{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://raw.githubusercontent.com/israeldi/quantlab/master/assets/images/Program-Logo.png\" width=\"400px\" align=\"right\">\n",
    "\n",
    "# Binomial Pricing Model\n",
    "### [(Go to Quant Lab)](https://israeldi.github.io/quantlab/)\n",
    "\n",
    "#### Source: Numerical Methods in Finance with C++\n",
    "\n",
    "&copy; Maciej J. Capinski, Tomasz Zastawniak\n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/israeldi/quantlab/master/assets/images/Numerical_Methods.jpg\" width=\"200px\" align=\"left\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Table of Contents\n",
    "\n",
    "1. [Random Numbers](#1.-Random-Numbers)\n",
    "2. [Plotting Random Samples](#2.-Plotting-Random-Samples)\n",
    "3. [Simulation](#3.-Simulation)\n",
    "    - 3.1 [Random Variables](#3.1-Random-Variables)\n",
    "    - 3.2 [Stochastic Processes](#3.2-Stochastic-Processes)\n",
    "        - 3.2.1 [Geometric Brownian Motion](#3.2.1-Geometric-Brownian-Motion)\n",
    "        - 3.2.2 [Square-Root Diffusion](#3.2.2-Square-Root-Diffusion)\n",
    "        - 3.2.3 [Stochastic Processes](#3.2-Stochastic-Processes)\n",
    "        - 3.2.4 [Stochastic Processes](#3.2-Stochastic-Processes)\n",
    "    - 3.3 [Variance Reduction](#3.3-Variance-Reduction)\n",
    "4. [Valuation](#4.-Valuation)\n",
    "    - 4.1 [European Options](#4.1-European-Options)\n",
    "    - 4.2 [American Options](#4.2-American-Options)\n",
    "5. [Risk Measures](#5.-Risk-Measures)\n",
    "    - 5.1 [Value-at-Risk](#5.1-Value-at-Risk)\n",
    "    - 5.2 [Credit Value Adjustments](#5.2-Credit-Value-Adjustments)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initially import all the modules we will be using for our notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import numpy.random as npr  \n",
    "# from pylab import plt, mpl\n",
    "import sys\n",
    "import os"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.1 Creating Main Program"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hi there\n"
     ]
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/ipykernel/kernelbase.py\u001b[0m in \u001b[0;36m_input_request\u001b[0;34m(self, prompt, ident, parent, password)\u001b[0m\n\u001b[1;32m    884\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 885\u001b[0;31m                 \u001b[0mident\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mreply\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msession\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrecv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstdin_socket\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    886\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/jupyter_client/session.py\u001b[0m in \u001b[0;36mrecv\u001b[0;34m(self, socket, mode, content, copy)\u001b[0m\n\u001b[1;32m    802\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 803\u001b[0;31m             \u001b[0mmsg_list\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msocket\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrecv_multipart\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmode\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    804\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mzmq\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mZMQError\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/zmq/sugar/socket.py\u001b[0m in \u001b[0;36mrecv_multipart\u001b[0;34m(self, flags, copy, track)\u001b[0m\n\u001b[1;32m    469\u001b[0m         \"\"\"\n\u001b[0;32m--> 470\u001b[0;31m         \u001b[0mparts\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrecv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtrack\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtrack\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    471\u001b[0m         \u001b[0;31m# have first part already, only loop while more to receive\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32mzmq/backend/cython/socket.pyx\u001b[0m in \u001b[0;36mzmq.backend.cython.socket.Socket.recv\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mzmq/backend/cython/socket.pyx\u001b[0m in \u001b[0;36mzmq.backend.cython.socket.Socket.recv\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mzmq/backend/cython/socket.pyx\u001b[0m in \u001b[0;36mzmq.backend.cython.socket._recv_copy\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/zmq/backend/cython/checkrc.pxd\u001b[0m in \u001b[0;36mzmq.backend.cython.checkrc._check_rc\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: ",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-2-7d983df3fa32>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0m__name__\u001b[0m\u001b[0;34m==\u001b[0m \u001b[0;34m\"__main__\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      2\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Hi there\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m     \u001b[0minput\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Provide a character: \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/ipykernel/kernelbase.py\u001b[0m in \u001b[0;36mraw_input\u001b[0;34m(self, prompt)\u001b[0m\n\u001b[1;32m    858\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parent_ident\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    859\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parent_header\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 860\u001b[0;31m             \u001b[0mpassword\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    861\u001b[0m         )\n\u001b[1;32m    862\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/ipykernel/kernelbase.py\u001b[0m in \u001b[0;36m_input_request\u001b[0;34m(self, prompt, ident, parent, password)\u001b[0m\n\u001b[1;32m    888\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mKeyboardInterrupt\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    889\u001b[0m                 \u001b[0;31m# re-raise KeyboardInterrupt, to truncate traceback\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 890\u001b[0;31m                 \u001b[0;32mraise\u001b[0m \u001b[0mKeyboardInterrupt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    891\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    892\u001b[0m                 \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "if __name__== \"__main__\":\n",
    "    print(\"Hi there\")\n",
    "    input(\"Provide a character: \")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Directory where we will save our plots\n",
    "directory = \"./images\"\n",
    "if not os.path.exists(directory):\n",
    "    os.makedirs(directory)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.2 Entering Data\n",
    "#### ([Back to Top](#Table-of-Contents))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/ipykernel/kernelbase.py\u001b[0m in \u001b[0;36m_input_request\u001b[0;34m(self, prompt, ident, parent, password)\u001b[0m\n\u001b[1;32m    884\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 885\u001b[0;31m                 \u001b[0mident\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mreply\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msession\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrecv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstdin_socket\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    886\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/jupyter_client/session.py\u001b[0m in \u001b[0;36mrecv\u001b[0;34m(self, socket, mode, content, copy)\u001b[0m\n\u001b[1;32m    802\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 803\u001b[0;31m             \u001b[0mmsg_list\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msocket\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrecv_multipart\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmode\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    804\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mzmq\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mZMQError\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/zmq/sugar/socket.py\u001b[0m in \u001b[0;36mrecv_multipart\u001b[0;34m(self, flags, copy, track)\u001b[0m\n\u001b[1;32m    469\u001b[0m         \"\"\"\n\u001b[0;32m--> 470\u001b[0;31m         \u001b[0mparts\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrecv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtrack\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtrack\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    471\u001b[0m         \u001b[0;31m# have first part already, only loop while more to receive\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32mzmq/backend/cython/socket.pyx\u001b[0m in \u001b[0;36mzmq.backend.cython.socket.Socket.recv\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mzmq/backend/cython/socket.pyx\u001b[0m in \u001b[0;36mzmq.backend.cython.socket.Socket.recv\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mzmq/backend/cython/socket.pyx\u001b[0m in \u001b[0;36mzmq.backend.cython.socket._recv_copy\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/zmq/backend/cython/checkrc.pxd\u001b[0m in \u001b[0;36mzmq.backend.cython.checkrc._check_rc\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: ",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-4-a1a06ba97b6b>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0m__name__\u001b[0m\u001b[0;34m==\u001b[0m \u001b[0;34m\"__main__\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m     \u001b[0mS0\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minput\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Enter S_0: \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      3\u001b[0m     \u001b[0mU\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minput\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Enter U: \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m     \u001b[0mD\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minput\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Enter D: \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m     \u001b[0mR\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minput\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Enter R: \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/ipykernel/kernelbase.py\u001b[0m in \u001b[0;36mraw_input\u001b[0;34m(self, prompt)\u001b[0m\n\u001b[1;32m    858\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parent_ident\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    859\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parent_header\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 860\u001b[0;31m             \u001b[0mpassword\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    861\u001b[0m         )\n\u001b[1;32m    862\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/ipykernel/kernelbase.py\u001b[0m in \u001b[0;36m_input_request\u001b[0;34m(self, prompt, ident, parent, password)\u001b[0m\n\u001b[1;32m    888\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mKeyboardInterrupt\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    889\u001b[0m                 \u001b[0;31m# re-raise KeyboardInterrupt, to truncate traceback\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 890\u001b[0;31m                 \u001b[0;32mraise\u001b[0m \u001b[0mKeyboardInterrupt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    891\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    892\u001b[0m                 \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "if __name__== \"__main__\":\n",
    "    S0 = float(input(\"Enter S_0: \"))\n",
    "    U = float(input(\"Enter U: \"))\n",
    "    D = float(input(\"Enter D: \"))\n",
    "    R = float(input(\"Enter R: \"))\n",
    "\n",
    "    # making sure that 0<S0, -1<D<U, -1<R\n",
    "    if (S0 <= 0.0 or U <= -1.0 or D <= -1.0 or U <= D or R <= -1.0):\n",
    "        print(\"Illegal data ranges\")\n",
    "        print(\"Terminating program\")\n",
    "        sys.exit()\n",
    "    \n",
    "    # checking for arbitrage\n",
    "    if (R >= U or R <= D):\n",
    "        print(\"Arbitrage exists\")\n",
    "        print(\"Terminating program\")\n",
    "        sys.exit()\n",
    "\n",
    "    print(\"Input data checked\")\n",
    "    print(\"There is no arbitrage\\n\")\n",
    "    \n",
    "    # compute risk-neutral probability\n",
    "    print(\"q = \", (R - D) / (U - D))\n",
    "    \n",
    "    # compute stock price at node n=3,i=2\n",
    "    n = 3; i = 2\n",
    "    print(\"n = \", n)\n",
    "    print(\"i = \", i)\n",
    "    print(\"S(n,i) = \", S0* math.pow(1 + U,i) * math.pow(1 + D, n - i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.3 Functions\n",
    "#### ([Back to Top](#Table-of-Contents))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "uuid": "4618b170-6bd3-4500-905a-0fe402f198c1"
