5. Pricing Interest Rate Options.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pricing Interst Rate Options"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Short Rate Models\n",
    "\n",
    "We will relax the assumption of contant risk-free continuously compounded interest rates and will model our interest rates using **stochastic differential equations**\n",
    "\n",
    "**Short Rate Models** explain the instantaneous continuously-compounded interest rate at time t, $r_t$\n",
    "\n",
    "**Meaning:**\n",
    "\n",
    "- An investment will grow approx. over a **very short period t**.\n",
    "\n",
    "$$Xe^{r_{t \\Delta t}}$$\n",
    "\n",
    "\n",
    "- Over a period from $[0, t]$ it will grow:\n",
    "\n",
    "$$Xe^{\\int_0^t r_s ds}$$\n",
    "\n",
    "\n",
    "- The Present Value of an asset will be equal to:\n",
    "\n",
    "$$Xe^{-\\int_0^t r_s ds}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vasicek Model\n",
    "\n",
    "The Vasicek model specifies a stochastic differential equation for the short rate $r_t$ as:\n",
    "\n",
    "$$ dr_t = \\alpha(b-r_t)dt + \\sigma dW_t, r_0=r(0) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where, $r(0)$ is the short rate at time 0 and $\\alpha, b, \\sigma$ are positive constants.\n",
    "\n",
    "\n",
    "- $b$ is the level to whcih the short rate will tend in the long run (mean-level)\n",
    "\n",
    "\n",
    "- $\\alpha$ is the rate at which the short rate will tend towards b \n",
    "\n",
    "\n",
    "- $\\sigma$ is the volatility of the short rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution to the SDE is: \n",
    "\n",
    "$$ r_t = r(0)e^{-\\alpha t} + b(1-e^{- \\alpha t}) + \\sigma e^{- \\alpha t} \\int_0^t e^{\\alpha s } dW_s $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This means that $r_t$ is **normally distributed**:\n",
    "\n",
    "$$ r_t \\sim N(r(0)e^{-\\alpha t} + b(1-e^{- \\alpha t}), \\frac{\\sigma^2}{2 \\alpha}(1 - e^{- 2\\alpha t})) $$\n",
    "\n",
    "- **Mean:** $r(0)e^{-\\alpha t} + b(1-e^{- \\alpha t})$\n",
    "\n",
    "\n",
    "- **Variance:** $\\frac{\\sigma^2}{2 \\alpha}(1 - e^{- 2\\alpha t})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**N.B.** as $t \\rightarrow \\infty$\n",
    "\n",
    "\n",
    "- $mean(r_t) \\rightarrow b$\n",
    "\n",
    "\n",
    "- $var(r_t) \\rightarrow \\frac{\\sigma^2}{2 \\alpha}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modelling a Path for the Short Rate\n",
    "\n",
    "if we know the value of $r_{t_1}$ and $t_2 > t_1, r_{t_2}$ is normally distributed with:\n",
    "\n",
    "- Mean: $r_{t_1}e^{- \\alpha(t_2 - t_1)} + b(1-e^{- \\alpha(t_2 - t_1)})$\n",
    "\n",
    "\n",
    "\n",
    "- Variance: $\\frac{\\sigma^2}{2 \\alpha}(1 - e^{- \\alpha(t_2 - t_1)})$\n",
    "\n",
    "We can treadt $t_1$ as a new starting point for our modelling, given the independence of increments of Brownian Motion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "import matplotlib.pyplot as plt\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# parameters\n",
    "\n",
    "r0 = 0.05\n",
    "alpha = 0.2\n",
    "b = 0.08\n",
    "sigma = 0.025"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def vasi_mean(r, t1, t2):\n",
    "    \"\"\"\n",
    "    Gives the mean under the Vasicek Model. Note that t2 > t1. r is the\n",
    "    interest rate from the beginning of the period.\n",
    "    \"\"\"\n",
    "    return np.exp(-alpha*(t2-t1))*r+b*(1-np.exp(-alpha*(t2-t1)))\n",
    "\n",
    "def vasi_var(t1,t2):\n",
    "    \"\"\"\n",
    "    Returns the variance under the Vasicek model with t2>t1\n",
    "    \"\"\"\n",
    "    return (sigma**2)*(1-np.exp(-2*alpha*(t2-t1)))/(2*alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Simulating Interest Rate Paths (on an annual basis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We create 10 simulations of possible short term interest rate paths over a period of 10 years. If done on a monthly basis can make the array t include fractional values.\n",
    "\n",
    "The simulated short rates, in terms of paths are of the form:\n",
    "\n",
    "$$ r_{i+1} = r_i e^{- \\alpha(t_{i+1} - t_i)} + b(1 - e^{\\alpha (t_{i+1} - t_i)}) + \\sqrt{\\frac{\\sigma^2}{2 \\alpha}(1 - e^{-2 \\alpha(t_{i+1} - t_i)})}Z_i $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `s_mean` array contains the expected value of the short rate at each point in time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "\n",
    "n_years = 10\n",
    "n_simulations = 10\n",
    "\n",
    "t = np.array(range(0,n_years+1))\n",
    "\n",
    "# standard nromal for each year and each simulation\n",
    "Z = norm.rvs(size=[n_simulations, n_years])\n",
    "\n",
    "# this array stores the paths\n",
    "r_sim = np.zeros([n_simulations, n_years+1])\n",
    "\n",
    "# set the first column (initial value of each simulation) to r(0)\n",
    "r_sim[:,0] = r0\n",
    "\n",
    "vasi_mean_vector = np.zeros(n_years+1)\n",
    "\n",
    "for i in range(n_years):\n",
    "    # interest rate paths\n",
    "    r_sim[:, i+1] = vasi_mean(r_sim[:,i], t[i], t[i+1]) + np.sqrt(vasi_var(t[i], t[i+1]))*Z[:,i]\n",
    "    \n",
    "s_mean = r0*np.exp(-alpha*t) + b*(1-np.exp(-alpha*t))\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 1080x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting the results\n",
    "plt.figure(figsize=(15,5))\n",
    "t_graph = np.ones(r_sim.shape)*t\n",
    "plt.plot(np.transpose(t_graph), np.transpose(r_sim*100), \"r\")\n",
    "plt.plot(t,s_mean*100)\n",
    "plt.xlabel(\"Year\")\n",
    "plt.ylabel(\"Short Rate\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**N.B.** the short rate can become negative (this is an unfavorable property in many interst rate modeling problems)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hull-White Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Specifies a SDE for the short rate as follows:\n",
    "\n",
    "$$ dr_t = (\\theta(t) - \\alpha(t)r_t)dt + \\sigma(t)dW_t. r_0 = r(0) $$\n",
    "\n",
    "Where $\\theta(t), \\alpha(t), \\sigma(t)$ can be time dependent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore we can see that the Vasicek Model is a **special case** of the Hull-White model where:\n",
    "\n",
    "- $\\theta(t) = \\alpha b$\n",
    "\n",
    "- $\\alpha(t) = \\alpha$\n",
    "\n",
    "- $\\sigma(t) = \\sigma$\n",
    "\n",
    "**Mean-Reversion Varying over Time:**\n",
    "\n",
    "In practice, the Hull-White model is often implemented with only $\\theta(t)$ being non-constant, which means that the mean reversion level is allowed to vary over time and the model can be better calibrated to prices or rates seen in the market.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cox-Ingersoll-Ross Model (CIR)\n",
    "\n",
    "Also suggests an SDE for short term rates:\n",
    "\n",
    "$$ dr_t = \\alpha(b-r_t)dt + \\sigma \\sqrt{r_t} dW_t, r_0 = r(0) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- $b$ is the mean level for the short term rate\n",
    "\n",
    "\n",
    "\n",
    "- $\\alpha$ is the rate of mean reversion\n",
    "\n",
    "\n",
    "\n",
    "- If $2ab \\geq \\sigma^2$ and $r(0) > 0$ then $r_t$ will **never become negative** (Feller Condition)\n",
    "\n",
    "\n",
