Options Calculator with Binomial model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Options Calculator with Binomial model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Binomial model is arguably the simplest techniques used for option pricing. In this calculator, the options price will be calculated by two binomial-tree methods,Cox-Ross-Rubinstein and Jarrow-Rudd (the equal-probability model). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each of the approaches has its advantages and disadvantages for pricing different types of options. However, they all involve a similar tree-step process.\n",
    "\n",
    "* There are only two possible prices for the underlying asset on the next day.\n",
    "* The two possible prices are the up-price and down-price\n",
    "* No dividends are paid out during the option’s life.\n",
    "* The risk-free rate is constant throughout the life of the option\n",
    "* There are no transaction and commissions costs in buying the option.\n",
    "* Investors are risk neutral\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ** All trees are built from four parameters (u, d, p and q). \n",
    "- The parameters u and d are how much the underlying will go up or down in each discrete time. \n",
    "- The parameters p and q are the probabilities for the price to go up and down respectively. Moreover, assume p+q=1.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "#### ** Note that the model assumes that the price of the equity underlying the option follows a random walk."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Input variables:\n",
    "  * Underlying price (per share): S; \n",
    "  * Strike price of the option (per share): K;\n",
    "  * Time to maturity (years): T;\n",
    "  * Continuously compounding risk-free interest rate: r;\n",
    "  * Volatility: sigma;\n",
    "  * Number of binomial steps: N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## import certain packages\n",
    "import math\n",
    "from datetime import datetime, date\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "from pandas import DataFrame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## define Cox_Ross_Rubinstein binomial model\n",
    "def Cox_Ross_Rubinstein_Tree (S,K,T,r,sigma,N, Option_type):\n",
    "    \n",
    "    # Underlying price (per share): S; \n",
    "    # Strike price of the option (per share): K;\n",
    "    # Time to maturity (years): T;\n",
    "    # Continuously compounding risk-free interest rate: r;\n",
    "    # Volatility: sigma;\n",
    "    # Number of binomial steps: N;\n",
    "\n",
    "        # The factor by which the price rises (assuming it rises) = u ;\n",
    "        # The factor by which the price falls (assuming it falls) = d ;\n",
    "        # The probability of a price rise = pu ;\n",
    "        # The probability of a price fall = pd ;\n",
    "        # discount rate = disc ;\n",
    "    \n",
    "    u=math.exp(sigma*math.sqrt(T/N));\n",
    "    d=math.exp(-sigma*math.sqrt(T/N));\n",
    "    pu=((math.exp(r*T/N))-d)/(u-d);\n",
    "    pd=1-pu;\n",
    "    disc=math.exp(-r*T/N);\n",
    "\n",
    "    St = [0] * (N+1)\n",
    "    C = [0] * (N+1)\n",
    "    \n",
    "    St[0]=S*d**N;\n",
    "    \n",
    "    for j in range(1, N+1): \n",
    "        St[j] = St[j-1] * u/d;\n",
    "    \n",
    "    for j in range(1, N+1):\n",
    "        if Option_type == 'P':\n",
    "            C[j] = max(K-St[j],0);\n",
    "        elif Option_type == 'C':\n",
    "            C[j] = max(St[j]-K,0);\n",
    "    \n",
    "    for i in range(N, 0, -1):\n",
    "        for j in range(0, i):\n",
    "            C[j] = disc*(pu*C[j+1]+pd*C[j]);\n",
    "            \n",
    "    return C[0]\n",
    "\n",
    "\n",
    "## define Jarrow_Rudd binomial model    \n",
    "def Jarrow_Rudd_Tree (S,K,T,r,sigma,N, Option_type):\n",
    "\n",
    "    # Underlying price (per share): S; \n",
    "    # Strike price of the option (per share): K;\n",
    "    # Time to maturity (years): T;\n",
    "    # Continuously compounding risk-free interest rate: r;\n",
    "    # Volatility: sigma;\n",
    "    # Steps: N;\n",
    "    \n",
    "        # The factor by which the price rises (assuming it rises) = u ;\n",
    "        # The factor by which the price falls (assuming it falls) = d ;\n",
    "        # The probability of a price rise = pu ;\n",
