Math623_Homework2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1625,
   "id": "9a56ae51-ed48-4b4d-b459-597e85a26650",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1626,
   "id": "dc358479-2d67-4866-9cc3-871670b8220f",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.set_printoptions(edgeitems=30, linewidth=100000, formatter=dict(float=lambda x: \"%.3g\" % x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bab42c1c-284b-42d8-96fa-d1597e49fcdd",
   "metadata": {},
   "source": [
    "## 1.1 ##"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1987,
   "id": "1c3aa6f4-dc36-4e7f-8c1c-e43591da08d5",
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma = 0.30\n",
    "a = 0.05\n",
    "deltat = 1/12\n",
    "V = sigma**2 * (1-np.exp(-2*a*deltat))/(2*a)\n",
    "deltau = np.sqrt(3*V)\n",
    "numMonths = 24\n",
    "P = lambda x: np.e**(-0.06 * x * deltat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1988,
   "id": "be3c1482-de63-4127-8a70-2c0a7c234026",
   "metadata": {},
   "outputs": [],
   "source": [
    "OU = np.zeros((numMonths*2 + 1,numMonths))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1989,
   "id": "d3105968-6f80-4ad6-88c3-27b4d759bd67",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(int(numMonths)):\n",
    "    index1 = int(numMonths - i - 1)\n",
    "    index2 = int(numMonths + i + 1)\n",
    "    OU[index1][:] = (i+1) * deltau\n",
    "    OU[index2][:] = -(i+1) * deltau"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42bc6865-f9d6-40e4-bd03-56d4b71c5527",
   "metadata": {},
   "source": [
    "## 1.2 ##"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1990,
   "id": "085c26c4-3993-4a8c-9197-c13f8a84c710",
   "metadata": {},
   "outputs": [],
   "source": [
    "intermediateParams = np.zeros((numMonths*2 + 1, 6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1991,
   "id": "638a3663-4782-465a-8355-e187452cd14f",
   "metadata": {},
   "outputs": [],
   "source": [
    "intermediateParams[:,0] = np.arange(numMonths, -numMonths-1,-1)\n",
    "intermediateParams[:,1] = np.arange(numMonths, -numMonths-1,-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1992,
   "id": "28ece8a5-f49f-4fa3-9fbc-5480320122d5",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    intermediateParams[i,2] = intermediateParams[i,0] * np.exp(-a * deltat) - math.floor(intermediateParams[i,0]*np.exp(-a * deltat) + 1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1993,
   "id": "dbea2a63-7dc6-4730-9900-19e18da42642",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    Beta = intermediateParams[i,2]\n",
    "    intermediateParams[i,3] = 1/6 + 1/2*(Beta**2 + Beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1994,
   "id": "41fdc07f-d1cb-4d44-984b-377a0008b98e",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    Beta = intermediateParams[i,2]\n",
    "    intermediateParams[i,4] = 2/3 - (Beta**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1995,
   "id": "a943a289-52b2-4bea-8fd6-19d047a16521",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    Beta = intermediateParams[i,2]\n",
    "    intermediateParams[i,5] = 1/6 + 1/2*(Beta**2 - Beta)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a26a3b7-04e1-4baf-8ad4-0cc19dd9d6f8",
   "metadata": {},
   "source": [
    "## 1.3/1.4 ##"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1996,
   "id": "c5226fc3-ceb3-4e8b-85c7-3b510d9dc4e9",
   "metadata": {},
   "outputs": [],
   "source": [
    "AD = np.zeros((numMonths*2 + 1,numMonths))\n",
    "AD[numMonths,0] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1997,
   "id": "3053963d-ad66-4acc-a433-15bd2cdc63f8",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "r = np.zeros((numMonths*2 + 1, numMonths))\n",
    "r[numMonths,0] = -np.log(P(1))/(1/12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1998,
   "id": "7a9844f2-f3cf-491a-876a-03e7bb0f6107",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "from sympy import symbols, Eq, solve, lambdify, nsolve\n",
    "\n",
    "for i in range(1,numMonths):\n",
    "#for i in range(1,2):\n",
    "    \n",
    "    r0 = symbols('r0')\n",
    "    P_sumAD = AD[numMonths,i-1]*np.e**(-r0*deltat)*intermediateParams[numMonths,4] + AD[numMonths+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+1,i-1])*deltat)*intermediateParams[numMonths,3] + AD[numMonths-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-1,i-1])*deltat)*intermediateParams[numMonths,5]\n",
    "    #-P(i) +\n",
    "    #update r values for one step ahead\n",
    "    for j in range(i):\n",
    "        # if i-j == 1\n",
    "        if i-j == 1:\n",
    "            P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "            P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        # if i-j == 2\n",
    "        if i-j == 2:\n",
    "            P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "            P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "            P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "            P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4]\n",
    "\n",
    "        # if i-j == 3\n",
    "        if i-j >= 3:\n",
    "            P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "            P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "            P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "            P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4] \n",
    "\n",
    "            P_sumAD += AD[numMonths-j-2,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-2,i-1])*deltat)*intermediateParams[numMonths-j-2,5]\n",
    "            P_sumAD += AD[numMonths+j+2,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+2,i-1])*deltat)*intermediateParams[numMonths+j+2,3]\n",
    "      \n",
    "    eq1 = Eq(P_sumAD,P(i))\n",
    "    solution = sympy.nsolve(eq1,0)\n",
    "    \n",
    "    #update ri(0)\n",
    "    r[numMonths, i-1] = solution                                             \n",
    "    #update r matrix based off ri(0) * exp(OUij)\n",
    "    for j in range(i-1):\n",
    "        r[numMonths + j + 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths + j + 1, i-1])\n",
    "        r[numMonths - j - 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths - j - 1, i-1])\n",
    "        \n",
    "    for j in range(i):\n",
    "        \n",
    "        Pd_down_iter = intermediateParams[numMonths + j,5]\n",
    "        Pm_down_iter = intermediateParams[numMonths + j,4]\n",
    "        Pu_down_iter = intermediateParams[numMonths + j,3]\n",
    "        \n",
    "        Pd_up_iter = intermediateParams[numMonths - j,5]\n",
    "        Pm_up_iter = intermediateParams[numMonths - j,4]\n",
    "        Pu_up_iter = intermediateParams[numMonths - j,3]\n",
    "        \n",
    "        #find one step forward Arrow-Debreu Prices\n",
    "        AD[numMonths - j - 1, i ] = abs(AD[numMonths - j , i - 1 ])*Pd_up_iter*np.exp(-r[numMonths - j, i - 1]*deltat) + abs(AD[numMonths - j - 1, i - 1 ])*Pm_up_iter*np.exp(-r[numMonths - j - 1, i - 1]*deltat) + abs(AD[numMonths - j - 2, i - 1 ])*Pu_up_iter*np.exp(-r[numMonths - j - 2, i - 1]*deltat)\n",
    "        AD[numMonths, i ] = abs(AD[numMonths + 1, i-1])*intermediateParams[numMonths,5]*np.exp(-r[numMonths + 1, i - 1]*deltat) + abs(AD[numMonths, i-1])*intermediateParams[numMonths,4]*np.exp(-r[numMonths , i - 1]*deltat) + abs(AD[numMonths +1, i-1])*intermediateParams[numMonths,3]*np.exp(-r[numMonths + 1, i - 1]*deltat)\n",
    "        AD[numMonths + j + 1, i ] = abs(AD[numMonths + j , i - 1 ])*Pd_down_iter*np.exp(-r[numMonths + j, i - 1]*deltat) + abs(AD[numMonths + j + 1, i - 1 ])*Pm_down_iter*np.exp(-r[numMonths + j + 1, i - 1]*deltat) + abs(AD[numMonths + j + 2, i - 1 ])*Pu_down_iter*np.exp(-r[numMonths + j + 2, i - 1]*deltat)   \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1999,
   "id": "df27216d-6f84-4685-8f51-3aa3221621b5",
   "metadata": {},
   "outputs": [],
   "source": [
    "i = 24\n",
    "r0 = symbols('r0')\n",