    "as $r_t \\rightarrow 0$:\n",
    "\n",
    "- The volatility of the short term rate becomes close to 0\n",
    "\n",
    "\n",
    "- Then the **upward drift** to the mean will exceed the **downward movement** of the volatility\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Disadvantage:** The CIR Model does **not** have a closed-form solution, but it can be simulated directly, since its conditional distribution is known."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Non-Constant Interest Rate Pricing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we want to price a derivative on some underlying process given by $X_t$. If the derivative has a maturity of T and a payoff function $\\phi(.)$, the price of the option at time 0 is given by:\n",
    "\n",
    "$$ P = E^Q[ e^{-\\int_0^T r_t dt} \\phi(X_T)] $$\n",
    "\n",
    "Where Q is the risk-neutral measure.\n",
    "\n",
    "**N.B.** The underlying process can be something related to interest rates, such that this formula is how one would price interest rate derivates (Interest Rate Swaps)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Applications of Short Rates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Price of a Zero-Coupon Bond\n",
    "\n",
    "This is equivalent to finding the average discount factor. The price of a ZCB at time $t$ with maturity $T$ is given by:\n",
    "\n",
    "$$ B(t,T) = E[e^{-\\int_t^Tr_s ds}] $$\n",
    "\n",
    "Where the expectation is taken under the risk-neutral measure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Applying the Vasicek Model to Price ZCBs\n",
    "\n",
    "$$ dr_t = \\alpha(b-r_t)dt + \\sigma dW_t, r_0=r(0) $$\n",
    "\n",
    "Under this model, the above expectation is given by:\n",
    "\n",
    "$$ B(t,T) = e^{-A(t,T)r_t + D(t,T)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where,\n",
    "\n",
    "$$A(t,T) = \\frac{1-e^{-\\alpha(T-t)}}{\\alpha} $$\n",
    "\n",
    "$$D(t,T) = (b - \\frac{\\sigma^2}{2 \\alpha^2})[A(t,T) - (T-t)] - \\frac{\\sigma^2 A(t,T)^2}{4 \\alpha}$$\n",
    "\n",
    "In this way we have a closed-form solution for our bond prices in the Vasicek Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "import matplotlib.pyplot as plt\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Parameters\n",
    "r0 = 0.05\n",
    "alpha = 0.2\n",
    "b = 0.08\n",
    "sigma = 0.025"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def vasi_mean(r, t1, t2):\n",
    "    \"\"\"\n",
    "    Gives the mean under the Vasicek Model. Note that t2 > t1. r is the\n",
    "    interest rate from the beginning of the period.\n",
    "    \"\"\"\n",
    "    return np.exp(-alpha*(t2-t1))*r+b*(1-np.exp(-alpha*(t2-t1)))\n",
    "\n",
    "def vasi_var(t1,t2):\n",
    "    \"\"\"\n",
    "    Returns the variance under the Vasicek model with t2>t1\n",
    "    \"\"\"\n",
    "    return (sigma**2)*(1-np.exp(-2*alpha*(t2-t1)))/(2*alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Analytical Bond price\n",
    "def A(t1, t2):\n",
    "    return (1-np.exp(-alpha*(t2-t1)))/alpha\n",
    "\n",
    "def D(t1, t2):\n",
    "    val_1 = (t2-t1-A(t1,t2))*(sigma**2/(2*alpha**2)-b)\n",
    "    val_2 = sigma**2*A(t1,t2)**2/(4*alpha)\n",
    "    return val_1 - val_2\n",
    "\n",
    "def bond_price(r,t,T):\n",
    "    return np.exp(-A(t,T)*r + D(t,T))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Joint Simulation\n",
    "\n",
    "In order to create estimates for our bond prices, we are going to be jointly simulating the short term rate, $r_t$ and a value $\\int_0^t r_s ds$ which we will denote as $Y_t$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since $r_t$ is Gaussian and $Y_t$ is an integral of Gaussians, they have a **Joint Gaussian Distribution**.\n",
    "\n",
    "Therefore, we can simulate them together once we have a vale for their means, variances and correlation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Gaussian Distribution of r\n",
    "\n",
    "$$ r_t \\sim N(r(0)e^{-\\alpha t} + b(1-e^{- \\alpha t}), \\frac{\\sigma^2}{2 \\alpha}(1 - e^{- 2\\alpha t})) $$\n",
    "\n",
    "##### Gaussian Distribution of Y\n",
    "\n",
    "$$ Y_{t_2} \\sim N(Y_{t_1} + (t_2 - t_1)b + (r_{t_1} - b)A(t_1, t_2), \\frac{\\sigma^2}{\\alpha^2}(t_2 - t_1 - A(t_1, t_2) - \\alpha \\frac{A(t_1, t_2)^2}{2}) ) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With $t_2 > t_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Covariance($Y_{t_2}, r_{t_2}$) $ = \\frac{\\sigma^2  A(t_1, t_2)^2}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Correlation($Y_{t_2}, r_{t_2}$) $ = \\frac{\\sigma^2  A(t_1, t_2)^2}{2 \\sigma(r_{t_2})\\sigma(Y_{t_2})}$\n",
    "\n",
    "Where sigma(Y) is the Standard Deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Functions for means, variance, and correlations\n",
    "\n",
    "## calculates the mean for Yt2 given Yt1 and rt1, where t1 < t2\n",
    "def Y_mean(Y, r, t1, t2):\n",
    "    return Y + (t2-t1)*b+(r-b)*A(t1,t2)\n",
    "\n",
    "## calculates the variance for Yt2\n",
    "def Y_var(t1,t2):\n",
    "    return sigma**2*(t2-t1-A(t1,t2)**2/2)/(alpha**2)\n",
    "\n",
    "## covariance bewteen yt2 and rt2\n",
    "def rY_var(t1, t2):\n",
    "    return sigma**2*2*(A(t1,t2)**2)/2\n",
    "   \n",
    "## correlation from the covariance\n",
    "def rY_rho(t1, t2):\n",
    "    return rY_var(t1,t2)/np.sqrt(vasi_var(t1,t2)*Y_var(t1,t2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can therefore, simulate joint paths for $r_t$ and $Y_t$ by simulating standard normal random variales with this correlation and then transforming them into normal variables with appropriate means and variances."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $Z_i \\sim N(0,1)$ for i = 1, 2 then $Z_1$ and $\\rho Z_1 + \\sqrt{1- \\rho ^2}Z_2$ have a correlation of $\\rho$. \n",
    "\n",
    "\n",
    "This is why we transform `Z_mont2` so that `Z_mont1` and `Z_mont2` have the correlation necessary to simulate $r_t$ using `Z_mont1` and $Y_t$ using `Z_monts`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$B(0,t) = E[e^{- \\int_0^t r_s ds}] = E[e^{-Y_t}]$$\n",
    "\n",
    "This is implemented at the end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initial Y Value\n",
    "Y0 = 0\n",
    "\n",
    "np.random.seed(0)\n",
    "\n",
    "# number of years simulated and number of simulations\n",
    "n_years = 10\n",
    "n_simulations = 100000\n",
    "\n",
    "# array of number of years\n",
    "t = np.array(range(0,n_years+1))\n",
    "\n",
    "# standard normal for the two variables\n",
    "Z_mont1 = norm.rvs(size= [n_simulations, n_years])\n",
    "Z_mont2 = norm.rvs(size= [n_simulations, n_years])\n",
    "r_simtemp = np.zeros([n_simulations, n_years+1])\n",
    "Y_simtemp = np.zeros([n_simulations, n_years+1])\n",
    "\n",
    "# Set the first column (the initial value of each simulation) to r(0) and Y(0)\n",
    "r_simtemp[:,0] = r0\n",
    "Y_simtemp[:,0] = Y0\n",
    "\n",
    "# Creating correlated standard normals\n",
    "correlations = rY_rho(t[0:-1], t[1:]) # 0 to last, from t to all\n",
    "Z_mont2 = correlations*Z_mont1 + np.sqrt(1-correlations**2)*Z_mont2\n",
    "\n",
    "# run simulation\n",
    "for i in range(n_years):\n",
    "    # generate the rt and yt paths\n",