    "        # The probability of a price fall = pd ;\n",
    "        # discount rate = disc ;\n",
    "        \n",
    "    u=math.exp((r-(sigma**2/2))*T/N+sigma*math.sqrt(T/N));\n",
    "    d=math.exp((r-(sigma**2/2))*T/N-sigma*math.sqrt(T/N));\n",
    "    pu=0.5;\n",
    "    pd=1-pu;\n",
    "    disc=math.exp(-r*T/N);\n",
    "\n",
    "    St = [0] * (N+1)\n",
    "    C = [0] * (N+1)\n",
    "    \n",
    "    St[0]=S*d**N;\n",
    "    \n",
    "    for j in range(1, N+1): \n",
    "        St[j] = St[j-1] * u/d;\n",
    "    \n",
    "    for j in range(1, N+1):\n",
    "        if Option_type == 'P':\n",
    "            C[j] = max(K-St[j],0);\n",
    "        elif Option_type == 'C':\n",
    "            C[j] = max(St[j]-K,0);\n",
    "    \n",
    "    for i in range(N, 0, -1):\n",
    "        for j in range(0, i):\n",
    "            C[j] = disc*(pu*C[j+1]+pd*C[j]);\n",
    "            \n",
    "    return C[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "What is the current stock price? 100\n",
      "What is the strike price? 110\n",
      "What is the expiration date of the options? (mm-dd-yyyy) 3-1-2020\n",
      "What is the continuously compounding risk-free interest rate in percentage(%)? 5\n",
      "What is the volatility in percentage(%)? 30\n"
     ]
    }
   ],
   "source": [
    "## input the current stock price and check if it is a number.\n",
    "S = input(\"What is the current stock price? \");\n",
    "while True:\n",
    "    try:\n",
    "        S = float(S)\n",
    "        break\n",
    "    except:\n",
    "        print(\"The current stock price has to be a NUMBER.\")\n",
    "        S = input(\"What is the current stock price? \")\n",
    "\n",
    "## input the strike price and check if it is a number.\n",
    "K = input(\"What is the strike price? \");\n",
    "while True:\n",
    "    try:\n",
    "        K = float(K)\n",
    "        break\n",
    "    except:\n",
    "        print(\"The the strike price has to be a NUMBER.\")\n",
    "        K = input(\"What is the strike price? \")\n",
    "\n",
    "\n",
    "## input the expiration_date and calculate the days between today and the expiration date.\n",
    "while True:\n",
    "    expiration_date = input(\"What is the expiration date of the options? (mm-dd-yyyy) \")\n",
    "    try:\n",
    "        expiration_date = datetime.strptime(expiration_date, \"%m-%d-%Y\")\n",
    "    except ValueError as e:\n",
    "        print(\"error: %s\\nTry again.\" % (e,))\n",
    "    else:\n",
    "        break\n",
    "T = (expiration_date - datetime.utcnow()).days / 365\n",
    "\n",
    "\n",
    "## input the continuously compounding risk-free interest rate and check if it is a number.\n",
    "r = input(\"What is the continuously compounding risk-free interest rate in percentage(%)? \");\n",
    "while True:\n",
    "    try:\n",
    "        r = float(r)\n",
    "        break\n",
    "    except:\n",
    "        print(\"The continuously compounding risk-free interest rate has to be a NUMBER.\")\n",
    "        r = input(\"What is the continuously compounding risk-free interest rate in percentage(%)? \")\n",
    "        \n",
    "\n",
    "## input the volatility and check if it is a number.\n",
    "sigma = input(\"What is the volatility in percentage(%)? \"); \n",
    "while True:\n",
    "    try:\n",
    "        sigma = float(sigma)\n",
    "        if sigma > 100 or sigma < 0:\n",
    "            print ( \"The range of sigma has to be in [0,100].\")\n",
    "            sigma = input(\"What is the volatility in percentage(%)? \")\n",
    "        break\n",
    "    except:\n",
    "        print(\"The volatility has to be a NUMBER.\")\n",
    "        sigma = input(\"What is the volatility in percentage(%)? \")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Symbol</th>\n",
       "      <th>Input</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Underlying price</th>\n",
       "      <td>S</td>\n",
       "      <td>100.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Strike price</th>\n",
       "      <td>K</td>\n",
       "      <td>110.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Time to maturity</th>\n",
       "      <td>T</td>\n",
       "      <td>2.221918</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Risk-free interest rate</th>\n",
       "      <td>r</td>\n",