    "P_sumAD = AD[numMonths,i-1]*np.e**(-r0*deltat)*intermediateParams[numMonths,4] + AD[numMonths+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+1,i-1])*deltat)*intermediateParams[numMonths,3] + AD[numMonths-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-1,i-1])*deltat)*intermediateParams[numMonths,5]\n",
    "#-P(i) +\n",
    "#update r values for one step ahead\n",
    "for j in range(i):\n",
    "    # if i-j == 1\n",
    "    if i-j == 1:\n",
    "        P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "        P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        # if i-j == 2\n",
    "    if i-j == 2:\n",
    "        P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "        P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "        P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4]\n",
    "\n",
    "        # if i-j == 3\n",
    "    if i-j >= 3:\n",
    "        P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "        P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "        P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4] \n",
    "\n",
    "        P_sumAD += AD[numMonths-j-2,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-2,i-1])*deltat)*intermediateParams[numMonths-j-2,5]\n",
    "        P_sumAD += AD[numMonths+j+2,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+2,i-1])*deltat)*intermediateParams[numMonths+j+2,3]\n",
    "      \n",
    "eq1 = Eq(P_sumAD,P(i))\n",
    "solution = sympy.nsolve(eq1,0)\n",
    "    \n",
    "    #update ri(0)\n",
    "r[numMonths, i-1] = solution                                             \n",
    "    #update r matrix based off ri(0) * exp(OUij)\n",
    "for j in range(i-1):\n",
    "    r[numMonths + j + 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths + j + 1, i-1])\n",
    "    r[numMonths - j - 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths - j - 1, i-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53693578-f6a7-4bac-8f71-9b066cf5402c",
   "metadata": {
    "tags": []
   },
   "source": [
    "## 1.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2081,
   "id": "e739b4e3-d548-4c21-9ecf-8d46dcbd1768",
   "metadata": {},
   "outputs": [],
   "source": [
    "deltat = 1/4\n",
    "P = lambda x: np.e**(-0.06 * x * deltat)\n",
    "numMonths = 8 #This is actually number of quarters but I am not changing the variable name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2082,
   "id": "e11023a4-dbdf-455c-b4b6-5c7f533c1c14",
   "metadata": {},
   "outputs": [],
   "source": [
    "intermediateParams = np.zeros((numMonths*2 + 1, 6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2083,
   "id": "28fd89ae-07d9-43cc-932a-1a61733b4234",
   "metadata": {},
   "outputs": [],
   "source": [
    "intermediateParams[:,0] = np.arange(numMonths, -numMonths-1,-1)\n",
    "intermediateParams[:,1] = np.arange(numMonths, -numMonths-1,-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2084,
   "id": "079b30a5-00b3-4dac-a4bc-ba1e93e99b83",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    intermediateParams[i,2] = intermediateParams[i,0] * np.exp(-a * deltat) - math.floor(intermediateParams[i,0]*np.exp(-a * deltat) + 1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2085,
   "id": "98746708-c335-48a9-8a00-4a8f1f4eb80f",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    Beta = intermediateParams[i,2]\n",
    "    intermediateParams[i,3] = 1/6 + 1/2*(Beta**2 + Beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2086,
   "id": "09b1154f-e91b-4465-af4c-45544f4c36e8",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    Beta = intermediateParams[i,2]\n",
    "    intermediateParams[i,4] = 2/3 - (Beta**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2087,
   "id": "44c5f113-0eff-4816-b947-cba28bae06f6",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths*2 + 1):\n",
    "    Beta = intermediateParams[i,2]\n",
    "    intermediateParams[i,5] = 1/6 + 1/2*(Beta**2 - Beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2088,
   "id": "f311a858-c30a-43b7-b228-a7c8dab9bd67",
   "metadata": {},
   "outputs": [],
   "source": [
    "AD = np.zeros((numMonths*2 + 1,numMonths))\n",
    "AD[numMonths,0] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2089,
   "id": "682b4501-7620-4b70-b48f-23b5a2fcf805",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "r = np.zeros((numMonths*2 + 1, numMonths))\n",
    "r[numMonths,0] = -np.log(P(1))/(1/4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2090,
   "id": "ecad2521-d9a1-4447-9ea3-1e71e6828feb",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "from sympy import symbols, Eq, solve, lambdify, nsolve\n",
    "\n",
    "for i in range(1,numMonths):\n",
    "#for i in range(1,2):\n",
    "    \n",
    "    r0 = symbols('r0')\n",
    "    P_sumAD = AD[numMonths,i-1]*np.e**(-r0*deltat)*intermediateParams[numMonths,4] + AD[numMonths+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+1,i-1])*deltat)*intermediateParams[numMonths,3] + AD[numMonths-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-1,i-1])*deltat)*intermediateParams[numMonths,5]\n",
    "    #-P(i) +\n",
    "    #update r values for one step ahead\n",
    "    for j in range(i):\n",
    "        # if i-j == 1\n",
    "        if i-j == 1:\n",
    "            P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "            P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        # if i-j == 2\n",
    "        if i-j == 2:\n",
    "            P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "            P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "            P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "            P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4]\n",
    "\n",
    "        # if i-j == 3\n",
    "        if i-j >= 3:\n",
    "            P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "            P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "            P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "            P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4] \n",
    "\n",
    "            P_sumAD += AD[numMonths-j-2,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-2,i-1])*deltat)*intermediateParams[numMonths-j-2,5]\n",
    "            P_sumAD += AD[numMonths+j+2,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+2,i-1])*deltat)*intermediateParams[numMonths+j+2,3]\n",
    "      \n",
    "    eq1 = Eq(P_sumAD,P(i))\n",
    "    solution = sympy.nsolve(eq1,0)\n",
    "    \n",
    "    #update ri(0)\n",
    "    r[numMonths, i-1] = solution                                             \n",
    "    #update r matrix based off ri(0) * exp(OUij)\n",
    "    for j in range(i-1):\n",
    "        r[numMonths + j + 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths + j + 1, i-1])\n",
    "        r[numMonths - j - 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths - j - 1, i-1])\n",
    "        \n",
    "    for j in range(i):\n",
    "        \n",
    "        Pd_down_iter = intermediateParams[numMonths + j,5]\n",
    "        Pm_down_iter = intermediateParams[numMonths + j,4]\n",
    "        Pu_down_iter = intermediateParams[numMonths + j,3]\n",
    "        \n",
    "        Pd_up_iter = intermediateParams[numMonths - j,5]\n",
    "        Pm_up_iter = intermediateParams[numMonths - j,4]\n",
    "        Pu_up_iter = intermediateParams[numMonths - j,3]\n",
    "        \n",
    "        #find one step forward Arrow-Debreu Prices\n",
    "        AD[numMonths - j - 1, i ] = abs(AD[numMonths - j , i - 1 ])*Pd_up_iter*np.exp(-r[numMonths - j, i - 1]*deltat) + abs(AD[numMonths - j - 1, i - 1 ])*Pm_up_iter*np.exp(-r[numMonths - j - 1, i - 1]*deltat) + abs(AD[numMonths - j - 2, i - 1 ])*Pu_up_iter*np.exp(-r[numMonths - j - 2, i - 1]*deltat)\n",
    "        AD[numMonths, i ] = abs(AD[numMonths + 1, i-1])*intermediateParams[numMonths,5]*np.exp(-r[numMonths + 1, i - 1]*deltat) + abs(AD[numMonths, i-1])*intermediateParams[numMonths,4]*np.exp(-r[numMonths , i - 1]*deltat) + abs(AD[numMonths +1, i-1])*intermediateParams[numMonths,3]*np.exp(-r[numMonths + 1, i - 1]*deltat)\n",