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'U' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-74dfc56e9700>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m     39\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0m__name__\u001b[0m\u001b[0;34m==\u001b[0m \u001b[0;34m\"__main__\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     40\u001b[0m     \u001b[0;31m# compute risk-neutral probability\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 41\u001b[0;31m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"q = \"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mRiskNeutProb\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mU\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mD\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     42\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     43\u001b[0m     \u001b[0moutput\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mGetInputData\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'U' is not defined"
     ]
    }
   ],
   "source": [
    "# computing risk-neutral probability\n",
    "def RiskNeutProb(U, D, R):\n",
    "    return (R - D) / (U - D)\n",
    "\n",
    "# computing the stock price at node n,i\n",
    "def S(S0, U, D, n, i):\n",
    "    return S0 * math.pow(1 + U,i) * math.pow(1 + D, n - i)\n",
    "\n",
    "def isValidInput(S0, U, D, R):\n",
    "    # making sure that 0<S0, -1<D<U, -1<R\n",
    "    if (S0 <= 0.0 or U <= -1.0 or D <= -1.0 or U <= D or R <= -1.0):\n",
    "        print(\"Illegal data ranges\")\n",
    "        print(\"Terminating program\")\n",
    "        return 0\n",
    "    \n",
    "    # checking for arbitrage\n",
    "    if (R >= U or R <= D):\n",
    "        print(\"Arbitrage exists\")\n",
    "        print(\"Terminating program\")\n",
    "        return 0\n",
    "    \n",
    "    return 1\n",
    "\n",
    "# inputting, displaying and checking model data\n",
    "def GetInputData():\n",
    "    #entering data\n",
    "    params = (\"S0\", \"U\", \"D\", \"R\")\n",
    "    S0, U, D, R = [float(input(\"Enter %s: \" % (var))) for var in params]\n",
    "    \n",
    "    if not isValidInput(S0, U, D, R):\n",
    "        return 0\n",
    "    \n",
    "    print(\"Input data checked\")\n",
    "    print(\"There is no arbitrage\\n\")\n",
    "    d = locals().copy()\n",
    "    return d\n",
    "\n",
    "\n",
    "if __name__== \"__main__\":\n",
    "    # compute risk-neutral probability\n",
    "    print(\"q = \", RiskNeutProb(U, D, R))\n",
    "    \n",
    "    output = GetInputData()\n",
    "    if output == 0:\n",
    "        sys.exit()\n",
    "    \n",
    "    # Update our parameters\n",
    "    locals().update(output)\n",
    "    \n",
    "    # compute stock price at node n=3,i=2\n",
    "    n = 3; i = 2\n",
    "    print(\"n = \", n)\n",
    "    print(\"i = \", i)\n",
    "    print(\"S(n,i) = \", S(S0,U,D,n,i))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.0 Object Oriented European and American\n",
    "#### ([Back to Top](#Table-of-Contents))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.19822404128088741"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Binomial Model (Parent class) ---------------------------------------------------\n",
    "class BinModel:\n",
    "    def __init__(self, S0=100, U=0.1, D=-0.1, R=0):\n",
    "        self.S0 = S0\n",
    "        self.U = U\n",
    "        self.D = D\n",
    "        self.R = R\n",
    "    \n",
    "    def RiskNeutProb(self):\n",
    "        return (self.R - self.D) / (self.U - self.D)\n",
    "    \n",
    "    def S(self, n, i):\n",
    "        return self.S0 * math.pow(1 + self.U, i) * math.pow(1 + self.D, n - i)\n",
    "        \n",
    "    \n",
    "# European Option class ------------------------------------------------------------------\n",
    "class EurOption():\n",
    "    \n",
    "    def PriceByCRR(self, Model, N=3):\n",
    "        q = Model.RiskNeutProb();\n",
    "        Price = np.zeros(N + 1)\n",
    "        \n",
    "        for i in range(0, N + 1):\n",
    "            Price[i] = self.Payoff(Model.S(N, i))\n",
    "        \n",
    "        for n in range(N - 1, -1, -1):\n",
    "            for i in range(0, n + 1):\n",
    "                Price[i] = (q * Price[i+1] + (1 - q) * Price[i]) / (1 + Model.R);\n",
    "\n",
    "        return Price[0]\n",
    "    \n",
    "# American Option class\n",
    "class AmOption():\n",
    "    def PriceBySnell(self, Model, N=3):\n",
