Monte Carlo Methods in Finance IIM Ahmedabad, Nov 6, 2005 Sandeep Juneja School of Technology and Computer Science Tata Institute of Fundamental Research

Talk Outline  Motivating Monte Carlo methods in finance through simple Binomial tree models for European options  Monte Carlo Method  Portfolio Credit Risk  Pricing Multi-dimensional American Options

European Call Option K Underlying price BUY CALL Option Payoff WRITE CALL K Option Payoff 0 0 An option (not an obligation) to purchase an underlying asset at a specified time T (expiration or maturity date) for a specified price K (strike price). Payoff G(S T )= (S T -K) + Payoff on the Maturity Date profit

European Put Option Underlying price Option Payoff Option Payoff K K 0 0 Underlying price BUY PUTWRITE PUT An option to sell an underlying asset at a specified time for a specified price. Payoff G(S T )= (K-S T ) + Payoff on the Maturity Date

Other Features  American option: Exercise at any time up to the expiration time  Bermudan option: Exercise allowed at a fixed number of times (Intermediate between European and American)

Examples of Options on Multiple Assets  Basket Option ([c 1 S 1 (T) + c 2 S 2 (T) c d S d (T)] - K) +  Out-performance Option (max{c 1 S 1 (T), c 2 S 2 (T),...,c d S d (T)} - K) +  Barrier Option I(min i=1,..,n {S 2 (t i ) <b}(K - S 1 (T)) +  Quantos S 2 (T)(S 1 (T) - K) + They all have an associated American version

Key Problems  The correct price of these options  How to hedge the risk of a portfolio containing options  No arbitrage principle: If 1 dollar = Rs. 40 and 1 pound = Rs. 60, ignoring transaction costs, 1 pound = 1.5 dollar, otherwise by buying low and selling high, an arbitrage may be created

Simple One Period Binomial Model to Price Options d < 1+r < u from no-arbitrage considerations S0S0 S 1 (H)= uS r S 1 (T)= dS 0 Two securities exist in this world V 0 ? V 1 (H), e.g., S 1 (H)-K V 1 (T), e.g., 0 Consider an option (If S 1 (T) <K<S 1 (H))

Create a Replicating Portfolio

The Risk Neutral Measure

Multi-Period Binomial Model  The analysis extends to multiple periods to more realistic models. S0S0 S 1 (H) S 1 (T) S 2 (HH) S 2 (HT,TH) S 2 (TT) S 3 (HHH) S 3 (HHT,HTH,THH) S 3 (TTH,HTT,THT) S 3 (TTT)

Solving for Option Price through Backward Recursion

A Numerical Example: Pricing a Lookback Option S 0 =4 S 1 (H)=8 S 1 (T)=2 S 2 (HH)=16 S 2 (HT,TH)=4 S 2 (TT)=1 S 3 (HHH)=32 S 3 (HHT,HTH,THH)=8 S 3 (TTH,HTT,THT) =2 S 3 (TTT)=0.5

The Discounted Price Process is a Martingale

Binomial Tree Model is Complete  Every security V N can be hedged using a replicating portfolio and hence has a unique price.  If the tree was trinomial, and there were two securities as before not every security could be replicated (incomplete market), only bounds could be developed on prices using the no- arbitrage condition

Fundamental Theorem in Option Pricing

Brownian Motion  A real valued process (W(t):t > 0), is standard Brownian motion if For t 0 < t 1...< t n, then W(t 1 )-W(t 0 ),..., W(t n )-W(t n-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).

Single Dimension Asset Pricing Model

Asset Price an Expectation under Equivalent Martingale Measure

Generating Sample Paths using Time Discretization  Suppose payoff depends on asset prices at times 0,1,2,...,n  Example: Asian Option  Approximately generate the trajectory of the asset price process using Euler’s scheme (finer discretizations improve accuracy) process dS t = r S t dt +  (t) S t dW(t)

Monte Carlo needed in Credit Risk Measurement  Consider a portfolio of loans having m obligors. We wish to manage probability of large losses due to credit defaults  Let Y k denote the loss from obligor k.  Our interest is in estimating P(Y Y m >u) for large u.  Note that P(Y Y m >u)= E[I(Y Y m >u)]  Loss given default E [Y Y m |Y Y m >u]=E[Y Y m I(Y Y m >u)]/P(Y Y m >u)

Monte Carlo Method  Motivating the Monte Carlo Approach  Monte Carlo Method Random number generation Generating random numbers from general distributions Popular variance reduction techniques

Illustrative Queueing Example  The inter-arrival times (A 1,A 2, …) are “independent identically distributed” with distribution function F A (x) =P(A < x). E.g. F A (x) = 1 - e - x  The service times (S 1,S 2, …) are independent identically distributed with distribution function F S (x) =P(S < x).