    "        AD[numMonths + j + 1, i ] = abs(AD[numMonths + j , i - 1 ])*Pd_down_iter*np.exp(-r[numMonths + j, i - 1]*deltat) + abs(AD[numMonths + j + 1, i - 1 ])*Pm_down_iter*np.exp(-r[numMonths + j + 1, i - 1]*deltat) + abs(AD[numMonths + j + 2, i - 1 ])*Pu_down_iter*np.exp(-r[numMonths + j + 2, i - 1]*deltat)   \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2091,
   "id": "602818e8-b9f1-4177-874a-cdb1d6b3ce53",
   "metadata": {},
   "outputs": [],
   "source": [
    "i = 8\n",
    "r0 = symbols('r0')\n",
    "P_sumAD = AD[numMonths,i-1]*np.e**(-r0*deltat)*intermediateParams[numMonths,4] + AD[numMonths+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+1,i-1])*deltat)*intermediateParams[numMonths,3] + AD[numMonths-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-1,i-1])*deltat)*intermediateParams[numMonths,5]\n",
    "#-P(i) +\n",
    "#update r values for one step ahead\n",
    "for j in range(i):\n",
    "    # if i-j == 1\n",
    "    if i-j == 1:\n",
    "        P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "        P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        # if i-j == 2\n",
    "    if i-j == 2:\n",
    "        P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "        P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "        P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4]\n",
    "\n",
    "        # if i-j == 3\n",
    "    if i-j >= 3:\n",
    "        P_sumAD += AD[numMonths-j,i-1]*np.e**(-r0*np.e**(OU[numMonths-j,i-1])*deltat)*intermediateParams[numMonths-j,3]\n",
    "        P_sumAD += AD[numMonths+j,i-1]*np.e**(-r0*np.e**(OU[numMonths+j,i-1])*deltat)*intermediateParams[numMonths+j,5]\n",
    "\n",
    "        P_sumAD += AD[numMonths-j-1,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-1,i-1])*deltat)*intermediateParams[numMonths-j-1,4]\n",
    "        P_sumAD += AD[numMonths+j+1,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+1,i-1])*deltat)*intermediateParams[numMonths+j+1,4] \n",
    "\n",
    "        P_sumAD += AD[numMonths-j-2,i-1]*np.e**(-r0*np.e**(OU[numMonths-j-2,i-1])*deltat)*intermediateParams[numMonths-j-2,5]\n",
    "        P_sumAD += AD[numMonths+j+2,i-1]*np.e**(-r0*np.e**(OU[numMonths+j+2,i-1])*deltat)*intermediateParams[numMonths+j+2,3]\n",
    "      \n",
    "eq1 = Eq(P_sumAD,P(i))\n",
    "solution = sympy.nsolve(eq1,0)\n",
    "    \n",
    "    #update ri(0)\n",
    "r[numMonths, i-1] = solution                                             \n",
    "    #update r matrix based off ri(0) * exp(OUij)\n",
    "for j in range(i-1):\n",
    "    r[numMonths + j + 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths + j + 1, i-1])\n",
    "    r[numMonths - j - 1 , i-1 ] = r[numMonths, i-1] * np.exp(OU[numMonths - j - 1, i-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2092,
   "id": "ade4f7c6-d02d-4626-bdcd-3f9d86446c5c",
   "metadata": {},
   "outputs": [],
   "source": [
    "#fixed rate bond\n",
    "T = 24\n",
    "Principal = 100\n",
    "quarter = 1/4\n",
    "coupon = 0.06"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2093,
   "id": "1379e280-e6ae-43e8-a355-8bae6859e237",
   "metadata": {},
   "outputs": [],
   "source": [
    "fixedRate = np.zeros(AD.shape,dtype = float)\n",
    "\n",
    "for i in range(fixedRate.shape[0]):\n",
    "    fixedRate[i,-1] = Principal + Principal*coupon*quarter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2094,
   "id": "2966de5e-fc03-4ba5-a816-c8d702287566",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths-1):\n",
    "    fixedRate[numMonths,numMonths-i-2] = fixedRate[numMonths,numMonths-1-i]*intermediateParams[numMonths,4]*np.e**(-r[numMonths,numMonths-i-2]*quarter) + fixedRate[numMonths+1,numMonths-1-i]*intermediateParams[numMonths,5]*np.e**(-r[numMonths,numMonths-i-2]*quarter) + fixedRate[numMonths-1,numMonths-1-i]*intermediateParams[numMonths,3]*np.e**(-r[numMonths,numMonths-i-2]*quarter) + 1.5\n",
    "    for j in range(numMonths-i-1):\n",
    "        fixedRate[numMonths - j -1,numMonths-i-2] = fixedRate[numMonths-j-1,numMonths-1-i]*intermediateParams[numMonths,4]*np.e**(-r[numMonths-j-1,numMonths-i-2]*quarter) + fixedRate[numMonths-j-2,numMonths-1-i]*intermediateParams[numMonths,3]*np.e**(-r[numMonths-j-1,numMonths-i-2]*quarter) + fixedRate[numMonths-j,numMonths-1-i]*intermediateParams[numMonths,5]*np.e**(-r[numMonths-j-1,numMonths-i-2]*quarter) + 1.5\n",
    "        fixedRate[numMonths + j +1,numMonths-i-2] = fixedRate[numMonths+j+1,numMonths-1-i]*intermediateParams[numMonths,4]*np.e**(-r[numMonths+j+1,numMonths-i-2]*quarter) + fixedRate[numMonths+j+2,numMonths-1-i]*intermediateParams[numMonths,5]*np.e**(-r[numMonths+j+1,numMonths-i-2]*quarter) + fixedRate[numMonths+j,numMonths-1-i]*intermediateParams[numMonths,3]*np.e**(-r[numMonths+j+1,numMonths-i-2]*quarter) + 1.5\n",
    "              "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2095,
   "id": "fcb097c8-d4d9-42b4-b9e2-3c4bafb7d2a3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "101.65079932028296"
      ]
     },
     "execution_count": 2095,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fixedRate[numMonths,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2096,
   "id": "bc24c531-2acb-4eb1-a5b5-b64c763ec44b",
   "metadata": {},
   "outputs": [],
   "source": [
    "r = r[1:-1,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2118,
   "id": "9eef0f1f-8224-4af0-80c6-51624aab0c76",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Cap Agreement\n",
    "T = 24\n",
    "Principal = 100\n",
    "Rk = 0.07"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2119,
   "id": "bda5f2d2-b81d-4d9f-ac84-d6b2046022f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "Cf = np.zeros((15,8),dtype = float)\n",
    "\n",
    "for i in range(Cf.shape[0]):\n",
    "    Cf[i,-1] = 100 + Principal * quarter * max(r[i,-1]-Rk,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2123,
   "id": "18f58fc8-e45d-431e-89d3-2f56993d4857",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(numMonths-1):\n",
    "    Cf[numMonths-1,numMonths-i-2] = Cf[numMonths-1,numMonths-1-i]*intermediateParams[numMonths,4]*np.e**(-r[numMonths-1,numMonths-i-2]*quarter) + Cf[numMonths,numMonths-1-i]*intermediateParams[numMonths,5]*np.e**(-r[numMonths-1,numMonths-i-2]*quarter) + Cf[numMonths-2,numMonths-1-i]*intermediateParams[numMonths,3]*np.e**(-r[numMonths-1,numMonths-i-2]*quarter) + Principal*quarter*max(r[numMonths-1,numMonths-i-2],0)\n",
    "    for j in range(numMonths-i-2):\n",
    "        Cf[numMonths - j -2,numMonths-i-2] = Cf[numMonths-j-2,numMonths-1-i]*intermediateParams[numMonths,4]*np.e**(-r[numMonths-j-3,numMonths-i-2]*quarter) + Cf[numMonths-j-3,numMonths-1-i]*intermediateParams[numMonths,3]*np.e**(-r[numMonths-j-2,numMonths-i-2]*quarter) + Cf[numMonths-j-1,numMonths-1-i]*intermediateParams[numMonths,5]*np.e**(-r[numMonths-j-2,numMonths-i-2]*quarter) + Principal*quarter*max(r[numMonths-j-2,numMonths-i-2],0)\n",
    "        Cf[numMonths + j,numMonths-i-2] = Cf[numMonths+j,numMonths-1-i]*intermediateParams[numMonths,4]*np.e**(-r[numMonths+j,numMonths-i-2]*quarter) + Cf[numMonths+j+1,numMonths-1-i]*intermediateParams[numMonths,5]*np.e**(-r[numMonths+j,numMonths-i-2]*quarter) + Cf[numMonths+j-1,numMonths-1-i]*intermediateParams[numMonths,3]*np.e**(-r[numMonths+j,numMonths-i-2]*quarter) + Principal*quarter*max(r[numMonths+j,numMonths-i-2],0)\n",
    "              "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2125,
   "id": "2b8b2b8f-1cb2-475c-9c02-5009dfe11a98",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100.56374970118692"
      ]
     },
     "execution_count": 2125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Cf[numMonths-1,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f69464b-cb58-4dc4-9183-84131340d7b4",
   "metadata": {},
   "source": [
    "## 2.b ##"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1762,
   "id": "ba19cbcf-d597-4a36-ae57-546b22dc5cec",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols , Eq, solve, diff, sqrt, simplify, evalf, log, exp\n",
    "k = symbols('k')\n",
    "T = symbols('T')\n",
    "t = symbols('t')\n",
    "h = sqrt(k**2 + 2*s**2)\n",
    "o = symbols('o')\n",
    "s = symbols('s')\n",
    "r = symbols('r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1763,
