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

docs/examples/Cython Magics.ipynb

 {
  "metadata": {
-  "name": "",
-  "signature": "sha256:72d92961ad5ed0edc2b71d8b835c31c35d82d67c6f7f2360d56c2c708f1d9767"
+  "name": "Cython Magics",
+  "signature": "sha256:c357b93e9480d6347c6677862bf43750745cef4b30129c5bc53cb879a19d4074"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -35,7 +35,7 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "%load_ext cython"
+      "%load_ext cythonmagic"
      ],
      "language": "python",
      "metadata": {},
@@ -164,12 +164,12 @@
       "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",
@@ -184,8 +184,8 @@
       "    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",
@@ -197,7 +197,7 @@
      "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": {},
@@ -213,11 +213,76 @@
      ],
      "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": [
+      "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(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
+      "%timeit black_scholes_cy(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)"
      ],
      "language": "python",
      "metadata": {},
@@ -226,11 +291,30 @@
        "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",
@@ -266,7 +350,7 @@
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -279,4 +363,4 @@
    "metadata": {}
   }
  ]
-}
+}
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