    "    r_simtemp[:, i+1] = vasi_mean(r_simtemp[:,i], t[i], t[i+1]) + np.sqrt(vasi_var(t[i], t[i+1]))*Z_mont1[:,i]\n",
    "    Y_simtemp[:, i+1] = Y_mean(Y_simtemp[:,i], r_simtemp[:,i], t[i], t[i+1]) + np.sqrt(Y_var(t[i], t[i+1]))*Z_mont2[:,i]\n",
    "\n",
    "# Estimate bond prices using Monte Carlo by taking the mean bond prices for each year over all simulations\n",
    "ZCB_prices = np.mean(np.exp(-Y_simtemp), axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quadrature Approach of Estimating the Values\n",
    "\n",
    "An alternative to simulating the $Y_t$ values is to estimate them using only $r_t$ as follows:\n",
    "\n",
    "$$ y_t = \\int_0^t r_s ds \\approx \\Sigma_{i=0}^{n-1}r_{t_i}\\delta_i \\approx \\sum_{i=0}^{n=1} \\frac{r_{t_i} + r_{t_{i+1}}}{2}\\delta_i$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where,\n",
    "\n",
    "- $0 = t_0 < t_1 < t_2 < ... < t_n = t$\n",
    "\n",
    "\n",
    "- $\\delta_i = t_{i+1}-t_i$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is an Example of a Quadrature, and we would expect the second approximation to be more accurate than the first this it has an O($\\delta^2$) error vs an O($\\delta$) error for the first. \n",
    "\n",
    "Simulating only the short rates uses **less computation time** than simulating the integrals and the short term rates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Y_t estimates using the approximations\n",
    "r_mat = np.cumsum(r_simtemp[:,0:-1], axis=1)*(t[1:] - t[0:-1])\n",
    "r_mat2 = np.cumsum(r_simtemp[:,0:-1] + r_simtemp[:,1:], axis=1)/2*(t[1:] - t[0:-1])\n",
    "\n",
    "# Bond Pirce Estimates\n",
    "## at time 0, bonds have a price of 1\n",
    "squad_prices = np.ones(n_years+1)\n",
    "trap_prices = np.ones(n_years+1)\n",
    "\n",
    "# find the means replacing Yt with r_mat\n",
    "squad_prices[1:] = np.mean(np.exp(-r_mat), axis=0)\n",
    "trap_prices[1:] = np.mean(np.exp(-r_mat2), axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find Closed-form bond prices and plot each one"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Closed-form bond prices\n",
    "bond_vec = bond_price(r0,0,t)\n",
    "\n",
    "# plotting bond prices\n",
    "plt.figure(figsize=(15,5))\n",
    "\n",
    "## Analytical Solution\n",
    "plt.plot(t,bond_vec)\n",
    "## Simulated Yt and rt\n",
    "plt.plot(t, ZCB_prices, \".\")\n",
    "# Simulated rt and estimated Yt\n",
    "plt.plot(t, squad_prices, \"x\")\n",
    "# Simulated rt and estimated Yt\n",
    "plt.plot(t, trap_prices, \"^\")\n",
    "plt.title(\"Bond Prices vs. Maturity\")\n",
    "plt.xlabel(\"Time to Maturity\")\n",
    "plt.ylabel(\"Bond Price\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding the Implied Yield\n",
    "\n",
    "for the given bond prices ($y_t$)\n",
    "\n",
    "$$ B(0,t) = e^{(-y_t)t} $$\n",
    "\n",
    "i.e. it is the constant interest rate which gives the correct bond price, which we get by solving for $y_t$:\n",
    "\n",
    "$$ y_t = - \\frac{\\ln(B(0,t))}{t} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Determining yields\n",
    "bond_yield = -np.log(bond_vec[1:])/t[1:]\n",
    "mont_yield = -np.log(ZCB_prices[1:])/t[1:]\n",
    "squad_yield = -np.log(squad_prices[1:])/t[1:]\n",
    "trap_yield = -np.log(trap_prices[1:])/t[1:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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C3i89R15eHunp6axbt46IiAjKy8spKCggKSmpibOcvBMJe4mJiaxdu5bo6Ohmfe62oLV8r4mIiEjr4nK5Z+fcM3NHh7l9ZdWNxgcH+HpCXM+oYHp2ORzmekR2Isj/xDc/yX/oYUrefBMOhT0Af38iJ0wg7kHN7v1c87Pnk9olFUecw3NbRn4G2UXZ3JB6QwtWdlir6LPXEVlr+cMf/sAHH3yAMYb777+fSZMmkZuby5gxY8jOzmbhwoW8/fbbHDx4kK1btzJ+/HiefPJJ7r33XiorKxk0aBD9+vVj0aJFnvPu27ePsLAwQkNDAXcoPPT51KlTGTNmDBMmTCAxMZHJkyezfPlynE4n8+bN449//CNbtmzhnnvuYfr06Xz22WfMnj2bd999F4AZM2aQlpbG1KlTG72WW265hTVr1lBZWcmECRN4+OGHee6559i9ezejRo0iOjqa5cuXNwp/Tz/9NPPnzwfgxhtv5M477yQ3N5dLL72U4cOH89VXX9GjRw/eeustOnXqdAr+i4iIiIg0j4M1tewsrmxyZm5n8UGqaw/PzhkD3SM6kRDViZEpMY1m5npGBRPVjG0JKrOyGgc9AKeTym++aZbzdzSpXVKZ9fksZqfPxhHnICM/w3Pc1nTcsLczA3JXQOL5kOD46fEnaMmSJWRlZfHtt99SWFjI0KFDGTFixFHjsrKy+OabbwgMDCQlJYWZM2fyxBNP8Pzzz5OVlXXU+IEDB9KtWzeSkpIYPXo0V111FWPHjm2yhoSEBFatWsVdd93F1KlT+fLLL6mqqqJfv35Mnz79hF/LY489RlRUFHV1dYwePZr169dz++238/TTT7N8+fKjZvYyMzNZsGABq1evxlrLsGHDSE9Pp3PnzmzevJl//OMf/PWvf2XixIm8+eabXHvttSdci4iIiIi3uVyWvWVVTc7M7SiupLC88excaKAfPaOCOT0mlAv6dm0U5rpHBp2y1gTJy5aekufpKBxxDmanz2bW57OYmDKRxTmLPcGvremYYW9nBvxtHNTVgG8ATHm72QLfypUrufrqq/H19aVbt26kp6ezZs0aBgwY0Gjc6NGjiYiIAODMM8/kxx9/JCEh4Zjn9fX15cMPP2TNmjV8+umn3HXXXWRmZvLQQw8dNXbcuHEA9O/fn/LycsLCwggLCyMoKIiSkpITfi2LFy9m3rx51NbWkp+fz4YNG456HUe+9vHjxxMSEgLAVVddxYoVKxg3bhxJSUkMGjQIgLPOOovc3NwTrkNERESkuZRX13pC3JG7W+YVV1JTd3h2zsdA98hO9IwK5sIzGoe5nlHBRAb7qz1BO+WIczAxZSJz189l2oBpbTLoQUcNe7kr3EHP1rk/5q5otrB3otdABgYGej739fWltrb2OKPdjDE4HA4cDgcXXXQR119/fZNh79C5fXx8Gj2Pj48PtbW1+Pn54XId/kFWVVV11Dm2b9/O7NmzWbNmDZ07d2bq1KlNjmvoeK/9yNdbWVl53HOJiIiInIw6l2XPAffsXFOtCooqahqNDwvyo1eXYPrGhnHRmd0ahbnukZ3wV4uCDikjP4PFOYuZNmAai3MW44h1tMnA1zHDXuL57hm9QzN7iec326lHjBjB3LlzmTJlCsXFxXzxxRc89dRTPxmUDvH398fpdOLv79/o9t27d7Nnzx6GDBkCuJeB9urV66Rq7NWrFxs2bKC6upqqqio+/fRThg8f3mjMgQMHCAkJISIigr179/LBBx8wcuRIAMLCwigrKztqGeeIESOYOnUq9957L9Zali5dyv/93/+dVI0iIiIix1JW5WxymeXO4oPk7T+Is+7wH6B9fQw96mfnLu4X23hTlKhgIoL9j/NM0hE1vEbPEefAEetodNyWdMywl+BwL91sxmv2amtrCQwMZPz48axatYqBAwdijOHJJ58kNjb2hJct3nzzzQwYMIAhQ4Y02qDF6XQya9Ysdu/eTVBQEDExMcyZM+ekak1ISGDixIkMGDCA3r17M3jw4KPGDBw4kMGDB9OvXz+Sk5M577zzGtV46aWXEhcXx/Llyz23DxkyhKlTp+JwuL+eN954I4MHD9aSTRERETmKc98+dt39e+Kfefqo9gC1dS7yS6uO2UR8/8HGm5FEBvvTMyqYM7uH86vUxoEuLiJIDcTlZ8kuym4U7A5dw5ddlN3mwp5aLzSTb7/9lptuuomMjIwWrUO8qzV8r4mIiLQHO/70IOVvvE7ZxeP45qqbGoW5XfsrqXUdfo/q52OI79zpqGvmEur/RXTS7Jx0LGq9cArNmTOH5557jmeffbalSxERERFpNWrrXOTtr2RbYTnbCirYWlDBtoJyinbsZvbSJQRaS+DH7zPXDMR0iSYhKpgB8ZGMGRDXqFVBbLhm50ROhsJeM5g+ffrPamkgIiIi0p7sr6hhW2F5fZhzB7pthRX8WFTR6Pq5zsH+JMeEcuuOz/Gv38QywAfeCs+h55+ubqHqRdovhT0RERER+Uk1tS52FFccFei2FZQ3uobO39fQq0sIydEhXHhGN5JjQjgtJoTk6FA6hwTg3LePrXO+wNa5dyI3tU4OvrWM2pm3HXXtnoj8Mgp7IiIiIgK42ygVlFfXh7nGgW7n/krqGlxHFxMWSHJ0CL9KjXOHufpAF9+503GXXBa++BK2QQsoAOtyUfDiS8Q9+IDXXptIR6SwJyIiItLBVDnr2F54dKDbVlBBWfXh3r+Bfj4kRYfQr3sEYwd29wS6pJgQwoNOblOUyqwscDbeTROnk8pvvvklL6lDm589n9QuqY12iszIzyC7KJsbUm9owcqkpSnsiYiIiLRD1lryS6vcgc6zQYr74+7SShpuyN49IojkmFDGD+lBcnQIyTGhJMeE0D2iEz4+plnrSl62tFnPJ5DaJbVRH7iGfeKkY1PYayaPPfYYr776Kr6+vvj4+DB37lyGDRvGjTfeyN13382ZZ575i58jNDSU8vLyEx5fWlrKzJkz+fLLLwE4++yzef755+ncufMvriU3N5cxY8aQnZ19wo8pKSnh1Vdf5dZbb/3Fzy8iIiJuFdW1nkB3aLfLbQUVbC+soNJZ5xkXEuBLUkwIZ/XqzG9i4kmOCeW0mBCSokMIDtBbwrbsUB+4WZ/PYmLKRBbnLG6TDcCl+Xn1/2xjTCTwMpAKWOAGa+2qBvdfA/xn/WE5cIu19ltv1uQNq1at4t1332XdunUEBgZSWFhITU0NAC+//HKL1fW73/2O1NRU/v73vwPw4IMPMnXqVN566y2vPWdtbS1+fk1/W5WUlPDiiy/+7LBXV1eHr69vc5QnIiLSJtW5LLtLKj0zc4dm6rYVVLDnQJVnnDEQ37kTydGhDEuOcge6+pm6buGBGNO8s3TSejjiHExMmcjc9XOZNmCagp4A3p/Z+zPwobV2gjEmAAg+4v7tQLq1dr8x5lJgHjDMmwV5Y01zfn4+0dHRBAYGAhAdHe25b+TIkcyePZu0tDRCQ0O57bbb+OSTT+jcuTOPP/44f/jDH9ixYwfPPvss48aNY+HChSxdupTq6mq2b9/O5MmTefDBB496zqeeeorFixdTXV3N+PHjefjhhxvdv2XLFjIzM3nttdc8tz3wwAOcdtpp5OTkkJ+fz+zZs3n33XcBmDFjBmlpaUydOpVHHnmEd955h8rKSs4991zmzp2LMYbMzExuuOEGgoODGT58uOe8Cxcu5L333qOqqoqKigrefvttrrjiCvbv34/T6eTRRx/liiuu4N5772Xr1q0MGjSIiy66iMsvv/yYNSQmJnLDDTfw0UcfMWPGDIYOHcptt91GQUEBwcHB/PWvf6Vv374n9d9LRESktSqtdHpm5hoGuu1FFdTUHt7UJDzIj+SYUM49vQunxYR6ll726hJMkL/+QNoRZeRnsDhnMdMGTGNxzmIcsQ4FPvFe2DPGhAMjgKkA1toaoKbhGGvtVw0OvwbivVXPId5Y03zxxRfzyCOP0KdPHy688EImTZpEenr6UeMqKioYOXIk//M//8P48eO5//77+fjjj9mwYQNTpkxh3LhxAGRkZJCdnU1wcDBDhw7l8ssvJy0tzXOejz76iM2bN5ORkYG1lnHjxvHFF18wYsQIz5gNGzYwaNCgRjNivr6+DB48mI0bNxIZGXnM1zNjxgweeMC9G9Z1113Hu+++y9ixY7n++uv5y1/+Qnp6Ovfcc0+jx6xatYr169cTFRVFbW0tS5cuJTw8nMLCQs4++2zGjRvHE088QXZ2NllZWQB89tlnx/26BgUFsXLlSgBGjx7NnDlz6N27N6tXr+bWW2/l3//+93EfLyIi0ho561zsLD54VKDbVlhOYfnht0p+PoaeUcEkx4SQnhLT6Fq6LiEBmqUTj4bvZx1xDhyxjkbH0nF5c2YvGSgAFhhjBgKZwB3W2opjjP8d8IEX6wG8s6Y5NDSUzMxMVqxYwfLly5k0aRJPPPEEU6dObTQuICCAX/3qVwD079+fwMBA/P396d+/P7m5uZ5xF110EV26dAHgqquuYuXKlUeFvY8++ojBgwcDUF5ezubNmxuFPWttk78EbMOrsY9h+fLlPPnkkxw8eJDi4mL69evHiBEjKCkp8YTY6667jg8+OPyf66KLLiIqKsrzHPfddx9ffPEFPj4+7Nq1i7179/7k8x5p0qRJntf31Vdf8Zvf/MZzX3V19c8+n4iIyKliraW4oqbRLpdb6wPdjqKD1DZoYdAlJIDkmBBG93X3pDsU6HpGBeN/nBYGIodkF2U3ej976P1udlG2wl4H582w5wcMAWZaa1cbY/4M3Av86ciBxphRuMPe8CPvq7//ZuBmgJ49e/7iwryxptnX15eRI0cycuRI+vfvz9/+9rejwp6/v78ngPn4+HiWffr4+FBbe3ib4yND2pHH1lr++Mc/Mm3atGPW069fP7755htcLhc+Pu5fFC6Xi/Xr1zNkyBB27NiBq0GPm6qqKs/HW2+9lbVr15KQkMBDDz1EVVXVMcPjISEhIZ7PFy1aREFBAZmZmfj7+5OYmOg5f0N+fn5N1nDkOV0uF5GRkZ4ZQRERkZbg3LePXXf/nvhnnvY0/66urePHooNsKyg/3Gy8frautPJwe4EAXx8So4Pp0zWMX/WL9QS606JDiQg+uRYGIoc0dSmSI07LOMW7YS8PyLPWrq4/fgN32GvEGDMA9yYul1pri5o6kbV2Hu7r+UhLS/vpqamf0NxrmnNycvDx8aF3794AZGVl0atXr5M+38cff0xxcTGdOnVi2bJlzJ8/v9H9l1xyCX/605+45pprCA0NZdeuXfj7+9O1a1fPmNNPP53Bgwfz6KOPepZkPvroo4wePZqePXtijGHDhg1UV1dTVVXFp59+yvDhwz2BKzo6mvLyct544w0mTJhAZGQkERERrFy5kuHDh7No0aJj1l9aWkrXrl3x9/dn+fLl/PjjjwCEhYVRVlbmGderV68mazhSeHg4SUlJvP766/zmN7/BWsv69esZOHDgSX+NRUREfo6i8mp2PvEMAZmZfPCHx3hzxGS2FVSQt/8gDSbp6BYeSHJ0KGMGxDUKdD06d8K3mVsYiIj8FK+FPWvtHmPMTmNMirU2BxgNbGg4xhjTE1gCXGet3eStWhryxprm8vJyZs6cSUlJCX5+fpx++unMmzfvpGscPnw41113HVu2bGHy5MmNlnCC+xrBjRs3cs455wDuZaSvvPJKo7AHMH/+fGbOnMnpp59OaWkpQ4cO5Z133gEgISGBiRMnMmDAAHr37u1ZEhoZGclNN91E//79SUxMZOjQoZ7zLViwwLNByyWXXHLM+q+55hrGjh1LWloagwYN8myk0qVLF8477zxSU1O59NJLeeqpp5qsoSmLFi3illtu4dFHH8XpdPLb3/5WYU9ERJpdlbOOzXvL+WHPAXL2lJGzt4wf9pRRW1DAgo/exVhLQsa/qexzIQN6defKwT04rUGj8dBAtTAQkdbDnMg1XCd9cmMG4Z61CwC2AdcDkwCstXOMMS8DvwZ+rH9IrbU2ralzHZKWlmbXrl3b6LaNGzdyxhlnnFBN3tiNszktXLiQtWvX8vzzzzfreXNycrjsssv4y1/+wmWXXdas5+5Ifs73moiItF51LsuO4oPk7DlJy5rfAAAgAElEQVTAD3vK3MFuTxm5RRWembpAPx/6dAsjJTaMyz75O7FffoypdYK/P5ETJhD34AMt+yJEpMMyxmT+VG4CL7desNZmAUcWMafB/TcCN3qzhiN11DXNKSkpbN26taXLEBEROeUKyqrJ2VPWaLZu094yqpzu68aNgV5RwaTEhjFmYHf6xroDXmKXEHx9DM59+9j66KfY2vpr8JxOSpcsIebWWzzX7omItEZaa9DKTJ069aiNXUREROSnVdbUsWlvWX2wOxzuiioOtzOIDg0gJTaMyY5enlDXu1sowQHHfktU+OJL2AYbigFYl4uCF1/S7N5JaO2rrETaE4U9ERERaVPqXJbcogpPqMupD3U/Fh/k0NUpnfx96dMtlNFndCUlNtwT7KJDA3/281VmZYHT2fhGp5PKb75phlfT8Xij57GINE1hT0RERFolay0FZdWea+p+2FNGzt4DbN5bTnWte6bNx0BilxDOiAtn/OB4UmLD6BsbRkJUcLPtfpm8bGmznEfcvNHzWESaprAnIiIiLa6iupacvYc3Sjm0BHP/wcMzajFhgfSNDeO6s3vVh7pwencLJcjftwUrl5PhjZ7HInI0hT0RERE5ZWrrXOQWVTSerdtTxo7ig54xwQG+9OkWxiX9YkmpX37ZNzacqJCAFqxcmlNz9zwWkaYp7DWDoqIiRo8eDcCePXvw9fUlpn53royMDAICWvaX0+rVq/nnP//JM888c9R98fHxZGdnExkZeczHH2vM/v37mTlzJl9//TXWWs4991z+8pe/HPdcJ2rLli1MmDCBrKysE35McXExixcvZvr06b/4+UVE5Jex1rL3QPXhHTDrg92WgnJq6pdg+voYkqJD6B8fwW/OiveEuvjOnfBRA/J2yxs9j0WkaQp7zaBLly6eUPLQQw8RGhrKrFmzGo2x1mKtxcfH55TXN2zYMIYNG9bs573++utJS0vjlVdeAeD+++/nd7/7HW+++WazP9chtbW1+Pk1/W1bXFzMnDlzfnbYO945RUTkp5VVOdm0t+yo2brSysNLMLuFB5ISG87w3tGk1PeuO72rlmB2RNlF2Y2C3aFr+LKLshX2RJpZh32H69y3j113/574Z572Wo+cLVu2cOWVVzJ8+HBWr17Nu+++y8MPP8y6deuorKxk0qRJPPCAe8vm+Ph4rr32Wv79739jjOEf//gHycnJ7N27l1tuuYUdO3bg4+PDc889x9lnn80ll1zC3r17Adi2bRsvvfQSV111FdOnT2fdunX4+/vz7LPPMmLECD755BOef/55li1bRkFBAZMnT6aoqIhhw4ZhD21bBowdO5bdu3dTVVXFXXfdxY03HrsFYk5ODtnZ2SxZssRz20MPPURycjJbt25l+/btnucEmD59OsOHD+faa6/lwQcf5P3336eyspLhw4fz0ksvYYxhzZo1/O53vyMkJITzzjvPc96XX36ZTz75hPLycqqrq3nzzTe58sorKSkpoba2lscff5wxY8Zw7733kpOTw6BBg/jVr37FhRdeeMwa4uPjmTZtGh9++CF33nkngwYNYsaMGRQWFhISEsLLL79Mnz59mu+bQUSkHXDWudheWNFoB8yN+WXsKqn0jAkN9KNPt1Au6x/n2QGzb2wYkcFagiluHbXnsUhL6LBhr/DFl6jMzPR6j5wNGzawYMEC5sxx95J/4okniIqKora2llGjRjFhwgTOPPNMADp37kxGRgbz58/n7rvvZtmyZdx+++384Q9/4OyzzyY3N5cxY8aQnZ3Nv/71L8C9TPSmm25i7NixPPfccwQEBPDdd9/x/fffc9lll7F58+ZG9Tz44IOMGjWK++67j7feestTF8Df/vY3oqKiOHjwIGlpafz617+mc+fOTb6u77//nsGDBzeaqfTz82PgwIFs3LiRoKCgY35N7rjjDh5++GGstUyePJkPP/yQSy+9lKlTpzJv3jzOO+887rrrrkaPWbVqFVlZWXTu3Bmn08lbb71FWFgY+/bt47zzzmPMmDE88cQTbNmyxTPL+sknnxz3v01ISAhffvklAKNGjeLll1/mtNNO48svv2TGjBl89NFHx328iEh7Za0lv7SqUWuDH/aUsa2ggpo69xJMPx9DckwIQ3p1ZvKwnp7ZuvjOnTBGSzBFRFqDDhn2nPv2Ubp0KVhL6ZIlxNx6i9dm90477TSGDh3qOf7HP/7B//7v/1JbW8vu3bvZsGGDJ+xdffXVAFxzzTXce++9gDuw5OTkeB6/f/9+Kisr6dSpE/v27WPKlCm8+eabhIeHs3LlSu655x4A+vXrR/fu3dmyZUujer744gvef/99AK644grCwsI89z3zzDO8/fbbAOTl5bF161bS0tKafF3W2iZ/mTecKTyWTz/9lKeeeoqqqioKCws566yzGDp0KJWVlZ4Zveuuu47ly5d7HnPxxRd7gqe1lv/8z/9k5cqV+Pj4sHPnTgoLC3/yeY80adIkAEpKSvj666/59a9/7bmvtrb2Z59PRKQtOlDlPKpfXc6eMg5UHf45GBcRREpsGOkpMZwRG05KbBjJMSEE+mkJpohIa9Yhw17hiy9hXe6/TFqXy6uzeyEhIZ7PN2/ezJ///GcyMjKIjIzk2muvpaqqynP/scJTU5u81NbWMmnSJP7rv/7LExZPJGgd63k++eQTvvjiC77++ms6derE8OHDG9V2pH79+rFu3TpcLpdndq+uro7s7GwGDx7M5s2bcdV/jQHPuQ4ePMiMGTNYt24dPXr04P777/fcd7y/BDf8Ov7973+ntLSUdevW4efnR3x8fJO1+vn5NVnDkee01hIdHf2zNoMREWntjrxcoabWxbbCck+w+yHfHex2lx7+2RgW6EdKbBhjB3avX4IZTkq3MCKC/VvwlYiIyMk69buFtDDPrJ6z/qJxp5PSJUuoLSjw+nMfOHCAsLAwwsPDyc/P9yzFPOS1114D3LN/h2a4LrzwQl544QXPmEOB5J577mHo0KFMmDDBc9+IESNYtGgRABs3biQ/P5/TTz+90XM0HPPOO+9QVlYGQGlpKVFRUXTq1Invv/+eNWvWHPe19O3blzPPPJMnnnjCc9vDDz/MpZdeSo8ePejVqxfff/89NTU17N+/n3//+98AVFZW4uPjQ3R0NGVlZZ7NXKKjowkKCmLVqlUAnhqbUlpaSteuXfHz8+Pjjz9m165dAISFhXleD3DMGo7UuXNn4uLiWLrU3TTX5XLx7bffHvf1i4i0ZkXl1Xz7+NMczMxk6V2PcMkzX3DmAx/yq2dXcMc/s3h5xTbyS6sYmhTFH36VwvypaXx57wWsf+hi3rjlXB4b35/rzknEkRSloCci0oZ1uJm9hrN6h3h7du+QIUOGcOaZZ5KamkpycnKjTUjAPevlcDg8G7QAvPDCC9xyyy0sWLDAc53fc889x7PPPktqaqrnurLHH3+cmTNnMm3aNPr374+/vz9///vfj5oRfPjhh7n66qtZvHgxo0aNokePHgBcfvnlzJs3j4EDB9K3b98T2r1z4cKFzJw5k9NPP53S0lLOOeccT2BKSkriyiuvpH///vTp04chQ4YA7p1Lp0yZQmpqKr169Wr0PAsWLODGG28kJCSEiy+++JjPe9111zF27FjS0tIYMmQIvXv3BqBbt26kpaXRv39/Lr/8cp544okma2jKP//5T2655RYeeughampquPbaaxk4cOBPfg1ERFpaaaWT7F2lfJtXwnd5pazPK+Xgnr0s+Og9jLX0+eZz+pw9jtEjkj2tDZKiQwjw63B/7xUR6XDMiS79ay3S0tLs2rVrG922ceNGzjjjjBN6/LYrx1P9ww9H3R7Yty/Jy5Y2S40n40T63bVmGzduZMyYMbz44otccsklLV2O1/yc7zURkeZWUV3L97sPsD6vhPV5pazPKyG36HAz8l5dgunfI4Lxny8i7suPMbVO8PcncsIEr/9BU0RETh1jTKa1tunNNRrocDN7LRno2rMzzjiDrVu3tnQZIiLtRpWzjg35BzyzdevzSthSUM6hv9F2jwhiQHwkv0lLYEB8BP17RBAZHIBz3z62/s+n2NrGlyt4czMyERFpnTpc2Gut8vLyWroEERFpITW1LjbtLWN9Xinf7Srh252lbNpbRq3LneyiQwMZGB/B5QPi6oNdJDFhgU2eqyUvV2iv5mfPJ7VLaqM+cBn5GWQXZTfZM05EpLVQ2BMRETmF6lyWLfvKDy/F3FXKxvwD1NS6A1pEJ38GxEcwrW8y/XtEMjAhgtjwoBPuXVeZlXV4E7JDnE4qv/mmuV9Kh5HaJZVZn89idvpsHHEOMvIzPMciIq1Zuwl7x+r7JtJc2tr1rSLS8lwuS25RBd/tKuXbne5Zu+xdB6h01gEQGuhHao9wpp6byID4CAb0iCQh6pc1JdflCs3PEedgdvpsZn0+i4kpE1mcs9gT/EREWrN2EfaCgoIoKiqiS5cuCnziFdZaioqKCAoKaulSRKSVstaSt7/SHezqd8b8blcpZfXNyYP8fejXPYJJQxMYmOBeipkcHYKPj35vtQWOOAcTUyYyd/1cpg2YpqAnIm1Cuwh78fHx5OXlUXAKeuVJxxUUFER8fHxLlyEircTeA1WejVPW1we74ooaAPx9DWfEhTNuYHcGxkfSPz6C3l1D8fNVu4O2KiM/g8U5i5k2YBqLcxbjiHUo8IlIq9cuwp6/vz9JSUktXYaIiLRTReXVrN9V2mhnzH1l1QD4+hh6dw3lwjO6MiA+kgHxEaTEhhHo59vCVUtzaXiNniPOgSPW0ehYRKS1ahdhT0REpLkcalLecGfMXSWVABgDydEhnHd6tPsau/gIzoyLoFOAgl17ll2U3SjYHbqGL7soW2FPRFo1rzZVN8ZEAi8DqYAFbrDWrmpwvwH+DFwGHASmWmvXHe+cTTVVFxERORkHa2rJ3uVuUv5dfcDbXljhub9nVDD94yMYWN/uILVHOGFB/i1YsYiISOtpqv5n4ENr7QRjTAAQfMT9lwK96/8NA16q/ygiItKsqpx1bMw/4Al16/NK2LKvnPpWdsRFBDEgPoIJZ8XTv4d71i4yOKBlixYREfkFvBb2jDHhwAhgKoC1tgaoOWLYFcDfrXt68WtjTKQxJs5am