   "id": "8a8134ce-5772-4e89-8e6a-03b8cadc56aa",
   "metadata": {},
   "outputs": [],
   "source": [
    "A = ((2*h * exp((T-t)*(k+h)/2))/(2*h + (k+h)*(exp((T-t)*h)-1)))**(2*k*o/(s**2))\n",
    "A_prime = diff(A,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1764,
   "id": "43d7645d-621a-427c-9eca-936307ec1abb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2 \\sqrt{k^{2} + 2 s^{2}} e^{\\frac{\\left(T - t\\right) \\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right)}{2}}}{\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}}\\right)^{\\frac{2 k o}{s^{2}}}$"
      ],
      "text/plain": [
       "(2*sqrt(k**2 + 2*s**2)*exp((T - t)*(k + sqrt(k**2 + 2*s**2))/2)/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2)))**(2*k*o/s**2)"
      ]
     },
     "execution_count": 1764,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1765,
   "id": "22ff124c-d868-402e-a777-ba3bcc4ed3b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{k o \\left(\\frac{2 \\sqrt{k^{2} + 2 s^{2}} e^{\\frac{\\left(T - t\\right) \\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right)}{2}}}{\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}}\\right)^{\\frac{2 k o}{s^{2}}} \\left(\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(\\frac{2 \\left(- \\frac{k}{2} - \\frac{\\sqrt{k^{2} + 2 s^{2}}}{2}\\right) \\sqrt{k^{2} + 2 s^{2}} e^{\\frac{\\left(T - t\\right) \\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right)}{2}}}{\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}} + \\frac{2 \\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(k^{2} + 2 s^{2}\\right) e^{\\frac{\\left(T - t\\right) \\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right)}{2}} e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}}}{\\left(\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}\\right)^{2}}\\right) e^{- \\frac{\\left(T - t\\right) \\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right)}{2}}}{s^{2} \\sqrt{k^{2} + 2 s^{2}}}$"
      ],
      "text/plain": [
       "k*o*(2*sqrt(k**2 + 2*s**2)*exp((T - t)*(k + sqrt(k**2 + 2*s**2))/2)/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2)))**(2*k*o/s**2)*((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2))*(2*(-k/2 - sqrt(k**2 + 2*s**2)/2)*sqrt(k**2 + 2*s**2)*exp((T - t)*(k + sqrt(k**2 + 2*s**2))/2)/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2)) + 2*(k + sqrt(k**2 + 2*s**2))*(k**2 + 2*s**2)*exp((T - t)*(k + sqrt(k**2 + 2*s**2))/2)*exp((T - t)*sqrt(k**2 + 2*s**2))/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2))**2)*exp(-(T - t)*(k + sqrt(k**2 + 2*s**2))/2)/(s**2*sqrt(k**2 + 2*s**2))"
      ]
     },
     "execution_count": 1765,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A_prime)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1766,
   "id": "81d5ceed-2463-4181-bb14-c193c7c13746",
   "metadata": {},
   "outputs": [],
   "source": [
    "B = (2 * (exp((T-t)*h)-1)) / (2*h + (k + h)*(exp((T-t)*h) - 1))\n",
    "B_prime = diff(B,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1767,
   "id": "eed4b904-becf-467e-97fd-dea848e4672b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 2}{\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}}$"
      ],
      "text/plain": [
       "(2*exp((T - t)*sqrt(k**2 + 2*s**2)) - 2)/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2))"
      ]
     },
     "execution_count": 1767,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1768,
   "id": "1bf78902-7273-4312-a24d-ade3817eb268",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\sqrt{k^{2} + 2 s^{2}} \\left(2 e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 2\\right) e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}}}{\\left(\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}\\right)^{2}} - \\frac{2 \\sqrt{k^{2} + 2 s^{2}} e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}}}{\\left(k + \\sqrt{k^{2} + 2 s^{2}}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 2 s^{2}}} - 1\\right) + 2 \\sqrt{k^{2} + 2 s^{2}}}$"
      ],
      "text/plain": [
       "(k + sqrt(k**2 + 2*s**2))*sqrt(k**2 + 2*s**2)*(2*exp((T - t)*sqrt(k**2 + 2*s**2)) - 2)*exp((T - t)*sqrt(k**2 + 2*s**2))/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2))**2 - 2*sqrt(k**2 + 2*s**2)*exp((T - t)*sqrt(k**2 + 2*s**2))/((k + sqrt(k**2 + 2*s**2))*(exp((T - t)*sqrt(k**2 + 2*s**2)) - 1) + 2*sqrt(k**2 + 2*s**2))"
      ]
     },
     "execution_count": 1768,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1769,
   "id": "d1c11ff1-7208-4094-b571-812fa77bac11",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "print(simplify(A_prime - k*o*B*A) == 0)\n",
    "print(simplify(B_prime + 1 - (1/2*(s**2)*B**2 + k*B))==0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1770,
   "id": "197e8c95-91ab-4c53-b7af-568d9d16aa26",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0\n"
     ]
    }
   ],
   "source": [
    "#terminal condition on A\n",
    "convert = lambdify( ['k','T','t','o','s','r'], A)\n",
    "sol_r = convert(1,30,30,0.03,0.1,0.01)\n",
    "print(sol_r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1771,
   "id": "5aee487a-ef5d-44eb-8fae-26300c8deb5f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
     ]
    }
   ],
   "source": [
    "#terminal condition on B\n",
    "convert = lambdify( ['k','T','t','o','s','r'], -B_prime + 1/2*(s**2)*(B**2) + k*B - 1)\n",
    "sol_r = convert(1,0,0,0.03,0.1,0.01)\n",
    "print(sol_r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1756,
   "id": "12bffcdf-56c7-40df-85ef-4973d14a3666",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(\\left(\\frac{2 \\sqrt{k^{2} + 0.02} e^{\\frac{\\left(T - t\\right) \\left(k + \\sqrt{k^{2} + 0.02}\\right)}{2}}}{\\left(k + \\sqrt{k^{2} + 0.02}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 0.02}} - 1\\right) + 2 \\sqrt{k^{2} + 0.02}}\\right)^{\\frac{2 k o}{s^{2}}} e^{- \\frac{r \\left(2 e^{\\left(T - t\\right) \\sqrt{k^{2} + 0.02}} - 2\\right)}{\\left(k + \\sqrt{k^{2} + 0.02}\\right) \\left(e^{\\left(T - t\\right) \\sqrt{k^{2} + 0.02}} - 1\\right) + 2 \\sqrt{k^{2} + 0.02}}} \\right)}$"
      ],
      "text/plain": [
       "log((2*sqrt(k**2 + 0.02)*exp((T - t)*(k + sqrt(k**2 + 0.02))/2)/((k + sqrt(k**2 + 0.02))*(exp((T - t)*sqrt(k**2 + 0.02)) - 1) + 2*sqrt(k**2 + 0.02)))**(2*k*o/s**2)*exp(-r*(2*exp((T - t)*sqrt(k**2 + 0.02)) - 2)/((k + sqrt(k**2 + 0.02))*(exp((T - t)*sqrt(k**2 + 0.02)) - 1) + 2*sqrt(k**2 + 0.02))))"
      ]
     },
     "execution_count": 1756,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "forwardRate = log(A*exp(-B*r))\n",
    "forwardRate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1757,
   "id": "75b8f007-6603-47aa-8559-afd22bbee8cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "instantFwdRate = -diff(forwardRate,T)\n",
    "simplify(instantFwdRate)\n",
    "\n",
    "instantFwdRateZero = instantFwdRate.subs('t',0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1758,
   "id": "81294451-9604-484b-8cfc-d94ecd04202e",
   "metadata": {},
   "outputs": [],
   "source": [
    "convert = lambdify( ['k','T','o','s','r'], instantFwdRateZero)\n",
    "\n",
    "storedValues = []\n",
    "x = []\n",
    "for i in range(31):\n",
    "    x.append(i)\n",
    "    storedValues.append(convert(1,i,0.03,0.1,0.01))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1759,
   "id": "261dfda8-38ee-44ae-b305-ca1142eb36f4",
   "metadata": {},
   "outputs": [],
   "source": [
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1760,
   "id": "8592ff68-4bad-41b3-8a09-f5a27bef4fd9",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/sethkelley/opt/anaconda3/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = sns.lineplot(x,storedValues)\n",
    "ax.set(xlabel = 'Time', ylabel = 'Price of instantaneous forward rate',title = \"Price of Instantaneous Fwd Rate vs. Time\")\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "181553f1-706f-4355-acf1-ddba3e5fa0e6",
   "metadata": {},
   "source": [
    "## 2.c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 720,
   "id": "7f9ca589-98bd-43c5-9fc6-8a939ede1147",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 30\n",
    "N = 1000\n",
    "terminalA = 1\n",
