Monte Carlo Methods for Pricing Financial Options Lecture 2-3 Sandeep Juneja School of Technology and Computer Science Tata Institute of Fundamental Research

Talk Outline Mathematical model for options pricing Motivating the Monte Carlo Method Monte Carlo Method Random Number Generation Random Variate Generation Variance Reduction Techniques Antithetic Variates Control Variates Importance Sampling: example from credit risk modelling

Brownian Motion A real valued process (W(t):t > 0), is standard Brownian motion if For t0 < t1...< tn, then W(t1)-W(t0),..., W(tn)-W(tn-1) are independent W(s+t)-W(s) is Normally distributed with mean 0 and variance t W(t) is a continuous function of t (with prob 1).

Asset Price Models

Equivalent Martingale Measure, Risk Neutral Pricing

Single Dimension Setting

Generating Sample Paths using Time Discretization Suppose payoff depends on asset prices at times 0,1,2,...,n (this is for notational convenience) Example: Asian Option Approximately generate the trajectory of the asset price process using Euler’s scheme (finer discretizations improve accuracy) process dSt = r Stdt + s(t) StdW(t) If the volatility is constant

Motivating Monte Carlo Method

Illustrative Queueing Example The inter-arrival times (A1,A2, …) are “independent identically distributed” with distribution function FA(x) =P(A < x). E.g. FA(x) = 1 - e-lx The service times (S1,S2, …) are independent identically distributed with distribution function FS(x) =P(S < x).

6 Solve or Run the Model ? To determine EW we could use deductive arguments, e.g. Wn+1= [ Wn + Sn - An+1 ]+ ==> …... ==> …… ==> EW = ……. Feasible only for simple models Or we could use the computer to simulate functioning of the queue for a large number of days and do statistical analysis

Monte Carlo Method To estimate the expectation a = EX Generate independent identically distributed samples of payoffs X1, X2,...,Xn of X and take their average as an estimator. It converges to correct value due to law of large numbers Due to central limit theorem, a 95% confidence interval is The variance may be empirically estimated T time Asset price

PDE versus Monte Carlo Method Partial Differential Equation (not described in these lectures) For sufficiently complex dynamics the PDE may fail to exist The computational effort grows exponentially in the number of underlying variables. Not competitive when more than 3-4 variables involved Monte Carlo Method Convergence rate proportional to Slow but for a given variance independent of problem dimension Motivates research in clever variance reduction techniques to speed up simulations

Now we discuss Random number generators Random variate generators Generating univariate and multi-variate normal random variables

Generating Uniform (0,1) Pseudo Random Numbers Requirement: Generate a sequence of numbers U1, U2,...so that Each Ui is uniformly distributed between 0 and 1. 2) The Ui are mutually independent 1/2 In practice, deterministic sequences generated that appear to satisfy The above two properties. Popular method: A linear congruential generator Given an initial integer seed x0 between 0 and m, set xi+1 = a xi mod m ui+1 = xi+1/m a < m is referred to as multiplier, m the modulus

Properties of a Good Random Number Generator

Linear Congruential Generators Consider the case where a=6, m=11. Starting from x0=1, the next value x1= 6 mod 11 =6, x2= 36 mod 11 =3... The sequence 1,6,3,7,9,10,5,8,4,2,1,6,... is generated Produces m-1=10 values before repeating. Has full period Consider a=3, m=11. Then x0=1 yields: 1,3,9,5,4,1... Then x0=2 yields: 2,6,7,10,8,2... In practice we want a generator that produces billions and billions of values before repeating

Achieving Full Period in an LCG Consider LCG xi+1 = (a xi) mod m If m is a prime, full period is obtained if a is a primitive root of m, i.e., am-1 – 1 is a multiple of m aj-1 – 1 is a not a multiple of m for j=1,2,...,m-2 Note that xi+1 = (a xi) mod m implies that xi = (ai x0) mod m Thus, xi = [(ai -1) x0+ x0] mod m. Only, xm-1 =x0 (need to show that ai x0 is also not a multiple of m) E.g. a=40014, m=214748563 (L’Ecuyer)