++tukREpP1z1rnI2VNWH+zcG6jk7GnYpDyAAfGRXJpa36Q8PoKuYdptV0RE2hdvzuwlAwXAAmPMQCATuMNaW9FgTA9gZ4PjvPrbGoU9Y8zNwM0APXv29GLJIiLS1tS5LFsLyhvtjLmhiSblN49Irr/OLpK4iBNvUi4iItJWeTPs+QFDgJnW2tXGmD8D9wJ/ajCmqd+0R11EaK2dB8wD9zV7XqhVRETaAJfL8mPxwcPtDvJKyd5dysEad5PykABfUntEMOWcXp6dMXtGBSvYiYhIh+TNsJcH5FlrV9cfv4E77B05JqHBcTyw24s1iYhIK+Tct49dd/+e+Geexi8mBnA3Kd9VUsl3eaV8W78z5vq8w03KA/186Nc9nIlpCZ6dMZOiQ/FVk3IRERHAi2HPWrvHGLPTGJNirc0BRgMbjhj2NjDDGPNP3BuzlOp6PRGRjqfwxZeozMzkm8f+Hysuu97T066oQZPyvrHhjB3Y3bMzZu9uofirSbmIiMgxeXs3zpnAovqdOLcB1xtjpgNYa+cA7+Nuu7AFd+uF671cj4iItALWWrYXVrAmt5j167cy8fU3CbAW/4/eZ5EZSEyvHlzQtysDEiIZ0COCvnFqUi4iIvJzeTXsWWuzgCP7P8xpcL8FbvNmDSIi0vLqXJYf9hwgY3sxa3KLydi+n8LyagDu/n4pPvWXawf4wLLwHBLuvLolyxUREWkXvD2zJyIiHVB1bR3f5ZWSkVtMxvZiMnP3U1btvtauR2Qnzu8dzdDEKIaGuai7+v/D1rnvM7VOKpYto3bGbZ5r90REROTkKOyJiMgvVlFdy7od+8nY7g53WTtLqK5vfXB611DGDOzOsKQohiZF0SOyk+dx+Q89TInL1ehc1uWi4MWXiHvwgVP6GtqL+dnzSe2SiiPO4bktIz+D7KJsbki9oQUrExGRU01hT0REfrbiihrW5BazZnsxGbnFfL/7AHUui4+B1B4RXHt2L/fMXWJnuoQGHvM8lVlZ4HQ2vtHppPKbb7z8Ctqv1C6pzPp8FrPTZ+OIc5CRn+E5FhGRjsW4L5trO9LS0uzatWtbugwRkQ5ld0kla3KLWb3dHfA27ysHIMDPh0EJke5Zu8QohvTqTGig/o7Y0g4FvIkpE1mcs9gT/EREpH0wxmRaa4/cG+Uo+o0sIiKNWGvZWlDhmblbvb2YXSWVAIQG+pGW2JkrB/fAkRTFgPgI7ZLZCjniHExMmcjc9XOZNmCagp6ISAelsCci0sHVuSwb8w94Zu3W5BZ7+ttFhwYwNDGK3w1PwpEUxRlx4Wpa3gZk5GewOGcx0wZMY3HOYhyxDgU+EZEOSGFPRKSDqXLWsT6vtL4FQjGZP+6nvH6nzPjOnUhPicGR6N5MJTk6BGMU7tqShtfoOeIcOGIdjY5FRKTjUNgTEWnnyqtryfxxPxnbi1izfT9ZeSXU1O+U2adbKFcM6o4jKQpHUhRxEZ1+4mzS2mUXZTcKdo44B7PTZ5NdlK2wJyLyc+zMgNwVkHg+JLTNn5/aoEVEpJ0pKq/2NC7PyC1iw+4DuCz4+hhSu4fjqN9MZWhiFJ1DAlq6XBERkdZnZwb8bRzU1YBvAEx5u1UFPm3QIiLSQeTtP+hZkpmxvZitBRUABPr5MLhnJDNGnY4jqQuDe0YSop0yRUREflruCnfQs3Xuj7krWlXYO1H6rS8i0oZYa9myr5yMQz3uthezu7QKgLAgP9J6dWbCWQk4kjqT2kM7ZYqIiJyUxPPdM3qHZvYSz2/pik6Kwp6ISCtWW+diQ/4Bz6zd2h/3U1y/U2ZMWCCOxChuTuyMI6kLKbFh2ilTRESkOSQ43Es32/g1ewp7IiKtSJWzjqydJe5Zu9xi1v24n4qaOgB6RgUzKqWru4F5UhSJXYK1U6aIiIi3JDjabMg7RGFPRKQFHahy1u+U6V6WuT6vlJo6906ZKd3CuGpIPEOTonAkRhEbEdTC1YqIiEhborAnInIKFZRVezZTWZNbzMZ8906Zfj6G1B4RTD0vEUdiFGmJnYkM1k6ZIiIicvIU9kREvMRaS97+Ss/1dmtyi9lW6N4pM8jfhyE9OzPzgt44kqIY3DOS4AD9SBYREZHmo3cWIiInwblvH7vu/j3xzzyNX0wMAC6XZUtBOau3H94pc88B906Z4UF+DE2MYuLQBBxJUaR2jyDAz6clX4KIiIi0cwp7IiInofDFl6jMzGTjk8/y9djfsXp7MWt/LKbkoBOArmGBOJKiPA3MU7qF4aOdMkVExJt2ZrT53SOleSnsiYicIGstWwsq+Cojh0FvvIm/tdS99w4vOlOJ6BHLRWd08wS8nlHaKVNERE6hnRnwt3GH+8JNeVuBTxT2RESOp7y6lq+2FPL5pgI+31RA3v5Kbst6E1zuHTP9fSxvhvxA8j1Xt3ClIiLSoeWucAc9W+f+mLtCYU8U9kREGrLWkrO3jM9zCvgsp4C1PxbjrLOEBPhy7unRzBgYycD314HL3fvOp7aWmnffovbOGZ5r90RERE65xPPdM3qHZvYSz2/piqQV8GrYM8bkAmVAHVBrrU074v4I4BWgZ30ts621C7xZk4jIkUornXy5pZDPc9yzd4c2VekbG8YN5yWRnhJDWq8oAvx8yH/oYUqsq9HjrctFwYsvEffgAy1Rfps2P3s+qV1SccQd/utzRn4G2UXZ3JB6QwtWJiLSxiQ43Es3dc2eNHAqZvZGWWsLj3HfbcAGa+1YY0wMkGOMWWStrTkFdYlIB+VyWTbkH+DzTQV8lrOPdTtKqHNZwoL8OL93NOl9YhjRJ4a4iE5HPbYyKwuczsY3Op1UfvPNKaq+fUntksqsz2cxO302jjgHGfkZnmMREfmZEhwKedJISy/jtECYce9iEAoUA7UtW5KItEf7K2pYsaWQz3L28cWmQgrLqwFI7RHO9PRkRqZ0ZVBCJP6+x2+HkLxs6akot8NwxDmYnT6bWZ/PYmLKRBbnLPYEPxEREfllvB32LPCRMcYCc6218464/3ngbWA3EAZMsvaI9VEiIiehzmVZn1dSP3tXwLd5JVgLkcH+nN87hpF9Yji/TzRdw4JautQOzxHnYGLKROaun8u0AdMU9ERERJqJt8Peedba3caYrsDHxpgfrLVfNLj/EiALuAA4rX7MCmvtgYYnMcbcDNwM0LNnTy+XLCJtVUFZNSs2u8Pdis0F7D/oxBgYGB/J7Rf0ZmRKDAPiI/FVv7tWJSM/g8U5i5k2YBqLcxbjiHUo8ImIiDQDr4Y9a+3u+o/7jDFLAQfQMOxdDzxhrbXAFmPMdqAvkHHEeeYB8wDS0tKsN2sWkbajts5F1s4SPqvfWOW7XaUARIcGMCqlK+kpMZzfO4aokIAWrlSOpeE1eo44B45YR6NjEREROXleC3vGmBDAx1pbVv/5xcAjRwzbAYwGVhhjugEpwDZv1SQibd+e0iq+qO95t2JzAQeqavH1MQzpGcmsi/uQ3qcr/bqH46PZuzYhuyi7UbA7dA1fdlG2wp6IiMgvZNyTal44sTHJwKGdDPyAV621jxljpgNYa+cYY7oDC4E4wOCe5XvleOdNS0uza9eu9UrNItL61NS6yPxxP59t2sfnOQX8sKcMgG7hgaT3iWFkSlfOOy2aiGD/Fq5URER+lp0ZahMgcpKMMZlHtrVritdm9qy124CBTdw+p8Hnu3HP+ImIeOTtP8jnmwr4PKeAr7YWUV5di5+PIS2xM/de2pf0PjH0jQ3DvZGviIi0OTsz4G/jDjcAn/K2Ap+IF7R06wUREaqcdazJLebznAI+21TAln3lAPSI7MS4Qd1J7xPDeadHExqoH1kiIu1C7gp30LN17o+5KxT2RLzgmO+cjDFRx3ugtba4+csRkY7ix6IKz8Yqq7YWUemsI8DXh2HJUfx2aAIjU2I4LSZUs3ciIu1R4vnuGb1DM3uJ57d0RSLt0vH+TJ6Ju0+eAXoC++s/j8S9sUqS16sTkXajsqaOr7cV1fe920du0UEAenUJZmJaPOkpMZyd3IXgAM3eiYi0ewkO99JNXbMn4lXHfFdlrU0CMMbMAd621r5ff3wpcOGpKU9E2iprLVsLKjzhbvX2YmpqXQT5+3BOchemnpvIyJSuJEaHtHSpIiLSEhIcCnkiXnYif0Ifaq2dfujAWvuBMea/vFiTiLRR5dW1fLWl0L25yqYC8vZXAnBaTAjXnd2L9D4xOJKiCPL3beFKRURERNq/Ewl7hcaY+4FXcC/rvBYo8mpVItImWGvJ2Vvm3lglp4C1PxbjrLOEBPhy7unRTE8/jfQ+MSREBbd0qSIiIiIdzomEvauBBzncM++L+ttEpAMqrXTy1ZZCz+Yqew5UAdA3NowbhieR3ieGtF5RBPj5tHClIiIiIh3bT4a9+l037zgFtYhIK+RyWTbkH/D0vcvcsZ86lyUsyI/ze0eT3ieG9D5diY0IaulSRURERKSB47VeeAf3ss0mWWvHeaUiEWlx+ytqWLGlkM9y9vHFpkIKy6sBSO0RzvT0ZEamdGVQQiT+vpq9ExEREWmtjjezN/uUVSEiLarOZVmfV1K/c2YB3+aVYC1EBvtzfu8YRvaJ4fw+0XQN0+ydiHRgOzPUKkBE2pTjtV74/NDnxphOQE9rbc4pqUpEmpVz3z523f174p95Gr+YGAAKyqpZsdkd7lZsLmD/QSfGwMD4SG6/oDcjU2IYEB+Jr4+amouIsDMD/jbucBPwKW8r8IlIq/eT1+wZY8binuULAJKMMYOAR7SMU6TtKHzxJSozM9nwP8/w0UVT+HxTAd/tKgUgOjSAUSldSU+J4fzeMUSFBLRwtSIirVDuCnfQs3Xuj7krFPZEpNU7kd04HwIcwGcA1tosY0yi1yoSkWZTU+viy9U/EP3Gm/hZi+v9d/mHawDJfXoy6+I+pPfpSr/u4fho9k5E5PgSz3fP6B2a2Us8v6UrEhH5SScS9mqttaXG6M2gSFtQU+viy62FvLc+n399v4f/+Po1LnG5APD3sSwJ+YFe03/bwlWKiLQxCQ730k1dsycibciJhL1sY8xkwNcY0xu4HfjKu2WJyM/hrHOxamsR763P58Pv91Ba6SQsyI8rEgK5fEkmxlUHgE9tLZVvL6P29ts81+7JzzM/ez6pXVJxxB1+o5eRn0F2UTY3pN7QgpWJiNclOBTyRKRNOZF902cC/YBq4B/AAeBObxYlIj+tts7Fl1sK+eOS73A89gn/MT+D977L54K+XXn5P9JYe/+F3Lbzc4xt3EHFulwUvPhSC1Xd9qV2SWXW57PIyM8A3EFv1uezSO2S2sKViYiIiDR2Ik3VDwL/X/0/EWlBdS5LxvZi3l2/mw+z91BUUUNwgC8XntGNywfEkd4nhiB/X8/4yqwscDobn8TppPKbb05x5e2HI87B7PTZzPp8FhNTJrI4ZzGz02c3mukTERERaQ2O11T9WWvtncdqrq7dOEVODZfLsia3mPe+y+f97/ZQWF5NJ39fLjijK2MHxDEypWujgNdQ8rKlp7jajsER52BiykTmrp/LtAHTFPRERESkVTrezN7/1X9Uc3WRU8zlsqzbsZ931+fz/nf57Cur/v/bu/cwu+r63uPv7+RCIIRbEpIJyRDut0kQHCYoRq6KEhppVVBoxYMWqMVSa6ytx2PVnraeNj21rQ8I5VCxEjQqKBLlomhIa8kQSEgGuYg4TEIm5AKEQCCZzHzPH3sHJpNkMrnsrJk979fz5Fl7r7X2Wp+E/ZD55PdbazFsSA3nHH8o0yaN4+zjR7Pf0N5ccqtKaGprYvaTs7lq8lXMfnI2jWMbLXySJKnP6emnxXMjYq0bhpYAAB2FSURBVFHXh6tLqpzMZOHSl5hTLnhta19n6OAazj5uNNMmj+Pc4w9l+D4WvKJtvkZv89TNxrGNW7yXJEnqK3r6yfFw4OGI+OPM/K+9FUgaSDKTxcvWMmdJG3MWt/HcS68xdFAN7zx2NJ99z/Gce8KhjBg2pOiY6qJ5TfMWxW7zNXzNa5ote5IkqU+JzK0ux3tzY8SpwL8CTwDXA52bt2XmIxVPtw0NDQ25YMGCIk4t7RGZyWPLX+ZHi5czZ3Eby158jSGDgqnHjGbapFreddIYDrDgSZIkaTsi4uHMbNjRfj3OCcvMRyLifwLfB47izRu1JHBOL0K0AOuADkoPZ98qUEScBXwVGAKszswzd3Rcqb/JTB5vW8ddi5czZ0kbz65Zz+Ca4IyjR3Htucfw7hPHcuB+FjxJe9DSJh8ALkkDXE934zwU+EfgSOCczHx0F89xdmau3s45DgKuA96Tma3lc0pVITN58vl1zFlcmqL5zOpXGVQTvP2okXzirKN494ljOXj40KJjSqpGS5vglunQsREGDYXL77TwSdIA1NPI3oPAV4CPZE9zPXfPpcDtmdkKkJkrK3Qeaa/59fPruGtxG3OWtPH0yleoCXjbUSP5+NQjOf+kMYzcf5+iI0qqdi3zSkUvO0rLlnmWPUkagHoqe1Myc9VuHj+BeyMigRsy88Zu248FhkTEL4ARwD9n5jd385zSXvebVa+8MYL35PPriIApRxzC5W+v5731YxllwZO0N02cWhrR2zyyN3Fq0YkkSQXYbtnbA0UP4IzMXF6ennlfRDyRmQ90O/9bgXOBfYH/jogHM/OprgeJiCuBKwHq6ur2QCxp9/129avMWbycuxa38cSKUsE77fBD+NL0k3jvpLEcOmJY0RElDVQTGktTN71mT5IGtIo+tCszl5eXKyPiDqAR6Fr2llG6KcurwKsR8QBwMvBUt+PcCNwIpbtxVjKz1JPWNeu5a0npLpqPLX8ZgLcefjB/9Tsn8t76WsYeaMGT1EdMaLTkSdIAV7GyFxHDgZrMXFd+/W7gy912+yHwtYgYDAwFpgD/VKlM0q5Y+sJ6frykdA3e4mVrATil7iA+P+0ELphUy7iD9i04oSRJkrS1nu7G+SPefNTCVjJz+g6OPQa4IyI2n2dWZt4dEVeXP//1zHw8Iu4GFlN6ht9Nmdm8k78HaY9b/tJr/HhJG3ctbmPR0pcAOHn8gXzuguO5YFIt4w/er+CEkiRJUs+2+1D1iNj8vLvfA8YC3yq//zDQkpmfq3y8rflQdVXKirWvlwvech5pLRW8+sMOYNqkcVw4uZYJh1jwJEmSVLzdfqh6Zs4tH+ivM/OdXTb9qHxtndTvrXz5dX7SvIK7Fi/noZYXATih9gA+c/5xTJtUy8RRwwtOKEmSJO2a3lyzNzoijszMZwAi4ghgdGVjSZWzat0G7m4uTdFsanmBTDh+7Ag+/a5juWByLUeN3r/oiJIkSdJu603Z+xTwi4h4pvx+InBVxRJJFbDmlQ3c/dgK5