    "terminalB = 0\n",
    "dt = 30/1000\n",
    "k = 1 \n",
    "o = 0.03\n",
    "s = 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 721,
   "id": "69534775-3723-4517-9b9b-570f0786d55d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The estimated value of A(0,T) is:  0.421\n",
      "The estimated value of B(0,T) is:  0.995\n"
     ]
    }
   ],
   "source": [
    "A = []\n",
    "B = []\n",
    "\n",
    "A.append(terminalA)\n",
    "B.append(terminalB)\n",
    "\n",
    "for i in range(N):\n",
    "    \n",
    "    insertA = A[0] - dt*(k*o*A[0]*B[0])\n",
    "    insertB = B[0] + dt - (1/2*(s**2)*(B[0]**2)+k*B[0])*dt\n",
    "    \n",
    "    A.insert(0,insertA)\n",
    "    B.insert(0,insertB)\n",
    "    \n",
    "print(\"The estimated value of A(0,T) is: \" ,np.round(A[0],3)) \n",
    "print(\"The estimated value of B(0,T) is: \" ,np.round(B[0],3))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 667,
   "id": "92dcf86a-bf16-45da-a987-2170cdae0539",
   "metadata": {},
   "outputs": [],
   "source": [
    "Asym = ((2*h * exp((T-t)*(k+h)/2))/(2*h + (k+h)*(exp((T-t)*h)-1)))**(2*k*o/(s**2))\n",
    "Bsym = (2 * (exp((T-t)*h)-1)) / (2*h + (k + h)*(exp((T-t)*h) - 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 668,
   "id": "3ea54d7d-e360-40c1-b42e-b293c71e12e1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The actual value of A(o,T) is:  0.421\n"
     ]
    }
   ],
   "source": [
    "#terminal condition on A\n",
    "convert = lambdify( ['k','T','t','o','s','r'], Asym)\n",
    "A_zero = convert(1,30,0,0.03,0.1,0.01)\n",
    "print(\"The actual value of A(o,T) is: \",np.round(A_zero,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 669,
   "id": "cff35e7a-5e06-427a-a678-5e6fb754d058",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The actual value of B(o,T) is:  0.995\n"
     ]
    }
   ],
   "source": [
    "#terminal condition on B\n",
    "convert = lambdify( ['k','T','t','o','s','r'], Bsym)\n",
    "B_zero = convert(1,30,0,0.03,0.1,0.01)\n",
    "print(\"The actual value of B(o,T) is: \",np.round(B_zero,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 670,
   "id": "1eb03ac7-1042-4005-9dfd-709628804450",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The error on A is:  0.00038023432859054894\n",
      "The error on A is:  2.5215874477736136e-14\n"
     ]
    }
   ],
   "source": [
    "print(\"The error on A is: \",np.abs(A_zero-A[0])/A_zero)\n",
    "print(\"The error on A is: \",np.abs(B_zero-B[0])/B_zero)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ea527cf-2929-4cd0-8dab-f66ab2e5bfb4",
   "metadata": {
    "tags": []
   },
   "source": [
    "## 2.d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 671,
   "id": "7beef4d5-cdb8-4441-af87-eac4b0bb2fff",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 30\n",
    "N = 1000\n",
    "terminalA = 1\n",
    "terminalB = 0\n",
    "dt = 30/1000\n",
    "k = 1 \n",
    "o = 0.03\n",
    "s = 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 672,
   "id": "d7a38a3f-3b8e-408e-97df-f061ad77a8ff",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The estimated value of A(0,T) is:  0.421\n",
      "The estimated value of B(0,T) is:  0.995\n"
     ]
    }
   ],
   "source": [
    "A = []\n",
    "B = []\n",
    "\n",
    "A.append(terminalA)\n",
    "B.append(terminalB)\n",
    "\n",
    "for i in range(N):\n",
    "    \n",
    "    insertB = (-(1 + k*dt) + np.sqrt((1 + k*dt)**2 + 4*(1/2 * dt * (s**2))*(B[0]+dt)))/(dt*(s**2))\n",
    "    insertA = A[0]/(1 + dt*k*o*insertB)\n",
    "    \n",
    "    A.insert(0,insertA)\n",
    "    B.insert(0,insertB)\n",
    "    \n",
    "print(\"The estimated value of A(0,T) is: \" ,np.round(A[0],3)) \n",
    "print(\"The estimated value of B(0,T) is: \" ,np.round(B[0],3))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 673,
   "id": "aab9054c-3d75-499b-8907-3b9d821e1deb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The error on A is:  0.0003797631052428798\n",
      "The error on A is:  1.4015005349658912e-11\n"
     ]
    }
   ],
   "source": [
    "print(\"The error on A is: \",np.abs(A_zero-A[0])/A_zero)\n",
    "print(\"The error on A is: \",np.abs(B_zero-B[0])/B_zero)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8627ab25-4176-459a-bc5b-020045a80681",
   "metadata": {
    "tags": []
   },
   "source": [
    "## 2.e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ff87fb6e-fbbc-4f76-8e1b-ae9d5d7a95f9",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 791,
   "id": "514857a9-9c2b-40e1-894a-1221b4f1ca11",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 30\n",
    "k = 12\n",
    "terminalA = 1\n",
    "terminalB = 0\n",
    "o = 0.03\n",
    "k=1\n",
    "s = 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 792,
   "id": "1ca26f2e-437d-4a74-86d4-8e480ab97d89",
   "metadata": {},
   "outputs": [],
   "source": [
    "Ak = []\n",
    "Bk = []\n",
    "Akval = []\n",
    "Bkval = []\n",
    "\n",
    "for j in range(1,16):\n",
    "    A = []\n",
    "    B = []\n",
    "\n",
    "    A.append(terminalA)\n",
    "    B.append(terminalB)\n",
    "    \n",
    "    insertA = 0 \n",
    "    insertB = 0\n",
    "    \n",
    "    N = 2**j\n",
    "    dt = T/N\n",
    "\n",
    "    for i in range(N):\n",
    "        insertA = A[0] - dt*(k*o*A[0]*B[0])\n",
    "        insertB = B[0] + dt - (1/2*(s**2)*(B[0]**2)+k*B[0])*dt\n",
    "\n",
    "        A.insert(0,insertA)\n",
    "        B.insert(0,insertB)\n",
    "        \n",
    "    Akval.append(A[0])\n",
    "    Bkval.append(B[0])\n",
    "    Ak.append(\"for k = {}, A[0,T] is {}\".format(j,np.round(A[0],5)))\n",
    "    Bk.append(\"for k = {}, B[0,T] is {}\".format(j,np.round(B[0],5)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 793,
   "id": "c4d3760a-7762-43f2-a302-2760580bb3cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['for k = 1, A[0,T] is -5.75',\n",
       " 'for k = 2, A[0,T] is 356.70171',\n",
       " 'for k = 3, A[0,T] is 2379830.05181',\n",
       " 'for k = 4, A[0,T] is 0.40559',\n",
       " 'for k = 5, A[0,T] is 0.41554',\n",
       " 'for k = 6, A[0,T] is 0.41814',\n",
       " 'for k = 7, A[0,T] is 0.41941',\n",
       " 'for k = 8, A[0,T] is 0.42004',\n",
       " 'for k = 9, A[0,T] is 0.42035',\n",
       " 'for k = 10, A[0,T] is 0.42051',\n",
       " 'for k = 11, A[0,T] is 0.42059',\n",
       " 'for k = 12, A[0,T] is 0.42063',\n",
       " 'for k = 13, A[0,T] is 0.42065',\n",
       " 'for k = 14, A[0,T] is 0.42066',\n",
       " 'for k = 15, A[0,T] is 0.42066']"
      ]
     },
     "execution_count": 793,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ak"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 794,
   "id": "4b9dcadc-954d-48df-9231-c4da514dc81b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['for k = 1, B[0,T] is -211.875',\n",
       " 'for k = 2, B[0,T] is -3210.74696',\n",
       " 'for k = 3, B[0,T] is -159029.9237',\n",
       " 'for k = 4, B[0,T] is 0.83123',\n",
       " 'for k = 5, B[0,T] is 0.99505',\n",
       " 'for k = 6, B[0,T] is 0.99505',\n",
       " 'for k = 7, B[0,T] is 0.99505',\n",
       " 'for k = 8, B[0,T] is 0.99505',\n",
       " 'for k = 9, B[0,T] is 0.99505',\n",
       " 'for k = 10, B[0,T] is 0.99505',\n",
       " 'for k = 11, B[0,T] is 0.99505',\n",
       " 'for k = 12, B[0,T] is 0.99505',\n",
       " 'for k = 13, B[0,T] is 0.99505',\n",
       " 'for k = 14, B[0,T] is 0.99505',\n",
       " 'for k = 15, B[0,T] is 0.99505']"
      ]
     },
     "execution_count": 794,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bk"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 806,
   "id": "aa86ceda-7151-414e-a58e-da3a17380490",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.73, 11.7, 21.2, -2.6, -7.61, -8.63, -9.64, -10.6, -11.6, -12.6, -13.6, -14.6, -15.6, -16.6, -17.6])"
      ]
     },
     "execution_count": 806,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logError = 1/np.log(2) * np.log(np.sqrt((Akval-A_zero)**2 + (Bkval-B_zero)**2))\n",
    "logError"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 810,