Random Numbers from LCG lie on a plane Spectral gap As a discrepancy measure Ui+1 Ui a=6, m=11

Generating Random Variates Given i.i.d. sequence of U(0,1) variables, generate independent samples from an arbitrary distribution F(x) = P(X < x) of X Inverse Transform Method Suppose X takes values 1,2 and 3 each with prob. 1/3. F(x) 1 1 F(x) 2/3 U 1/3 1 2 3 F-1(U) x F-1(U) has distribution function F(x)

Inverse Transform Method 1 F-1(u)= inf{x: F(x) > u}. Then, P(F-1(U) < y) = P(U < F(y))=F(y) Also F(X) has Unif (0,1) distribution F(x) U a b Example: F(X) = 1-exp(-a X). Thus, X is exponentially distributed with rate a. Then, X= -log(1-U)/a has the correct distribution

Acceptance Rejection Method c*g(x) f(x) Need to generate X with pdf f(x) There exists a pdf g(x) so that f(x) < c g(x) for all x Algorithm: generate Y using pdf g. Generate independent uniformly generated rv U. Accept the sample if f(Y)/c g(Y) > U Otherwise, reject and repeat.

Rationale Strategy: generate a sample X from f. Spread it uniformly between 0 and f(X) f(x) Lx x Prob density of being in rectangular strip = f(x)dx * Lx/f(x)= Lxdx Prob of being in the region= area of the region This property is retained by the acceptance rejection method

Generating Normally Distributed Random Variates

Generating Multivariate Normal Random Variates

Ordinary simulation can be computationally expensive Large computation requirement as Generating each sample may be expensive Variance of each sample is high. Large number of samples needed to get a reasonable confidence interval We discuss the following variance reduction techniques Antithetic variates Control variates Importance sampling

Antithetic Variates Consider the estimator Xn = ( X1 + X2 + … + Xn)/n Var (X1 + X2) = Var (X1) + Var (X2) + 2 Cov (X1, X2) To reduce variance we need Cov (X1, X2) < 0 Theorem Given any distribution of rv X and Y (FX-1(U), FY-1(U)) has the maximum covariance (FX-1(U), FY-1(1-U)) has the minimum covariance

Example of Antithetic Technique Example: Asian Option Antithetic Result: E [f(N1,...,Nn) f(-N1,...,-Nn)] < E [f(N1,...,Nn) ] E [f(-N1,...,-Nn)] If f is increasing in each of its arguments

Control Variates Consider estimating EX via simulation Along with X, suppose that C is also generated and EC is known If C is correlated with X, then knowing C is useful in improving our estimate Let Y = X - b ( C - EC) be our new estimate. Note that EY = EX Best b* = Cov (X,C)/Var(C) Then Var (Y) = (1- r2)Var (X) (r: correlation coefficient) In practice , b = sample covariance(X,C)/sample variance(C) = and the estimate is Xn + b (Cn - EC)

Pricing Asian Options Option pay-off Control variate Stock price K strike price T=0 Stock price Option pay-off Control variate

A Simple Credit Risk Model Consider a portfolio of loans having m obligors. We wish to manage probability of large losses due to credit defaults Let Xk denote the centered and scaled firm values in a fixed time period. We assume that its distribution is standard normal Obligor k defaults if Xk < xk. This threshold is chosen so that P(Xk < xk) = pk (a probability that may be estimated using historical data) The rv (X1, X2,...Xm) are assumed to be correlated (to capture dependence)...the correlation structure may be estimated using equity prices If obligor k defaults, a loss Yk is incurred Our interest is in estimating P(Y1I(X1 < x1)+...+YmI(Xm < xm)>u)

Importance Sampling Assuming Independence

Rare Event Simulation problem through an Example

Importance Sampling for this