ixu48Fn1tCZcPSh+3PtuccwbVItx4wZUXRESZIkaY/aYdkr31TlGOD48qonMnNDZWNJu+/FVzdyz2MrmLOkjV/+Zg0dncmRo4ZzzdlHM23yOI4dsz/lGwhJkiRJVae3j154K6URvcHAyRFBZn6zYqmkXbR2fTv3/Ko0gvdfT69mU2cyceR+XH3mkVw4eRzHjx1hwZMkSdKAsMOyFxH/ARwFLAI6yqsTsOypT3j59Xbue+x55ixpY96vV9HekUw4ZF/+8J1HMm1SLSeNO8CCJ0mSpAGnNyN7DcCJub1nNEgFWPd6Oz99/HnmLG7jgadWs7Gjk8MO2pcrzjiCaZNrmXTYgRY8SZIkDWi9KXvNlJ6z11bhLFKPXt2w6Y2C94unVrFxUye1Bw7jI287nGmTa3nLhIMseJIkSVJZb8reKOBXEdEEvHFjlsycXrFUGvDaV67kuT/7NCP//h+Yu7qTOYvbuP+JlWzY1MmYA/bhsil1XDi5llMmHExNjQVP6veWNkHLPJg4FSY0Fp1GkqSq0Juy98VKh5C6au/oZOHf/CP7P/wwt//h5/mXSb/L6BH78KHTJnDhyeN4a50FT6oqS5vglunQsREGDYXL77TwSZK0B/Tm0QtzI2IMcFp5VVNmrqxsLA1Ey15cz3ceWspP5jYz874fE5m869mHOONLf07DqccwyIK3025uvpn6kfU01r75g3NTWxPNa5q5ov6KApNJXbTMKxW97CgtW+ZZ9iRJ2gNqdrRDRFwMNAEfBC4G5kfEByodTANDR2fys8ef54pvPMTUv/85X/v501zxzP0MLX8zB0dSN+fbFr1dVD+ynhlzZ9DU1gSUit6MuTOoH1lfcDKpi4lTSyN6Mai0nDi16ESSJFWF2NFNNiPiUeBdm0fzImI08NPMPHkv5NtKQ0NDLliwoIhTaw96/uXX+c5DS/l2UyvL177OoeVpmh88cl9eff90csMbl4cS++zD0T+9j8GjRxeYuP/aXPAuPu5iZj85m5lnztxipE/qE7xmT5KkXouIhzOzYUf79eaavZpu0zbX0IsRQam7zs7kP59eza3zn+Wnj6+kozOZeswovvA7J3LuCWMYMqiGti9+iezs3OJz2dnJquuup/avvlBQ8v6tsbaRi4+7mBsW38BVk6+y6KlvmtBoyZMkaQ/rTdm7OyLuAW4rv78E+HHlIqnarH5lA99dsIzbmlppfWE9I4cP5eNTj+DDp9UxcdTwLfZ9bdEiaG/f8gDt7by2cOFeTFxdmtqamP3kbK6afBWzn5xN49hGC58kSdIA0JsbtHwmIn4PeAcQwI2ZeUfFk6lfy0wefOYFbp3/LPc8toL2jmTKEYcw4/zjOP+kMewzeNA2P3fkD/xq7Umbp3BunrrZOLZxi/eSJEmqXr0Z2SMzbwduj4hRlKZxStv00vqNfO/hZcxqauWZVa9ywLDB/MHpE7l0ygSOPnRE0fEGnOY1zVsUu8baRmaeOZPmNc2WPUmSpCq33Ru0RMTpwFeAF4C/Bv6D0gPWa4CPZObdeytkV96gpe/JTB5pfZFb57dy1+I2Nm7q5NS6g7hsyuFMm1zLsCHbHsWTJEmStPP2xA1avgZ8DjgQuB94b2Y+GBHHU7p+r5Cyp77j5dfb+eHC57h1fitPrFjH/vsM5pKGCVw6pY4Tag8oOp4kSZI0oPVU9gZn5r0AEfHlzHwQIDOfiPCZZwPZ4mUvMWt+Kz9ctJzX2juoP+wA/u73JjH95HEM36dXM4MlSZIkVVhPP5l3vf/9a9229fxwPlWdVzds4kePLufW+a0seW4t+w4ZxPSTx3HZ6XVMHn9Q0fEkSZIkddNT2Ts5Il6mdAfOfcuvKb8f1puDR0QLsA7oADZtb15pRJwGPAhckpnf62V27QWPt73MrPmt3LHwOV7ZsInjxozgy+87iYtOOYwDhg0pOp4kSZKk7dhu2cvMPXVXjbMzc/X2NkbEIOD/APfsofNpN73e3sGcxW3cOv9ZHml9iaGDa7hwUi2XnV7HqXUH4zReSZIkqe/rCxdYfRL4PnBa0UEGuqdXvsKs+a18/5FlrH2tnSNHDefz007g/aeO5+DhQ4uOJ0mSJGknVLrsJXBvRCRwQ2be2HVjRBwG/C5wDpa9QmzY1ME9jz3PrPnP8uAzLzBkUHD+SWO5bMrhnH7kIY7iSZIkSf1UpcveGZm5PCIOBe6LiCcy84Eu278KfDYzO3oqFRFxJXAlQF1dXUUDDxSta9Yzq6mV7y5YyppXNzLhkH357HuO54MN4xm1/z5Fx5MkSZK0m7b7UPU9fqKILwKvZObMLut+S+mGL1B6YPt64MrM/MH2juND1Xdde0cnP3v8eW6d38q8X69mUE1w3gmHcumUw5l69ChqahzFkyRJkvq6PfFQ9d0NMByoycx15dfvBr7cdZ/MPKLL/t8A7uqp6GnXPPfSa3ynqZVvP7SUles2UHvgMD513rFcctoExh7YqxurSupuaRO0zIOJU2FCY9FpJEmStlLJaZxjgDvK0zMHA7My8+6IuBogM79ewXMPeB2dydynVnLrg638/MmVJHDWsaP52ymHc9Zxoxk8qKboiFL/tbQJbpkOHRth0FC4/E4LnyRJ6nMqVvYy8xng5G2s32bJy8yPVirLQLLy5df5zkNL+fZDS3nupdcYPWIfPnHW0XyocQLjD96v6HhSdWiZVyp62VFatsyz7EmSpD6nLzx6QbupszP5r9+s5tYHW7nv8efp6EzecfQoPj/tBM47cQxDHMWT9qyJU0sjeptH9iZOLTqRJEnSVix7/diaVzbw3YeXcVtTK8+uWc/B+w3h4+84gg831jFx1PCi40nVa0Jjaeqm1+xJkqQ+zLLXz2Qm83/7ArfOb+We5hVs7Oik8YhD+LN3Hct76seyz+BBRUeUBoYJjZY8SZLUp1n2+omX1m/k+488x6z5z/KbVa9ywLDBXHZ6HZc21nHMmBFFx5MkSZLUx1j2+rDM5JHWl5g1v5W7Fi9nw6ZOTqk7iH/4wGQunDyOfYc6iidJkiRp2yx7fdC619v5waLl3PrgszyxYh3Dhw7iA28dz6VT6jhp3IFFx5MkSZLUD1j2+pAly9Yyq+lZfrhoOes3dnDSuAP429+dxPS3jGP/ffxPJUmSJKn3bBAFW79xEz96dDm3zm9l8bK1DBtSw/STx3HZlMOZPP5Ayg+llyRJkqSdYtkryBMrXmbW/FbueOQ51m3YxLFj9udL00/iolMO48B9hxQdT5IkSVI/Z9nbi15v7+DHS9qYNb+VBc++yNDBNUybVMulU+poOPxgR/EkSZIk7TGWvb3gN6teYdb8Vr7/yDJeWt/OEaOG8/lpJ/D+U8dz8PChRceTJEmSVIUsexWycVMn9zy2glnzW/nvZ9YwuCY4v34slzXW8bajRjqKJ0mSJKmiLHt7WOua9dz2UCvfXbCU1a9sZPzB+/KZ84/jgw3jOXTEsKLjSZIkSRogLHt7wOsrnufxT/wJN571MX6yvJ2agHNPGMNlU+p45zGjqalxFE+SJEnS3mXZ2wN++pd/xxG/WsJJHd/juE/O4JLTJlB74L5Fx+qXbm6+mfqR9TTWNr6xrqmtieY1zVxRf0WBySRJkqT+paboAP1d+8qVHPXwL6ghOa+liWtOPtiitxvqR9YzY+4MmtqagFLRmzF3BvUj6wtOJkmSJPUvlr3dtPq66yE7AcjOTlZdd33Bifq3xtpGZp45kxlzZ/C1hV9jxtwZzDxz5hYjfZIkSZJ2zLK3G9pXrmTtHXdAe3t5RTtrb7+dTatWFRusn2usbeTi4y7mhsU3cPFxF1v0JEmSpF1g2dsNq6+7nuzs3GKdo3u7r6mtidlPzuaqyVcx+8nZb0zplCRJktR7lr3d8NqiRW+O6m3W3s5rCxcWE6gKbL5Gb+aZM7nmlGvemNJp4ZMkSZJ2TmRm0Rl2SkNDQy5YsKDoGKoQ78ZZIUuboGUeTJwKE5wWK0mS1J9FxMOZ2bDD/Sx7UpVb2gS3TIeOjTBoKFx+p4VPkiSpH+tt2avoNM6IaImIJRGxKCK2amgRcVlELC7/+mVEnFzJPNKA1DKvVPSyo7RsmVd0IkmSJO0Fe+Oh6mdn5urtbPstcGZmvhgR7wVuBKbshUzSwDFxamlEb/PI3sSpRSeSJEnSXrA3yt52ZeYvu7x9EBhfVBapak1oLE3d9Jo9SZKkAaXSZS+BeyMigRsy88Ye9v0Y8JMK55EGpgmNljxJkqQBptJl74zMXB4RhwL3RcQTmflA950i4mxKZe8d2zpIRFwJXAlQV1dXybySJEmSVBUqeoOWzFxeXq4E7gC2GlqIiMnATcD7MnPNdo5zY2Y2ZGbD6NGjKxlZkiRJkqpCxcpeRAyPiBGbXwPvBpq77VMH3A78QWY+VakskiRJkjTQVHIa5xjgjojYfJ5ZmXl3RFwNkJlfB74AjASuK++3qTfPi5AkSZIk9axiZS8znwG2em5eueRtfv1x4OOVyiBJkiRJA1VFr9mTJEmSJBXDsidJkiRJVciyJ0mSJElVyLInSZIkSVXIsidJkiRJVciyJ0mSJElVyLInSZIkSVXIsidJkiRJVciyJ0mSJElVyLInSZIkSVXIsidJkiRJVciyJ0mSJElVyLInSZIkSVXIsidJkiRJVciyJ0mSJElVyLInSZIkSVXIsidJkiRJVciyJ0mSJElVyLKnvmlpE8z7x9JSkiRJ0k4bXHQAaStLm+CW6dCxEQYNhcvvhAmNRaeSJEmS+hVH9tT3tMwrFb3sKC1b5hWdSJIkSep3Klr2IqIlIpZExKKIWLCN7RER/xIRT0fE4og4tZJ51E9MnFoa0YtBpeXEqUUnkiRJkvqdvTGN8+zMXL2dbe8Fjin/mgJcX172Czc330z9yHoaa9+cYtjU1kTzmmauqL+iwGT93ITG0tTNlnmloucUTkmSJGmnFT2N833AN7PkQeCgiKgtOFOv1Y+sZ8bcGTS1lW4i0tTWxIy5M6gfWV9wsiowoRGmftqiJ0mSJO2iSo/sJXBvRCRwQ2be2G37YcDSLu+Xlde1VTjXHtFY28jMM2cyY+4MLj7uYmY/OZuZZ87cYqRPkiRJkopQ6bJ3RmYuj4hDgfsi4onMfKDL9tjGZ7L7ioi4ErgSoK6urjJJd1FjbSMXH3cxNyy+gasmX2XRkyRJktQnVHQaZ2YuLy9XAncA3ZvQMmBCl/fjgeXbOM6NmdmQmQ2jR4+uVNxd0tTWxOwnZ3PV5KuY/eTsN6Z0SpIkSVKRKlb2ImJ4RIzY/Bp4N9Dcbbc7gY+U78p5OrA2M/vFFE548xq9mWfO5JpTrnljSqeFT5IkSVLRKjmyNwb4z4h4FGgC5mTm3RFxdURcXd7nx8AzwNPAvwGfqGCePa55TfMW1+htvoaveU33TitJkiRJe1dkbnWJXJ/W0NCQCxZs9cg+SZIkSRoQIuLhzGzY0X5FP3pBkiRJklQBlj1JkiRJqkKWPUmSJEmqQpY9SZIkSapClj1JkiRJqkKWPUmSJEmqQpY9SZIkSapClj1JkiRJqkKWPUmSJEmqQpY9SZIkSapClj1JkiRJqkKWPUmSJEmqQpY9SZIkSapClj1JkiRJqkKWPUmSJEmqQpY9SZIkSapClj1JkiRJqkKWPUmSJEmqQpY9SZIkSapClj1JkiRJqkKWPUmSJEmqQpY9SZIkSapCFS97ETEoIhZGxF3b2FYXET8vb18cERdUOo8kSZIkDQR7Y2TvWuDx7Wz7PDA7M08BPgRctxfySJIkSVLVq2jZi4jxwDTgpu3sksAB5dcHAssrmUeSJEmSBorBFT7+V4E/B0ZsZ/sXgXsj4pPAcOC8CuepjKVN0DIPJk6FCY1Fp5EkSZKkyo3sRcSFwMrMfLiH3T4MfCMzxwMXAP8REVtliogrI2JBRCxYtWpVhRLvoqVNcMt0uP9vSsulTUUnkiRJkqSKTuM8A5geES3At4FzIuJb3fb5GDAbIDP/GxgGjOp+oMy8MTMbMrNh9OjRFYy8C1rmQcdGyI7SsmVe0YkkSZIkqXJlLzP/MjPHZ+ZESjdfuT8zf7/bbq3AuQARcQKlstfHhu52YOJUGDQUYlBpOXFq0YkkSZIkqeLX7G0lIr4MLMjMO4FPA/8WEZ+idLOWj2Zm7u1Mu2VCI1x+p9fsSZIkSepTor91q4aGhlywYEHRMSRJkiSpEBHxcGY27Gi/vfGcPUmSJEnSXmbZkyRJkqQqZNmTJEmSpCpk2ZMkSZKkKmTZkyRJkqQqZNmTJEmSpCpk2ZMkSZKkKmTZkyRJkqQqZNmTJEmSpCoUmVl0hp0SEauAZ4vOob1iFLC66BBSD/yOqq/zO6q+zu+o+rq++h09PDNH72inflf2NHBExILMbCg6h7Q9fkfV1/kdVV/nd1R9XX//jjqNU5IkSZKqkGVPkiRJkqqQZU992Y1FB5B2wO+o+jq/o+rr/I6qr+vX31Gv2ZMkSZKkKuTIniRJkiRVIcue+pSImBARP4+IxyPisYi4tuhM0rZExKCIWBgRdxWdRdqWiDgoIr4XEU+U/5/6tqIzSV1FxKfKf9c3R8RtETGs6Ewa2CLi5ohYGRHNXdYdEhH3RcSvy8uDi8y4syx76ms2AZ/OzBOA04E/jogTC84kbcu1wONFh5B68M/A3Zl5PHAyfl/Vh0TEYcCfAA2ZWQ8MAj5UbCqJbwDv6bbuL4CfZeYxwM/K7/sNy576lMxsy8xHyq/XUfrh5LBiU0lbiojxwDTgpqKzSNsSEQcA7wT+H0BmbszMl4pNJW1lMLBvRAwG9gOWF5xHA1xmPgC80G31+4Bbyq9vAS7aq6F2k2VPfVZETAROAeYXm0TayleBPwc6iw4ibceRwCrg38vTjW+KiOFFh5I2y8zngJlAK9AGrM3Me4tNJW3TmMxsg9KgBHBowXl2imVPfVJE7A98H/jTzHy56DzSZhFxIbAyMx8uOovUg8HAqcD1mXkK8Cr9bOqRqlv5uqf3AUcA44DhEfH7xaaSqo9lT31ORAyhVPRuzczbi84jdXMGMD0iWoBvA+dExLeKjSRtZRmwLDM3z4z4HqXyJ/UV5wG/zcxVmdkO3A68veBM0rY8HxG1AOXlyoLz7BTLnvqUiAhK15g8npn/t+g8UneZ+ZeZOT4zJ1K6mcD9mem/RqtPycwVwNKIOK686lzgVwVGkrprBU6PiP3Kf/efizcRUt90J3B5+fXlwA8LzLLTBhcdQOrmDOAPgCURsai87nOZ+eMCM0lSf/RJ4NaIGAo8A/yPgvNIb8jM+RHxPeARSnfiXgjcWGwqDXQRcRtwFjAqIpYBfwV8BZgdER+j9I8UHywu4c6LzCw6gyRJkiRpD3MapyRJkiRVIcueJEmSJFUhy54kSZIkVSHLniRJkiRVIcueJEmSJFUhy54kqc+IiJERsaj8a0VEPNfl/S8rcL63RMQFO/mZsyIiy7fh3rzulPK6GTv47EURceIu5BxXvk39LmWWJA1Mlj1JUp+RmWsy8y2Z+Rbg68A/bX6fmW+vwCnfAuxKcVoCXNLl/YeAR3vxuYuAnSp7ETE4M5dn5gfKq3Y1syRpgLHsSZL6hYh4pbw8KyLmRsTsiHgqIr4SEZdFRFNELImIo8r7jY6I70fEQ+VfZ3Q73lDgy8Al5ZHDSyLikIj4QUQsjogHI2LyduK0AsMiYkxEBPAe4Cddjv2H5XM+Ws6wX0S8HZgO/EP5fEdFxC8ioqH8mVER0VJ+/dGI+G5E/Ai4NyImRkTzdjL/OiJGlz9XExFPR8SoPfXnLknqvwYXHUCSpF1wMnAC8ALwDHBTZjZGxLXAJ4E/Bf6Z0sjgf0ZEHXBP+TMAZObGiPgC0JCZ1wBExL8CCzPzoog4B/gmpZG0bfke8EFgIfAIsKHLttsz89/Kx/zfwMcy818j4k7grszcPCWzp9/j24DJmflCREzsIfPxwGXAV4HzgEczc3XPf3ySpIHAsidJ6o8eysw2gIj4DXBvef0S4Ozy6/OAE7sUqgMiYkRmruvhuO8A3g+QmfeXryE8MDPXbmPf2cB3gOOB24Cu00zryyXvIGB/SkVzZ92XmS/0Yr+bgR9SKntXAP++C+eSJFUhy54kqT/qOorW2eV9J2/+3VYDvC0zX9uJ425rqC23tWNmroiIduBdwLVsWfa+AVyUmY9GxEeBs7Zzvk28eUnFsG7bXu1N4MxcGhHPl0cip1Aa5ZMkyWv2JElV617gms1vImJb0zHXASO6vH+AclmKiLOA1Zn5cg/n+ALw2czs6LZ+BNAWEUPYsnx1P18L8Nby6w/QO92PAXAT8C1g9jaySJIGKMueJKla/QnQUL7Zyq+Aq7exz88pTfVcFBGXAF/c/BngK8DlPZ0gM3+ZmT/Yxqb/BcwH7gOe6LL+28BnImJh+UYyM4E/Kj9Worc3VemeGeBOStNFncIpSXpDZG5zdookSeonynf0/KfMnFp0FklS3+E1e5Ik9WMR8RfAH+G1epKkbhzZkyRJkqQq5DV7kiRJklSFLHuSJEmSVIUse5IkSZJUhSx7kiRJklSFLHuSJEmSVIUse5IkSZJUhf4/XkwTlyZdQ98AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 1080x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting the yields\n",
    "plt.figure(figsize=(15,5))\n",
    "plt.plot(t[1:], bond_yield*100, label=\"Closed-Form\")\n",
    "plt.plot(t[1:], mont_yield*100, \".\", label=\"Joint Simulation\")\n",
    "plt.plot(t[1:], squad_yield*100, \"x\", label=\"Simple Quadrature\")\n",
    "plt.plot(t[1:], trap_yield*100, \"^\", label=\"Trapezoidal Quadrature\")\n",
    "plt.title(\"Yield Curve\")\n",
    "plt.xlabel(\"Time to Maturity\")\n",
    "plt.ylabel(\"Bond Yield\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. The LIBOR Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The LIBOR Forward Market Model (LFMM)\n",
    "\n",
    "Is seen as the benchmark interest rate model (as the BS is for equity option pricing).\n",
    "\n",
    "**Market Models** attempt to model financial instruments that are actually traded in the market instead of focusing on an idealization of a financial variable (such as the short rate) which drives the market.\n",
    "\n",
    "**The LFMM results in the analytical Black-Scholes Price for Market Caps**\n",
    "\n",
    "The model can be calibrated easier using observable market instruments.\n",
    "\n",
    "(There aremany ways to make the model complex, like introducing stochastic volatility)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Problem with Continuous Rates\n",
    "\n",
    "They are not directly observable in the market. The LFMM instead is a market model, and it consistently models simple rates which are directly observable in markets.\n",
    "\n",
    "#### Simple Rates\n",
    "\n",
    "Is just an interest rate where there is **no compounding** on the interest earned\n",
    "\n",
    "$$(1 + rT)$$\n",
    "\n",
    "Therefore, the LFMM models the **simple forward rates in the market**\n",
    "\n",
    "#### Forward Rates\n",
    "\n",
    "A forward rate is an interest rate relevant to a future period that you can agree on today."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LFMM Stochastic Differential Equation\n",
    "\n",
    "Suppose we have a set of dates $\\{ T_0, T_0, ..., T_N \\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $P_j(t)$ be the price of a bond at time t with maturity $T_j$\n",
    "\n",
    "Then the forward rate bewteen times $t_j$ and $t_{j+1}$ at time $t$ is:\n",
    "\n",
    "$$ F(t,T_j, T_{j+1}1) = F_j(t) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The differential between the two dates can be expressed as:\n",
    "$$\\delta_j = T_{j+1} - T_j$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The LFMM says that each market forward rate has the following SDE:\n",
    "\n",
    "$$ dF_j(t) = F_j(t)\\mu_j(t)dt + F_j(t) \\sigma_j (t) dW_t $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where:\n",
    "\n",
    "$\\mu_j(t)$ is the time dependent drift associated with $F_j(t)$\n",
    "\n",
    "$\\sigma_j(t)$ is the time dependent volatility associated with $F_j(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Assumptions\n",
    "\n",
    "We can assume that each forward rate has its own Brownian Motion. This would imply that there is a source of **noise** or randomness for every forward rate in the market.\n",
    "\n",
    "**Simplifying assumptions:** \n",
    "\n",
    "We are going to assume there be only one source of noise in the market, therefore **only one Brownian Motion**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Time-Dependent Drift\n",
    "\n",
    "$$ \\mu_(t) = \\sum_{k=\\tau(t)}^{j} \\frac{\\delta_k F_k(t) \\sigma_k(t) \\sigma_j(t)}{1 + \\delta_k F_k(t)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where,\n",
    "\n",
    "$\\tau(t) = \\min \\{ i:t < T_i \\}$ which explains how the forward rates evolve over time\n",
    "\n",
    "**N.B.** previous Forward Rates are not useful at present time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution to the Libor SDE Paths and their Approximation\n",
    "\n",
    "### Basic Approximation\n",
    "\n",
    "We can refactor the solution to the Libor SDE to include interest rate paths in the following way:\n",
    "\n",
    "$$ \\hat{F}_j(t_i) = \\hat{F}_j(t_{i-1}) e^{\\big[\\hat{\\mu}_j(t_{i-1}) - \\frac{1}{2}\\sigma^2)\\delta_{i-1} + \\sigma_j \\sqrt{\\delta_{i-1}}Z_i \\big]} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where, \n",
    "\n",
    "$\\hat{F}_j(t_i)$ is the approximation for the $j^{th}$ forward rate at time $t_i$\n",
    "\n",
    "\n",
    "$Z \\sim N(0,1)$\n",
    "\n",
    "$$ \\hat{\\mu}_j (t_{i-1}) = \\sum_{k=1}^{j} \\frac{\\delta_k \\hat{F}_j(t_{i-1}) \\sigma_k \\sigma_j}{1 + \\delta_k \\hat{F}_j(t_{i-1})} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need to then apply this using Monte Carlo Simulation using the values for $\\hat{F}_j(t_0)$\n",
    "\n",
    "We can initialize these variables in the following way:\n",
    "\n",
    "$$ \\hat{F}_j(t_0) = F_j(t_0) $$\n",
    "\n",
    "Where $F_j(t_0)$ are implied from the market ZCB prices and the following formula:\n",
    "\n",
    "$$ F_j(t_0) = \\frac{P_j(0) - P_{j+1}(0)}{\\delta_0 P_{j+1}(0)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predictor-Corrector Method for Improving the Accuracy of Approximations\n",
    "\n",
    "We can use this method that revolves around estimating an **initial** and a **terminal** drift, then using an average of these to **project future forward rates**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialising the Estimator is done in the same way as above:\n",
    "\n",
    "$$ \\hat{F}_j(t_0) = F_j(t_0)$$\n",
    "\n",
    "##### Estimate the Forward Rate for the Next Price Period:\n",
    "\n",
    "$$ \\tilde{F}_j(t_i) = \\bar{F}_j(t_{i-1}) e^{\\big[\\mu^1_j(t_{i-1}) - \\frac{1}{2}\\sigma^2)\\delta_{i-1} + \\sigma_j \\sqrt{\\delta_{i-1}}Z_i \\big]} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where, \n",
    "\n",