   "id": "53f5e415-37e2-43ed-b3ba-5a4f13cdd68d",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/sethkelley/opt/anaconda3/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax2 = sns.lineplot([i for i in range(1,16)],logError)\n",
    "ax2.set(xlabel = 'k', ylabel = 'Logarithmic Error',title = \"Logarithmic Error vs. k\")\n",
    "ax2.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 811,
   "id": "f5aae151-c00f-4f29-b1c2-f9d5593343ca",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/sethkelley/opt/anaconda3/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax3 = sns.lineplot([2**i for i in range(1,16)],np.sqrt((Akval-A_zero)**2 + (Bkval-B_zero)**2))\n",
    "ax3.set(xlabel = 'N', ylabel = 'Error',title = \"Error vs. N\")\n",
    "ax3.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 824,
   "id": "a81810cc-7fb4-4de6-be77-43eb72d8ed8d",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/sethkelley/opt/anaconda3/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "test = np.sqrt((Akval-A_zero)**2 + (Bkval-B_zero)**2)\n",
    "ax4 = sns.lineplot([2**i for i in range(10,16)],test[9:])\n",
    "ax4.set(xlabel = 'N', ylabel = 'Error',title = \"Error vs. N\")\n",
    "ax4.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "893eec35-f981-4017-a1ec-6b77fb9792c7",
   "metadata": {},
   "source": [
    "The local truncation error of backward Euler Method is O(h^2). This seems to be confirmed by what we see when we analyze the graph of Error vs. N."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43e76cbb-06b1-459b-882c-19bc5a734756",
   "metadata": {},
   "source": [
    "## 2.f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 882,
   "id": "57e0acdb-f29c-42f6-976c-9abfbda1a271",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 30\n",
    "N = 1000\n",
    "terminalA = 1\n",
    "terminalB = 0\n",
    "dt = 30/1000\n",
    "k = 1\n",
    "t = np.arange(0,30,30/1000)\n",
    "st = 0.1 + 0.05*np.sin(2*np.pi*t/T)\n",
    "ot = 0.02 + 0.06*t/30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 883,
   "id": "750023ad-1331-41c2-84c5-1f0da3af7d7d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The estimated value of A(0,T) is:  0.243\n",
      "The estimated value of B(0,T) is:  0.994\n"
     ]
    }
   ],
   "source": [
    "A = []\n",
    "B = []\n",
    "\n",
    "A.append(terminalA)\n",
    "B.append(terminalB)\n",
    "\n",
    "for i in range(N):\n",
    "    \n",
    "    s = st[N-i-1]\n",
    "    o = ot[N-i-1]\n",
    "    \n",
    "    insertA = A[0] - dt*(k*o*A[0]*B[0])\n",
    "    insertB = B[0] + dt - (1/2*(s**2)*(B[0]**2)+k*B[0])*dt\n",
    "    \n",
    "    A.insert(0,insertA)\n",
    "    B.insert(0,insertB)\n",
    "    \n",
    "print(\"The estimated value of A(0,T) is: \" ,np.round(A[0],3)) \n",
    "print(\"The estimated value of B(0,T) is: \" ,np.round(B[0],3))  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9cdbf9a-618b-4a46-8a53-ecfaed61283f",
   "metadata": {},
   "source": [
    "## 2.g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 889,
   "id": "0fb29ca6-3f4d-4f80-a9dc-731b6416a843",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = np.array([dt*i for i in range(1,1001)])\n",
    "N = 1000\n",
    "terminalA = 1\n",
    "terminalB = 0\n",
    "dt = 30/1000\n",
    "o = 0.03\n",
    "k=1\n",
    "s = 0.1\n",
    "r0 = 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 895,
   "id": "4ab903d8-4fbd-4f77-a971-ef9c956392b1",
   "metadata": {},
   "outputs": [],
   "source": [
    "Ak = []\n",
    "Bk = []\n",
    "Pk = []\n",
    "\n",
    "Akval = []\n",
    "Bkval = []\n",
    "Pkval = []\n",
    "\n",
    "diffLogP = []\n",
    "\n",
    "for maturity in T:\n",
    "    A = []\n",
    "    B = []\n",
    "    P = []\n",
    "    \n",
    "    A.append(terminalA)\n",
    "    B.append(terminalB)\n",
    "    P.append(terminalA * np.exp(-terminalB*r0))\n",
    "    \n",
    "    for i in range(int(maturity/(30/1000))):\n",
    "        insertA = A[0] - dt*(k*o*A[0]*B[0])\n",
    "        insertB = B[0] + dt - (1/2*(s**2)*(B[0]**2)+k*B[0])*dt\n",
    "        \n",
    "        P.insert(0,insertA*np.exp(-insertB * r0))     \n",
    "        A.insert(0,insertA)\n",
    "        B.insert(0,insertB)\n",
    "    \n",
    "    for e in range(len(P)-1):\n",
    "        diffLogP.insert(0,(np.log(P[len(P)-1-e]) - np.log(P[len(P)-e-2]))/dt)\n",
    "    \n",
    "    Pkval.append(diffLogP[0])\n",
    "    Akval.append(A[0])\n",
    "    Bkval.append(B[0])\n",
    "    Ak.append(\"for T = {}*dt, A[0,{}*dt] is {}\".format(int(maturity/(30/1000)),int(maturity/(30/1000)),np.round(A[0],5)))\n",
    "    Bk.append(\"for k = {}, B[0,{}*dt] is {}\".format(int(maturity/(30/1000)),int(maturity/(30/1000)),np.round(B[0],5)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 891,
   "id": "574aab79-6081-4e1b-8796-d481ae4e5294",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(['for T = 1*dt, A[0,1*dt] is 1.0',\n",
       "  'for T = 2*dt, A[0,2*dt] is 0.99997',\n",
       "  'for T = 3*dt, A[0,3*dt] is 0.99992',\n",
       "  'for T = 4*dt, A[0,4*dt] is 0.99984',\n",
       "  'for T = 5*dt, A[0,5*dt] is 0.99974',\n",
       "  'for T = 6*dt, A[0,6*dt] is 0.99961',\n",
       "  'for T = 7*dt, A[0,7*dt] is 0.99946',\n",
       "  'for T = 8*dt, A[0,8*dt] is 0.99929',\n",
       "  'for T = 9*dt, A[0,9*dt] is 0.99909',\n",
       "  'for T = 10*dt, A[0,10*dt] is 0.99888'],\n",
       " ['for T = 991*dt, A[0,991*dt] is 0.42391',\n",
       "  'for T = 992*dt, A[0,992*dt] is 0.42353',\n",
       "  'for T = 993*dt, A[0,993*dt] is 0.42315',\n",
       "  'for T = 994*dt, A[0,994*dt] is 0.42277',\n",
       "  'for T = 995*dt, A[0,995*dt] is 0.42239',\n",
       "  'for T = 996*dt, A[0,996*dt] is 0.42202',\n",
       "  'for T = 997*dt, A[0,997*dt] is 0.42164',\n",
       "  'for T = 998*dt, A[0,998*dt] is 0.42126',\n",
       "  'for T = 999*dt, A[0,999*dt] is 0.42088',\n",
       "  'for T = 1000*dt, A[0,1000*dt] is 0.42051'])"
      ]
     },
     "execution_count": 891,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ak[:10],Ak[990:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 892,
   "id": "223bf7fc-ebea-47f0-a335-21d53f97c281",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(['for k = 1, B[0,1*dt] is 0.03',\n",
       "  'for k = 2, B[0,2*dt] is 0.0591',\n",
       "  'for k = 3, B[0,3*dt] is 0.08733',\n",
       "  'for k = 4, B[0,4*dt] is 0.11471',\n",
       "  'for k = 5, B[0,5*dt] is 0.14126',\n",
       "  'for k = 6, B[0,6*dt] is 0.16702',\n",
       "  'for k = 7, B[0,7*dt] is 0.19201',\n",
       "  'for k = 8, B[0,8*dt] is 0.21624',\n",
       "  'for k = 9, B[0,9*dt] is 0.23975',\n",
       "  'for k = 10, B[0,10*dt] is 0.26255'],\n",
       " ['for k = 991, B[0,991*dt] is 0.99505',\n",
       "  'for k = 992, B[0,992*dt] is 0.99505',\n",
       "  'for k = 993, B[0,993*dt] is 0.99505',\n",
       "  'for k = 994, B[0,994*dt] is 0.99505',\n",
       "  'for k = 995, B[0,995*dt] is 0.99505',\n",
       "  'for k = 996, B[0,996*dt] is 0.99505',\n",
       "  'for k = 997, B[0,997*dt] is 0.99505',\n",
       "  'for k = 998, B[0,998*dt] is 0.99505',\n",
       "  'for k = 999, B[0,999*dt] is 0.99505',\n",
       "  'for k = 1000, B[0,1000*dt] is 0.99505'])"
      ]
     },
     "execution_count": 892,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bk[:10],Bk[990:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 896,
   "id": "6f215fc8-eba0-48db-8fd9-c9cbd136dcdf",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/sethkelley/opt/anaconda3/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax5 = sns.lineplot(T,Pkval)\n",
    "ax5.set(xlabel = 'Time', ylabel = 'Price of instantaneous forward rate',title = \"Estimated Price of Instantaneous Fwd Rate vs. Time\")\n",
    "ax5.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f8419fdc-d171-44cd-9ef3-d3589f10f8a8",
   "metadata": {},
   "source": [
    "## 3.e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1150,
   "id": "3ea81e5b-b0b1-445d-b6ff-5b30e74d189e",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 1\n",
    "K = 1\n",
    "r = 0.02\n",
    "delta = 0.3\n",
    "N = 1000\n",
    "\n",
    "S0 = K*np.exp(3*delta*np.sqrt(T/K))\n",
