double cp):\n",
       "    \"\"\"Price an option using the Black-Scholes model.\n",
       "    \n",
       "    s : initial stock price\n",
       "    k : strike price\n",
       "    t : expiration time\n",
       "    v : volatility\n",
       "    rf : risk-free rate\n",
       "    div : dividend\n",
       "    cp : +1/-1 for call/put\n",
       "    \"\"\"\n",
       "    cdef double d1, d2, optprice\n",
       "    with nogil:\n",
       "        d1 = (log(s/k)+(rf-div+0.5*pow(v,2))*t)/(v*sqrt(t))\n",
       "        d2 = d1 - v*sqrt(t)\n",
-      "        optprice = cp*s*exp(-div*t)*std_norm_cdf(cp*d1) - \\\n",
-      "            cp*k*exp(-rf*t)*std_norm_cdf(cp*d2)\n",
+      "        optprice = cp*s*exp(-div*t)*std_norm_cdf_cy(cp*d1) - \\\n",
+      "            cp*k*exp(-rf*t)*std_norm_cdf_cy(cp*d2)\n",
       "    return optprice"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
      "prompt_number": 6
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "black_scholes(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
+      "black_scholes_cy(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
+       "metadata": {},
        "output_type": "pyout",
        "prompt_number": 7,
        "text": [
         "10.327861752731728"
        ]
       }
      ],
      "prompt_number": 7
     },
     {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "For comparison, the same code is implemented here in pure python."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from math import exp, sqrt, pow, log, erf\n",
+      "\n",
+      "def std_norm_cdf_py(x):\n",
+      "    return 0.5*(1+erf(x/sqrt(2.0)))\n",
+      "\n",
+      "def black_scholes_py(s, k, t, v, rf, div, cp):\n",
+      "    \"\"\"Price an option using the Black-Scholes model.\n",
+      "    \n",
+      "    s : initial stock price\n",
+      "    k : strike price\n",
+      "    t : expiration time\n",
+      "    v : volatility\n",
+      "    rf : risk-free rate\n",
+      "    div : dividend\n",
+      "    cp : +1/-1 for call/put\n",
+      "    \"\"\"\n",
+      "    d1 = (log(s/k)+(rf-div+0.5*pow(v,2))*t)/(v*sqrt(t))\n",
+      "    d2 = d1 - v*sqrt(t)\n",
+      "    optprice = cp*s*exp(-div*t)*std_norm_cdf_py(cp*d1) - \\\n",
+      "        cp*k*exp(-rf*t)*std_norm_cdf_py(cp*d2)\n",
+      "    return optprice"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 8
+    },
+    {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "%timeit black_scholes(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
+      "black_scholes_py(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 9,
+       "text": [
+        "10.327861752731728"
+       ]
+      }
+     ],
+     "prompt_number": 9
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Below we see the runtime of the two functions: the Cython version is nearly a factor of 10 faster."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%timeit black_scholes_cy(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "1000000 loops, best of 3: 821 ns per loop\n"
+        "1000000 loops, best of 3: 319 ns per loop\n"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 10
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%timeit black_scholes_py(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "100000 loops, best of 3: 2.28 \u00b5s per loop\n"
+       ]
+      }
+     ],
+     "prompt_number": 11
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
       "External libraries"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
       "Cython allows you to specify additional libraries to be linked with your extension, you can do so with the `-l` flag (also spelled `--lib`).  Note that this flag can be passed more than once to specify multiple libraries, such as `-lm -llib2 --lib lib3`.  Here's a simple example of how to access the system math library:"
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "%%cython -lm\n",
       "from libc.math cimport sin\n",
       "print 'sin(1)=', sin(1)"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
         "sin(1)= 0.841470984808\n"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
       "You can similarly use the `-I/--include` flag to add include directories to the search path, and `-c/--compile-args` to add extra flags that are passed to Cython via the `extra_compile_args` of the distutils `Extension` class.  Please see [the Cython docs on C library usage](http://docs.cython.org/src/tutorial/clibraries.html) for more details on the use of these flags."
      ]
     }
    ],
    "metadata": {}
   }
  ]
-}
+}
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