    "$$ \\mu^1_j (t) = \\sum_{k=1}^{j} \\frac{\\delta_k \\bar{F}_j(t_{i-1}) \\sigma_k \\sigma_j}{1 + \\delta_k \\bar{F}_j(t_{i-1})} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then use these forward estimates to estimate the drift at time $t_i$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\mu^2_j (t_{i-1}) = \\sum_{k=1}^{j} \\frac{\\delta_k \\tilde{F}_j(t_{i-1}) \\sigma_k \\sigma_j}{1 + \\delta_k \\tilde{F}_j(t_{i-1})} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Final step is to compute the final estimate for the forward rate in the next period using:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\bar{F}_j(t_i) = \\bar{F}_j(t_{i-1}) e^{\\big[\\frac{1}{2}(\\mu^1_j(t_{i-1}) + \\mu^2_j(t_{i-1}) - \\sigma^2)\\delta_{i-1} + \\sigma_j \\sqrt{\\delta_{i-1}}Z_i \\big]} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**N.B.** $Z_i$ is used both for the preliminary and the final forward estimates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulating Bond Prices\n",
    "\n",
    "We can do this by:\n",
    "\n",
    "- **simulating forward rates**\n",
    "\n",
    "\n",
    "- using these to **imply capitalisation factors**\n",
    "\n",
    "\n",
    "- then **inverting these** to imply **discount factors** which should be equal to the bond price under no arbitrage.\n",
    "\n",
    "**N.B.** when taking averages to get our MC estimate, we only take averages when we have all simulations for the bond prices. This is a computationally expensive simulation, therefore we will be vectorizing as much as possible for maximum efficiency."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "import matplotlib.pyplot as plt\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "# parameters\n",
    "r0 = 0.05\n",
    "alpha = 0.2\n",
    "b = 0.08\n",
    "sigma = 0.025\n",
    "\n",
    "# problem parameters\n",
    "## we are dealing with bond maturities that are 2 years apart\n",
    "t = np.linspace(0,40,21)\n",
    "## volatility for the LFMM model\n",
    "sigmaj = 0.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Generate Bond Prices Using Vasicek Dynamics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Analytical Bond price\n",
    "def A(t1, t2):\n",
    "    return (1-np.exp(-alpha*(t2-t1)))/alpha\n",
    "\n",
    "def C(t1, t2):\n",
    "    val_1 = (t2-t1-A(t1,t2))*(sigma**2/(2*alpha**2)-b)\n",
    "    val_2 = sigma**2*A(t1,t2)**2/(4*alpha)\n",
    "    return val_1 - val_2\n",
    "\n",
    "def bond_price(r,t,T):\n",
    "    return np.exp(-A(t,T)*r + C(t,T))\n",
    "\n",
    "vasi_bond = bond_price(r0,0,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Applying the algorithms\n",
    "np.random.seed(0)\n",
    "# is the number of MC simulations we are going to do at each point\n",
    "n_simulations = 100000\n",
    "\n",
    "# stores the number of time steps we are simulating over (number of time increments) \n",
    "n_steps = len(t)\n",
    "\n",
    "# pre-allocate space fpr forward rates under both methods\n",
    "## We need to pre-allocate entire matrices for the forward rates and not vectors\n",
    "## We also initialize the forward rates which is why we are multipyling a matrix of ones by the\n",
    "## forward rates at time t0 implied bu the bond prices\n",
    "### Each row of the matrix consitutes one sample path\n",
    "\n",
    "mc_forward = np.ones([n_simulations,n_steps-1])*(vasi_bond[:-1]-vasi_bond[1:])/(2*vasi_bond[1:])\n",
    "predcorr_forward = np.ones([n_simulations,n_steps-1])*(vasi_bond[:-1]-vasi_bond[1:])/(2*vasi_bond[1:])\n",
    "predcorr_capfac = np.ones([n_simulations,n_steps])\n",
    "mc_capfac = np.ones([n_simulations,n_steps])\n",
    "\n",
    "# store the time increments bewteen bond maturities\n",
    "delta = np.ones([n_simulations,n_steps-1])*(t[1:]-t[:-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\mu^2_j (t_{i-1}) = \\sum_{k=1}^{j} \\frac{\\delta_k \\tilde{F}_j(t_{i-1}) \\sigma_k \\sigma_j}{1 + \\delta_k \\tilde{F}_j(t_{i-1})} $$\n",
    "\n",
    "$$ \\bar{F}_j(t_i) = \\bar{F}_j(t_{i-1}) e^{\\big[\\frac{1}{2}(\\mu^1_j(t_{i-1}) + \\mu^2_j(t_{i-1}) - \\sigma^2)\\delta_{i-1} + \\sigma_j \\sqrt{\\delta_{i-1}}Z_i \\big]} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`np.cumsum` returns the cumulative sum of the elements in a matrix with `axis=1` returning the cumprod of each row of the matrix.\n",
    "\n",
    "For Example: if we have a vector, then the second element of the returned vector is the sum of the first and second element in the original vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(1,n_steps):\n",
    "    Z = norm.rvs(size = [n_simulations,1])\n",
    "    \n",
    "    # Explicit Monte Carlo simulation\n",
    "    muhat = np.cumsum(delta[:,i:]*mc_forward[:,i:]*sigmaj**2/(1+delta[:,i:]*mc_forward[:,i:]),axis = 1)\n",
    "    mc_forward[:,i:] = mc_forward[:,i:]*np.exp((muhat-sigmaj**2/2)*delta[:,i:]+sigmaj*np.sqrt(delta[:,i:])*Z)\n",
    "    \n",
    "    # Predictor-Corrector Montecarlo simulation\n",
    "    mu_initial = np.cumsum(delta[:,i:]*predcorr_forward[:,i:]*sigmaj**2/(1+delta[:,i:]*predcorr_forward[:,i:]),axis = 1)\n",
    "    for_temp = predcorr_forward[:,i:]*np.exp((mu_initial-sigmaj**2/2)*delta[:,i:]+sigmaj*np.sqrt(delta[:,i:])*Z)\n",
    "    # refers to the above equation 1\n",
    "    mu_term = np.cumsum(delta[:,i:]*for_temp*sigmaj**2/(1+delta[:,i:]*for_temp),axis = 1)\n",
    "    # refers to the above equation 2\n",
    "    predcorr_forward[:,i:] = predcorr_forward[:,i:]*np.exp((mu_initial+mu_term-sigmaj**2)*delta[:,i:]/2+sigmaj*np.sqrt(delta[:,i:])*Z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Implying Capitalisation Factors\n",
    "\n",
    "$$ C(t_0, t_n) = \\Pi_{k=1}^{n} (1 + \\delta_k F_k(t_k)) $$\n",
    "\n",
    "Where, \n",
    "\n",
    "$C(t,T)$ is the capitalisation factor for the period $t_0$ and $t_n$\n",
    "\n",
    "We then take the inverse of the capitalisation factors to get the bond prices (same as the discount factors) and take the mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Implying capitalisation factors from the forward rates\n",
    "mc_capfac[:,1:] = np.cumprod(1+delta*mc_forward, axis = 1)\n",
    "predcorr_capfac[:,1:] = np.cumprod(1+delta*predcorr_forward, axis = 1)\n",
    "\n",
    "# Inverting the capitalisation factors to imply bond prices (discount factors)\n",
    "mc_price = mc_capfac**(-1)\n",
    "predcorr_price = predcorr_capfac**(-1)\n",
    "\n",
    "# Taking averages\n",
    "mc_final = np.mean(mc_price,axis = 0)\n",
    "predcorr_final = np.mean(predcorr_price,axis = 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1080x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(15,5))\n",
    "plt.xlabel(\"Maturity\")\n",
    "plt.ylabel(\"Bond Price\")\n",
    "plt.plot(t, vasi_bond, label=\"Vasicek Bond Prices\")\n",
    "plt.plot(t, mc_final, \"o\", label=\"Simple Monte Carlo Bond Prices\")\n",
    "plt.plot(t, predcorr_final, \"x\", label=\"Predictor-Corrector Bond Prices\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numpy Appendix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 3, 6])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr = [1,2,3]\n",
    "np.cumsum(arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3., 4.])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = np.array([[2,3],[4,5]])\n",
    "np.mean(m, axis=0)"
   ]
  }
 ],
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