    "sLower = K*np.exp(-3*delta*np.sqrt(T/K))\n",
    "M = int(1/((S0/1000)**2))\n",
    "dx = sBar/N\n",
    "dt = T/M\n",
    "\n",
    "S = np.arange(S0,sLower,-dx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1151,
   "id": "c2fb05f6-fac0-4a88-8d14-ff6d5af01d9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "V = np.zeros((len(S),M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1152,
   "id": "aecb547b-2fa8-4032-a86a-7a705455980c",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(M):\n",
    "    V[-1,i] = K*np.exp(-r*(T-i*dt)) - K*np.exp(-3*delta*np.sqrt(T/K))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1153,
   "id": "09e45691-f4b8-40db-b485-8984dcf445f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "V[:,-1] = K - S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1154,
   "id": "8eba1bb6-10fe-4799-97cf-44bc1f5d0e32",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(len(V[:,-1])):\n",
    "    V[i,-1] = max(V[i,-1],0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1156,
   "id": "72a9ff35-c45e-4cda-9b6e-d39be746d5ed",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(V.shape[1]-1):\n",
    "    for j in range(V.shape[0]-2):\n",
    "        sigma = delta * S[j]**(-1/2)\n",
    "        pUp = (sigma**2)*dt/(2*(dx**2)) + (r - (sigma**2)/2)*dt/(2*dx)\n",
    "        p = 1 - (sigma**2)*dt/(dx**2)\n",
    "        pDown = (sigma**2)*dt/(2*(dx**2)) - (r - (sigma**2)/2)*dt/(2*dx)\n",
    "        \n",
    "        V[1+j,V.shape[1]-2-i] = V[j,V.shape[1]-1-i]*(pUp) + V[1+j,V.shape[1]-1-i]*(p-r*dt) + V[2+j,V.shape[1]-1-i]*(pDown)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1157,
   "id": "bb596437-8fb1-42f4-93ee-e730a676aaa2",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/sethkelley/opt/anaconda3/lib/python3.9/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
      "  warnings.warn(\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax6 = sns.lineplot(S,V[:,0])\n",
    "ax6.set(xlabel = 'Time', ylabel = 'Price of instantaneous forward rate',title = \"Estimated Price of Instantaneous Fwd Rate vs. Time\")\n",
    "ax6.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e89a155-cb2b-467d-bad9-d5b58b9f7b07",
   "metadata": {
    "tags": []
   },
   "source": [
    "## 3.f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1158,
   "id": "65b61e58-73ac-4d72-8118-8dbdef1813cf",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 1\n",
    "K = 1\n",
    "r = 0.02\n",
    "delta = 0.3\n",
    "N = 1000\n",
    "\n",
    "S0 = K*np.exp(3*delta*np.sqrt(T/K))\n",
    "sLower = K*np.exp(-3*delta*np.sqrt(T/K))\n",
    "M = 1000\n",
    "dx = sBar/N\n",
    "dt = T/M\n",
    "\n",
    "S = np.arange(S0,sLower,-dx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1159,
   "id": "a7ad23fa-f244-4136-bec4-1e6bbb1f39c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "V = np.zeros((len(S),M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1160,
   "id": "12e330ab-6ef9-43d3-8951-4b8e483bc5fc",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(M):\n",
    "    V[-1,i] = K*np.exp(-r*(T-i*dt)) - K*np.exp(-3*delta*np.sqrt(T/K))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1161,
   "id": "1d500473-3ac0-4616-8759-f45934ee81dc",
   "metadata": {},
   "outputs": [],
   "source": [
    "V[:,-1] = K - S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1162,
   "id": "00f67085-5246-468a-8c88-9b7ac96fbb02",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(len(V[:,-1])):\n",
    "    V[i,-1] = max(V[i,-1],0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1163,
   "id": "14a172e1-2085-47fe-a77b-aaa756f5584b",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/bv/j678dvtd34bc5mtymgsd2c6w0000gn/T/ipykernel_9023/2722469056.py:8: RuntimeWarning: overflow encountered in double_scalars\n",
      "  V[1+j,V.shape[1]-2-i] = V[j,V.shape[1]-1-i]*(pUp) + V[1+j,V.shape[1]-1-i]*(p-r*dt) + V[2+j,V.shape[1]-1-i]*(pDown)\n",
      "/var/folders/bv/j678dvtd34bc5mtymgsd2c6w0000gn/T/ipykernel_9023/2722469056.py:8: RuntimeWarning: invalid value encountered in double_scalars\n",
      "  V[1+j,V.shape[1]-2-i] = V[j,V.shape[1]-1-i]*(pUp) + V[1+j,V.shape[1]-1-i]*(p-r*dt) + V[2+j,V.shape[1]-1-i]*(pDown)\n"
     ]
    }
   ],
   "source": [
    "for i in range(V.shape[1]-1):\n",
    "    for j in range(V.shape[0]-2):\n",
    "        sigma = delta * S[j]**(-1/2)\n",
    "        pUp = (sigma**2)*dt/(2*(dx**2)) + (r - (sigma**2)/2)*dt/(2*dx)\n",
    "        p = 1 - (sigma**2)*dt/(dx**2)\n",
    "        pDown = (sigma**2)*dt/(2*(dx**2)) - (r - (sigma**2)/2)*dt/(2*dx)\n",
    "        \n",
    "        V[1+j,V.shape[1]-2-i] = V[j,V.shape[1]-1-i]*(pUp) + V[1+j,V.shape[1]-1-i]*(p-r*dt) + V[2+j,V.shape[1]-1-i]*(pDown)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1164,
   "id": "3b3f45d0-1b46-479a-aa6e-c4b2b7824e02",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
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       "       [-inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [inf, -inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [-inf, inf, -inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [inf, -inf, inf, -inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [-inf, inf, -inf, inf, -inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [inf, -inf, inf, -inf, inf, -inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       [-inf, inf, -inf, inf, -inf, inf, -inf, inf, -inf, inf, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ..., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "       ...,\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 1.09e+49, -5.68e+46, 2.96e+44, -1.54e+42, 8.06e+39, -4.21e+37, 2.2e+35, -1.15e+33, 6.01e+30, -3.15e+28, 1.65e+26, -8.66e+23, 4.55e+21, -2.4e+19, 1.27e+17, -6.7e+14, 3.55e+12, -1.89e+10, 1.01e+08, -5.45e+05, 2.95e+03, -15.5, 0.64, 0.551, 0.551, 0.551, 0.551, 0.551, 0.551, 0.551],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -1.13e+49, 5.83e+46, -3.02e+44, 1.56e+42, -8.09e+39, 4.19e+37, -2.17e+35, 1.12e+33, -5.8e+30, 3e+28, -1.55e+26, 8.03e+23, -4.15e+21, 2.15e+19, -1.11e+17, 5.76e+14, -2.99e+12, 1.55e+10, -8.07e+07, 4.21e+05, -2.21e+03, 12.3, 0.49, 0.553, 0.553, 0.553, 0.553, 0.553, 0.553, 0.552],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 1.14e+49, -5.86e+46, 3.01e+44, -1.55e+42, 7.95e+39, -4.08e+37, 2.09e+35, -1.07e+33, 5.48e+30, -2.8e+28, 1.43e+26, -7.25e+23, 3.68e+21, -1.86e+19, 9.36e+16, -4.69e+14, 2.34e+12, -1.16e+10, 5.67e+07, -2.75e+05, 1.31e+03, -5.54, 0.582, 0.554, 0.554, 0.554, 0.554, 0.554, 0.554, 0.554],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -1.06e+49, 5.77e+46, -2.98e+44, 1.53e+42, -7.8e+39, 3.98e+37, -2.03e+35, 1.03e+33, -5.24e+30, 2.65e+28, -1.34e+26, 6.74e+23, -3.38e+21, 1.68e+19, -8.31e+16, 4.07e+14, -1.97e+12, 9.35e+09, -4.33e+07, 1.93e+05, -806, 3.51, 0.548, 0.556, 0.556, 0.556, 0.556, 0.555, 0.555, 0.555],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -2.21e+48, -4.35e+46, 2.84e+44, -1.5e+42, 7.68e+39, -3.92e+37, 2e+35, -1.02e+33, 5.16e+30, -2.62e+28, 1.33e+26, -6.69e+23, 3.37e+21, -1.69e+19, 8.42e+16, -4.18e+14, 2.05e+12, -9.98e+09, 4.79e+07, -2.25e+05, 1.03e+03, -3.93, 0.575, 0.557, 0.557, 0.557, 0.557, 0.557, 0.557, 0.557],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 1.67e+50, -1.88e+47, -2.2e+43, 1.27e+42, -7.46e+39, 3.86e+37, -1.97e+35, 1.01e+33, -5.15e+30, 2.63e+28, -1.35e+26, 6.9e+23, -3.54e+21, 1.81e+19, -9.29e+16, 4.76e+14, -2.44e+12, 1.25e+10, -6.45e+07, 3.33e+05, -1.72e+03, 9.56, 0.511, 0.559, 0.559, 0.558, 0.558, 0.558, 0.558, 0.558],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -1.83e+51, 3.05e+48, -4.14e+45, 3.57e+42, 3.28e+39, -3.55e+37, 1.9e+35, -9.76e+32, 5.02e+30, -2.58e+28, 1.34e+26, -6.92e+23, 3.6e+21, -1.88e+19, 9.84e+16, -5.18e+14, 2.74e+12, -1.46e+10, 7.82e+07, -4.22e+05, 2.29e+03, -12, 0.63, 0.56, 0.56, 0.56, 0.56, 0.56, 0.56, 0.56],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 1.56e+52, -3.07e+49, 5.33e+46, -7.76e+43, 8.59e+40, -4.27e+37, -1.42e+35, 8.98e+32, -4.6e+30, 2.36e+28, -1.22e+26, 6.32e+23, -3.29e+21, 1.73e+19, -9.09e+16, 4.82e+14, -2.58e+12, 1.39e+10, -7.5e+07, 4.09e+05, -2.24e+03, 12.9, 0.494, 0.562, 0.561, 