    "        q=Model.RiskNeutProb()\n",
    "        Price = np.zeros(N + 1)\n",
    "        \n",
    "        for i in range(0, N + 1):\n",
    "            Price[i] = self.Payoff(Model.S(N, i))\n",
    "\n",
    "        for n in range(N - 1, -1, -1):\n",
    "            for i in range(0, n + 1):\n",
    "                ContVal = (q * Price[i+1] + (1 - q) * Price[i]) / (1 + Model.R)\n",
    "                Price[i] = self.Payoff(Model.S(n,i))\n",
    "                if (ContVal > Price[i]):\n",
    "                    Price[i] = ContVal\n",
    "        \n",
    "        return Price[0]\n",
    "    \n",
    "class AsianOption()\n",
    "    def PriceAsian(self, Model, N=3):\n",
    "        q = Model.RiskNeutProb();\n",
    "        Price = np.zeros(N + 1)\n",
    "        \n",
    "        for i in range(0, 2 * N + 1):\n",
    "            \n",
    "            Price[i] = self.Payoff(Model.S(N, i))\n",
    "        \n",
    "        for n in range(N - 1, -1, -1):\n",
    "            for i in range(0, n + 1):\n",
    "                Price[i] = (q * Price[i+1] + (1 - q) * Price[i]) / (1 + Model.R);\n",
    "\n",
    "        return Price[0]\n",
    "\n",
    "# Payoff Classes ---------------------------------------------------------------------------- \n",
    "class Call(EurOption, AmOption):\n",
    "    \n",
    "    def __init__(self, K=1):\n",
    "        self.K = K\n",
    "    \n",
    "    def Payoff(self, z):\n",
    "        if (z > self.K): \n",
    "            return z - self.K\n",
    "        \n",
    "        return 0\n",
    "\n",
    "class Put(EurOption, AmOption):\n",
    "    \n",
    "    def __init__(self, K=1):\n",
    "        self.K = K\n",
    "        \n",
    "    def Payoff(self, z):\n",
    "        if (z < self.K): \n",
    "            return self.K - z\n",
    "        \n",
    "        return 0\n",
    "\n",
    "class Digital_Call(EurOption, AmOption):\n",
    "    def __init__(self, K=1):\n",
    "        self.K = K\n",
    "        \n",
    "    def Payoff(self, z):\n",
    "        if self.K < z:\n",
    "            return 1\n",
    "        return 0\n",
    "    \n",
    "class BullSpread(EurOption, AmOption):\n",
    "    def __init__(self, K1=1, K2 = 2):\n",
    "        self.K1 = K1\n",
    "        self.K2 = K2\n",
    "    \n",
    "    def Payoff\n",
    "    \n",
    "\n",
    "model = BinModel(S0=10, U=0.2, D=-0.1, R=0.1)\n",
    "Option1 = Digital_Call(K=22)\n",
    "Option1.PriceByCRR(model, N=6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises:\n",
    "\n",
    "1. Modify the `PriceByCRR()` function in `EurOption` Class to compute the time $0$ price of a European option using the **Cox–Ross–Rubinstein (CRR)** formula:\n",
    "$$H(0)=\\frac{1}{(1+R)^{N}}\\sum_{i=0}^{N}\\frac{N!}{i!(N-i)!}q^{i}(1-q)^{N-i}h(S(N,I))$$\n",
    "\n",
    "\n",
    "2. The payoff of a **digital call** with strike price $K$ is:\n",
    "$$h^{digit\\thinspace call}(z)=\\begin{cases}\n",
    "1 & \\textrm{{if} }K<z,\\\\\n",
    "0 & \\textrm{otherwise.}\n",
    "\\end{cases}$$\n",
    "Include the ability to price digital calls in the program developed in the present section by adding the new payoff Class `DigitCall` just as was done for calls and puts.\n",
    "\n",
    "\n",
    "3. Add the ability to price bull spreads and bear spreads by introducing new subclasses `BullSpread` and `BearSpread` of the `EurOption` and `AmOption` classes defined in our current program. The payoffs of a bull spread and a bear spread, which depend on two parameters $K1 < K2$, are given by:\n",
    "$$h^{bull}(z)=\\begin{cases}\n",
    "0 & \\textrm{if }z\\leq K_{1},\\\\\n",
    "z-K_{1} & \\textrm{if }K_{1}<z<K_{2}\\\\\n",
    "K_{2}-K_{1} & \\textrm{if }K_{2}\\leq z,\n",
    "\\end{cases}$$ and\n",
    "$$h^{bear}(z)=\\begin{cases}\n",
    "K_{2}-K_{1} & \\textrm{if }z\\leq K_{1},\\\\\n",
    "K_{2}-z & \\textrm{if }K_{1}<z<K_{2}\\\\\n",
    "0 & \\textrm{if }K_{2}<z,\n",
    "\\end{cases}$$\n",
    "\n",
    "\n",
    "4. Consider the **fixed-strike** Asian call option with process,\n",
    "$$A_{N}=\\frac{1}{N}\\sum_{k=0}^{N-1}S(k)$$ and payoff, $$C(N)=max\\{A_{N},0\\}$$\n",
    "Add a new Class `AsianOption` to price an Asian style Call Option. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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