       "      <td>5.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Volatility</th>\n",
       "      <td>sigma</td>\n",
       "      <td>30.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                        Symbol       Input\n",
       "Underlying price             S  100.000000\n",
       "Strike price                 K  110.000000\n",
       "Time to maturity             T    2.221918\n",
       "Risk-free interest rate      r    5.000000\n",
       "Volatility               sigma   30.000000"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = {'Symbol': ['S', 'K', 'T', 'r', 'sigma'],\n",
    "        'Input': [S, K, T , r , sigma]}\n",
    "input_frame = DataFrame(data, columns=['Symbol', 'Input'], \n",
    "                   index=['Underlying price', 'Strike price', 'Time to maturity', 'Risk-free interest rate', 'Volatility'])\n",
    "input_frame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Output"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Option</th>\n",
       "      <th>Price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Cox-Ross-Rubinstein</th>\n",
       "      <td>Call</td>\n",
       "      <td>18.350952</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Cox-Ross-Rubinstein</th>\n",
       "      <td>Put</td>\n",
       "      <td>16.784775</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Jarrow-Rudd</th>\n",
       "      <td>Call</td>\n",
       "      <td>18.343573</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Jarrow-Rudd</th>\n",
       "      <td>Put</td>\n",
       "      <td>16.777730</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                    Option      Price\n",
       "Cox-Ross-Rubinstein   Call  18.350952\n",
       "Cox-Ross-Rubinstein    Put  16.784775\n",
       "Jarrow-Rudd           Call  18.343573\n",
       "Jarrow-Rudd            Put  16.777730"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = r/100; sigma = sigma/100;\n",
    "binomial_model_pricing = {'Option' : ['Call', 'Put', 'Call', 'Put'],\n",
    "                          'Price': [Cox_Ross_Rubinstein_Tree(S, K, T, r, sigma,1000,'C'), Cox_Ross_Rubinstein_Tree(S, K, T, r, sigma,1000,'P'),\n",
    "                                     Jarrow_Rudd_Tree(S, K, T, r, sigma,1000,'C'), Jarrow_Rudd_Tree(S, K, T, r, sigma,1000,'P')]}\n",
    "binomial_model_pricing_frame = DataFrame(binomial_model_pricing, columns=[ 'Option', 'Price'], \n",
    "                   index = ['Cox-Ross-Rubinstein','Cox-Ross-Rubinstein', 'Jarrow-Rudd', 'Jarrow-Rudd'])                                        \n",
    "binomial_model_pricing_frame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Plot call / put options price with different steps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DdNso70gFucXlHDxSSX5JJcYYvG4XIW4X8VGh9OoeTmpsWP3e10qpE8YxAcHj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\nP7K+mTchNjyEO6cP4oaJGTz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kvaVSSp0OWg0IxpilQGHjw0C0fT8G2NfEdcYYU2I/9No3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\nKKWUAjQgKKWUsmlAUEopBcD/ByCiQKrOFNgsAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119d9eeb8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## call option with different steps \n",
    "runs1 = list(range(50,5000,50))\n",
    "CRR1 = []\n",
    "JR1 = []\n",
    "\n",
    "for i in runs1:\n",
    "    CRR1.append(Cox_Ross_Rubinstein_Tree(S, K, T, r, sigma,i ,'C'))\n",
    "    JR1.append(Jarrow_Rudd_Tree(S, K, T, r, sigma,i ,'C'))\n",
    "\n",
    "plt.plot(runs1, CRR1, label='Cox_Ross_Rubinstein')\n",
    "plt.plot(runs1, JR1, label='Jarrow_Rudd')\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UUiHddtttXTbx67vf/S6ffPJJvWN33HEHN954Y5eUxy5sEhB0cTt1ajLGtLpNvTuZN29e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\nlFIqQAOCUkopAP4fztNJykROBtUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1197ceda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## put option with different steps \n",
    "runs2 = list(range(50,5000,50))\n",
    "CRR2 = []\n",
    "JR2 = []\n",
    "\n",
    "for i in runs2:\n",
    "    CRR2.append(Cox_Ross_Rubinstein_Tree(S, K, T, r, sigma,i ,'P'))\n",
    "    JR2.append(Jarrow_Rudd_Tree(S, 110, T, r, sigma,i ,'P'))\n",
    "\n",
    "plt.plot(runs2, CRR2, label='Cox_Ross_Rubinstein')\n",
    "plt.plot(runs2, JR2, label='Jarrow_Rudd')\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}