What Is That – Monte Carlo? <ul><li>Supposing we have an integral </li></ul>where g(y) is an arbitrary function, f(y) is a probability density function, and A is the range of integration. To obtain an estimate of E(g) , we pick a number N of sample values (y t ) at random from the probability density function f(y) .
<ul><li>The estimate of E(g) is then given by </li></ul>This simple equation is the quintessence of Monte Carlo method: An integral is interpreted as the average value of a function over an interval that contains a fixed number of points chosen in random.
History of The Method <ul><li>The Monte Carlo method was first utilized by scientists at Los Alamos to study some neutron characteristics during the Manhattan Project in 1940s. </li></ul><ul><li>With the progress in computing technology and development of random number generating algorithms, simplicity of the method has led to its wide spread application to a variety of problems where a closed-form solution is either unattainable or extremely hard to find, especially in evaluating multi-dimensional integrals. </li></ul>
<ul><li>Although Monte Carlo is applicable to a wide range of deterministic problems, it comes as a natural choice in modeling stochastic processes which explains why the method is so popular in financial industry. Indeed, the method’s very name implies its origin in games of chance and random numbers. </li></ul><ul><li>In this project, I have attempted to demonstrate how the method can simplify pricing of certain derivative securities. </li></ul>
Example: European Call Option <ul><li>To price a European call option on a stock, we first the stock price equation under Q-measure in the discretized manner: </li></ul><ul><li>S i+1 = S i exp(r – σ 2 /2 + σ Z) </li></ul><ul><li>Where S i is the current stock price; r is the risk-free rate; </li></ul><ul><li>σ 2 is the constant variance of the return ; Z is a normally distributed random number </li></ul>
<ul><li>Let S 0 = 75, the strike price K = 50, σ 2 = 0.025 (per time interval), r = 0.015 (per interval), and number of intervals equal to 20. </li></ul><ul><li>For this data, a simulation with a number of paths N = 75000 produces call option price X = 39.62 as the next slide illustrates. The option price is computed by averaging (S N – K) + over the number of paths and discounting result to the present time. </li></ul><ul><li>(For comparison, a 20-period binomial tree yields price for the same option X = 40.99) </li></ul>
Pricing A Caplet <ul><li>The option pricing theory tells us that a caplet is a put option expiring at time T i-1 on a bond that expires at time T i and can therefore be priced by the following equation: </li></ul>Where k is the fixed rate; δ is the a time interval (LIBOR tenor); B t is a risk-free bond (for t = 0, B t = 1); B -1 T i-1 is a discounting factor; K = 1/(1+k δ ) is the strike price; B(T i-1 , T i ) is the underlying asset (bond)
Computing Bond Prices From CIR Model <ul><li>Cox Ingersoll Ross model belongs to the family of affine models, and this circumstance allows for bond prices to be expressed in a relatively simple closed form, such as the following: </li></ul>Where r(t) is a short term rate obtained from the model; Y(t,T) and Z(t,T) are deterministic functions defined as follows:
The Discounting Factor B Ti-1 -1 The factor which discounts the option price to time 0 is determined from the following equation: Which can be approximated using the trapezoid method as
Generating Short Term Rates Simulation from CIR Model In its discretized form, the Cox Ingersoll Ross model looks the following way: Where σ , θ , and κ are deterministic parameters, and W is a normally distributed random number. Given the parameters σ , θ , κ , and r(t= 0) one can easily construct an iteration procedure to obtain short term rates in a given interval starting from zero.
<ul><li>Having generated a series of short term rates for a period T i by using Monte Carlo simulation, one can compute the underlying bond price from the rate r(T i-1 ) as </li></ul>and the discounting factor as
<ul><li>Then it becomes possible to compute the discounted put option price by averaging </li></ul>over ALL the sample paths (5000,10000,…75000?) That gives us our E Q ! The following two slides display a few sample paths chosen in random (by me) with their corresponding short rate term structures and bond prices
Effect Of CIR Parameters On Short Term Rates <ul><li>From the discretized version of CIR model, it follows that a change in parameter κ changes the short rate for one unit of discretization in time by </li></ul>The following slide shows sensitivity of short rates to changes in Kappa (all other parameters fixed)
<ul><li>If we change θ by Δθ , the effect on short rates will be </li></ul>and the following slide displays the same sample path with three different Thetas
<ul><li>Following the same approach with the volatility, we obtain the following relationship between a change in σ and the corresponding change in short rates: </li></ul>The following slide displays the same path with three different Sigmas
References <ul><li>Boyle, Phelim P. (1976) Options: A Monte Carlo Approach, The University of British Columbia, Vancouver, BC, Canada </li></ul><ul><li>Baxter, M. and Rennie A. (1996) Financial Calculus , Cambridge University Press </li></ul><ul><li>Zeytun, S. and Gupta, A. (2007) A Comparative Study of the Vasicek and the CIR Model of the Short Rate, Berichte des Fraunhofer ITWM, Nr. 124 </li></ul>