0.561, 0.561, 0.561, 0.561, 0.561],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -1.1e+53, 2.49e+50, -5.08e+47, 9.13e+44, -1.39e+42, 1.68e+39, -1.33e+36, -9.18e+31, 3.93e+30, -1.98e+28, 1e+26, -5.08e+23, 2.59e+21, -1.33e+19, 6.83e+16, -3.55e+14, 1.85e+12, -9.75e+09, 5.17e+07, -2.77e+05, 1.49e+03, -7.5, 0.607, 0.563, 0.563, 0.563, 0.563, 0.563, 0.563, 0.562],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 6.69e+53, -1.69e+51, 3.92e+48, -8.28e+45, 1.54e+43, -2.45e+40, 3.11e+37, -2.77e+34, 1.05e+31, 1.59e+28, -7.67e+25, 3.68e+23, -1.76e+21, 8.34e+18, -3.94e+16, 1.85e+14, -8.6e+11, 3.96e+09, -1.8e+07, 8.05e+04, -354, 2.12, 0.558, 0.564, 0.564, 0.564, 0.564, 0.564, 0.564, 0.564],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -3.52e+54, 9.76e+51, -2.54e+49, 6.1e+46, -1.33e+44, 2.58e+41, -4.26e+38, 5.65e+35, -5.35e+32, 2.62e+29, 6.65e+25, -2.96e+23, 1.28e+21, -5.29e+18, 2.02e+16, -6.65e+13, 1.38e+11, 4.12e+08, -8.28e+06, 7.67e+04, -582, 4.6, 0.539, 0.566, 0.566, 0.566, 0.566, 0.565, 0.565, 0.565],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 1.63e+55, -4.9e+52, 1.4e+50, -3.75e+47, 9.34e+44, -2.12e+42, 4.26e+39, -7.34e+36, 1.02e+34, -1.01e+31, 5.41e+27, 3.69e+23, -1.6e+21, 6.65e+18, -2.57e+16, 8.62e+13, -1.92e+11, -4.01e+08, 9.76e+06, -9.34e+04, 727, -4.58, 0.602, 0.567, 0.567, 0.567, 0.567, 0.567, 0.567, 0.567],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -6.7e+55, 2.16e+53, -6.7e+50, 1.97e+48, -5.46e+45, 1.41e+43, -3.31e+40, 6.93e+37, -1.25e+35, 1.81e+32, -1.9e+29, 1.08e+26, 2.87e+21, -1.33e+19, 6.08e+16, -2.73e+14, 1.19e+12, -5e+09, 1.97e+07, -6.79e+04, 167, 0.739, 0.562, 0.569, 0.569, 0.569, 0.568, 0.568, 0.568, 0.568],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 2.46e+56, -8.46e+53, 2.81e+51, -8.96e+48, 2.72e+46, -7.8e+43, 2.08e+41, -5.09e+38, 1.11e+36, -2.1e+33, 3.2e+30, -3.55e+27, 2.15e+24, 2.37e+19, -1.17e+17, 5.75e+14, -2.84e+12, 1.41e+10, -6.97e+07, 3.47e+05, -1.74e+03, 9.34, 0.525, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -8.12e+56, 2.95e+54, -1.04e+52, 3.57e+49, -1.17e+47, 3.68e+44, -1.09e+42, 3.02e+39, -7.7e+36, 1.76e+34, -3.47e+31, 5.59e+28, -6.56e+25, 4.24e+22, 1.75e+17, -8.87e+14, 4.52e+12, -2.31e+10, 1.19e+08, -6.18e+05, 3.23e+03, -16.5, 0.662, 0.571, 0.571, 0.571, 0.571, 0.571, 0.571, 0.571],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 2.42e+57, -9.25e+54, 3.45e+52, -1.25e+50, 4.41e+47, -1.49e+45, 4.85e+42, -1.49e+40, 4.3e+37, -1.14e+35, 2.73e+32, -5.68e+29, 9.66e+26, -1.2e+24, 8.32e+20, 1.1e+15, -5.63e+12, 2.88e+10, -1.49e+08, 7.69e+05, -4e+03, 21.5, 0.463, 0.574, 0.573, 0.573, 0.573, 0.573, 0.573, 0.573],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -6.56e+57, 2.61e+55, -1.02e+53, 3.92e+50, -1.46e+48, 5.3e+45, -1.86e+43, 6.23e+40, -1.99e+38, 5.98e+35, -1.66e+33, 4.17e+30, -9.13e+27, 1.65e+25, -2.19e+22, 1.63e+19, 5.91e+12, -2.99e+10, 1.51e+08, -7.67e+05, 3.9e+03, -19.3, 0.675, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 1.61e+58, -6.69e+55, 2.73e+53, -1.09e+51, 4.3e+48, -1.65e+46, 6.17e+43, -2.23e+41, 7.78e+38, -2.58e+36, 8.1e+33, -2.36e+31, 6.22e+28, -1.44e+26, 2.77e+23, -3.95e+20, 3.18e+17, 2.74e+10, -1.35e+08, 6.62e+05, -3.24e+03, 16.4, 0.499, 0.576, 0.576, 0.576, 0.576, 0.576, 0.575, 0.575],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -3.62e+58, 1.55e+56, -6.57e+53, 2.75e+51, -1.13e+49, 4.56e+46, -1.8e+44, 6.95e+41, -2.6e+39, 9.41e+36, -3.25e+34, 1.07e+32, -3.25e+29, 9.07e+26, -2.23e+24, 4.58e+21, -7.05e+18, 6.17e+15, 1.15e+08, -5.43e+05, 2.55e+03, -11.3, 0.632, 0.577, 0.577, 0.577, 0.577, 0.577, 0.577, 0.577],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 7.43e+58, -3.28e+56, 1.43e+54, -6.2e+51, 2.65e+49, -1.12e+47, 4.64e+44, -1.89e+42, 7.54e+39, -2.92e+37, 1.1e+35, -3.96e+32, 1.36e+30, -4.36e+27, 1.29e+25, -3.38e+22, 7.45e+19, -1.24e+17, 1.2e+14, 4.84e+05, -2.23e+03, 10.7, 0.534, 0.579, 0.579, 0.579, 0.578, 0.578, 0.578, 0.578],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -1.39e+59, 6.3e+56, -2.83e+54, 1.26e+52, -5.6e+49, 2.45e+47, -1.06e+45, 4.53e+42, -1.9e+40, 7.82e+37, -3.14e+35, 1.23e+33, -4.63e+30, 1.67e+28, -5.65e+25, 1.77e+23, -4.97e+20, 1.19e+18, -2.16e+15, 2.31e+12, 2.4e+03, -10.6, 0.632, 0.58, 0.58, 0.58, 0.58, 0.58, 0.58, 0.58],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 2.38e+59, -1.1e+57, 5.08e+54, -2.33e+52, 1.06e+50, -4.8e+47, 2.15e+45, -9.55e+42, 4.19e+40, -1.81e+38, 7.73e+35, -3.22e+33, 1.31e+31, -5.17e+28, 1.96e+26, -7.03e+23, 2.35e+21, -7.09e+18, 1.84e+16, -3.71e+13, 4.44e+10, 14.4, 0.514, 0.582, 0.581, 0.581, 0.581, 0.581, 0.581, 0.581],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -3.71e+59, 1.75e+57, -8.24e+54, 3.86e+52, -1.8e+50, 8.39e+47, -3.88e+45, 1.78e+43, -8.11e+40, 3.66e+38, -1.64e+36, 7.2e+33, -3.12e+31, 1.32e+29, -5.47e+26, 2.18e+24, -8.32e+21, 2.98e+19, -9.74e+16, 2.77e+14, -6.23e+11, 8.51e+08, 0.662, 0.583, 0.583, 0.583, 0.583, 0.583, 0.583, 0.583],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 5.24e+59, -2.52e+57, 1.21e+55, -5.76e+52, 2.75e+50, -1.31e+48, 6.2e+45, -2.93e+43, 1.38e+41, -6.44e+38, 2.99e+36, -1.38e+34, 6.28e+31, -2.83e+29, 1.25e+27, -5.43e+24, 2.29e+22, -9.3e+19, 3.58e+17, -1.28e+15, 4.02e+12, -1.02e+10, 1.63e+07, 0.585, 0.584, 0.584, 0.584, 0.584, 0.584, 0.584],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -6.69e+59, 3.25e+57, -1.58e+55, 7.68e+52, -3.73e+50, 1.81e+48, -8.75e+45, 4.23e+43, -2.04e+41, 9.81e+38, -4.7e+36, 2.25e+34, -1.07e+32, 5.04e+29, -2.36e+27, 1.09e+25, -4.97e+22, 2.22e+20, -9.64e+17, 4.02e+15, -1.57e+13, 5.55e+10, -1.62e+08, 3.1e+05, 0.586, 0.586, 0.586, 0.586, 0.585, 0.585],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 7.6e+59, -3.74e+57, 1.84e+55, -9.05e+52, 4.45e+50, -2.19e+48, 1.08e+46, -5.31e+43, 2.61e+41, -1.29e+39, 6.32e+36, -3.1e+34, 1.52e+32, -7.46e+29, 3.64e+27, -1.77e+25, 8.56e+22, -4.11e+20, 1.95e+18, -9.07e+15, 4.11e+13, -1.78e+11, 7.15e+08, -2.47e+06, 5.87e+03, 0.587, 0.587, 0.587, 0.587, 0.587],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -7.52e+59, 3.73e+57, -1.85e+55, 9.19e+52, -4.57e+50, 2.27e+48, -1.13e+46, 5.66e+43, -2.83e+41, 1.42e+39, -7.09e+36, 3.56e+34, -1.79e+32, 9.01e+29, -4.54e+27, 2.29e+25, -1.16e+23, 5.84e+20, -2.95e+18, 1.49e+16, -7.44e+13, 3.69e+11, -1.79e+09, 8.32e+06, -3.5e+04, 112, 0.589, 0.588, 0.588, 0.588],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., 6.13e+59, -3.05e+57, 1.53e+55, -7.63e+52, 3.82e+50, -1.92e+48, 9.64e+45, -4.86e+43, 2.46e+41, -1.24e+39, 6.31e+36, -3.22e+34, 1.64e+32, -8.44e+29, 4.35e+27, -2.25e+25, 1.17e+23, -6.13e+20, 3.23e+18, -1.71e+16, 9.16e+13, -4.94e+11, 2.69e+09, -1.48e+07, 8.15e+04, -440, 2.68, 0.59, 0.59, 0.59],\n",
       "       [nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, ..., -3.46e+59, 1.73e+57, -8.69e+54, 4.36e+52, -2.19e+50, 1.11e+48, -5.59e+45, 2.84e+43, -1.44e+41, 7.35e+38, -3.76e+36, 1.93e+34, -9.98e+31, 5.18e+29, -2.7e+27, 1.42e+25, -7.52e+22, 4.02e+20, -2.17e+18, 1.19e+16, -6.58e+13, 3.73e+11, -2.16e+09, 1.3e+07, -8.15e+04, 549, -3.53, 0.631, 0.591, 0.591],\n",
       "       [0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, 0.574, ..., 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593, 0.593]])"
      ]
     },
     "execution_count": 1164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f3d676fd-1fe5-483b-ad3f-7d74a90b7846",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "543dbdaa-65b4-424a-a01b-dd5e6119d813",
   "metadata": {},
   "source": [
    "We can see that if M is not suitable for producing stability then we have errors in computing the option price. More generally, we will not have convergence to the true solution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ef5a01a-86ac-476a-8448-4d3d33e08a06",
   "metadata": {},
   "source": [
    "In some instances we have nan's as our values and in other instances we have produced extremely large numbers that eventually lead to overflow errors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8f38ea3d-c597-4dc1-a64e-2d298971ce00",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "43d64b0a-d6e5-42dc-904c-fe3031e92603",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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