Solve or Run the Model ?  To determine EW we could use deductive arguments, e.g. W n+1 = [ W n + S n - A n+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 6

Key Statistical Ideas  Law of large numbers: If X 1, X 2, … are independent identically distributed random variables with mean m = EX, then For dice  =1*1/6 +2*1/6 +3*1/6 +4*1/6+5*1/6 +6*1/6 = 7/2  Central limit theorem  2 is the variance of each X i determines the convergence rate

Pricing Asian Option through Monte Carlo k+1 k Asset price

Constructing Estimators…

Now we discuss  Uniformly distributed random number generators: Building blocks for creating randomness  General random number generators  Generating uni-variate and multi-variate normal random variables

Generating Uniform (0,1) Pseudo Random Numbers Requirement: Generate a sequence of numbers U 1, U 2,...so that 1)Each U i is uniformly distributed between 0 and 1 2) The U i ’s are mutually independent 1/2 0

Linear Congruential Generators Popular method: A linear congruential generator Given an initial integer seed x 0 between 0 and m, set x i+1 = a x i mod m u i+1 = x i+1 /m a < m is referred to as multiplier, m the modulus 1/2 0

Properties of a Good Random Number Generator

Periodicity of Linear Congruential Generators  Consider the case where a=6, m=11. Starting from x 0 =1, the next value x 1 = 6 mod 11 =6, x 2 = 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 x 0 =1 yields: 1,3,9,5,4,1... Then x 0 =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 x i+1 = (a x i ) mod m  If m is a prime, full period is obtained if a is a primitive root of m, i.e., a m-1 – 1 is a multiple of m a j-1 – 1 is a not a multiple of m for j=1,2,...,m-2 Example of good LCG a=40014, m=

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

General Random Numbers  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 -1 (U) has distribution function F(x) 1 2/3 1 1/3 23 F(x) 1 U F -1 (U) x

Inverse Transform Method 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 P(F -1 (U) < y) = P(U < F(y))=F(y) Also F(X) has U(0,1) distribution 1 U F(x) F -1 (U)

Acceptance Rejection Method f(x) c*g(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. Accept the sample if f(Y) < c g(Y). Otherwise, reject and repeat.

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

Generating Normally Distributed Random Numbers

Algorithm for Normal Distribution

Generating Multivariate Normal Random Numbers

Algorithm for Multivariate Normal Random Numbers

Recap of Monte Carlo Method for Pricing Multi-dimensional European Options  Identify the risk neutral probability measure. Estimate the model from the data Replace drift with the risk free rate  Discretize the state space. Generate sample paths of the assets using the multi-variate Normal random vectors  Collect independent identically distributed samples of option payoffs  Use central limit theorem to develop confidence interval of the price estimate

Ordinary simulation can be computationally expensive  Convergence rate proportional to  Slow but for a given variance independent of problem dimension  Generating each sample may be expensive  Motivates research in clever variance reduction techniques to speed up simulations

Common Variance Reduction Techniques  We discuss the following variance reduction techniques Common random numbers Antithetic variates Control variates Importance sampling

Using Common Random Numbers  Often we need to compare two systems, so we need to estimate EX - EY = E(X-Y)  One way is to estimate EX by its sample mean X n estimate EY by its sample mean Y n two sample means are independently generated.  Note that Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) Positive correlation between X and Y helps The variations in X-Y cancel

Common Random Numbers to Estimate Sensitivity

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

Example of Antithetic Technique  Example: Asian Option Antithetic

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-  2 )Var (X) (  : correlation coefficient)  In practice, b = sample covariance(X,C)/sample variance(C) = and the estimate is X n + b (C n - EC)

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

Rare Event Simulation Problem

Importance Sampling to Rescue

Importance Sampling in Abstract Setting

Importance Sampling (contd.)

Importance Sampling for Sums of continuous RV taking Large Values

Portfolio Credit Risk with Extremal Dependence

Credit Risk  Credit Risk: The risk of loss due to obligor defaulting on payments. More generally, due to change in obigor’s credit quality  Market Risk: The risk of losses due to changes in market prices.  In credit risk: Lack of liquidity, time horizons are typically large Relevant model input information: probability of default, loss given default. Market risk measurement is more concerned with measures such as price volatilities

Credit Risk: Heavier Loss Tails

Portfolio Credit Risk  We focus on measurement of portfolio credit risk  The portfolio may comprise loans, defaultable bonds, letters of credit, credit default swaps (CDS) etc.  Motivation Basel II accord permit the use of internal models for calculating credit risk The emergence of collateralized debt obligations, where portfolio risk measurement is crucial  Accurate dependence modeling is critical Literature suggests that extremal dependence may exist among obligor losses

Section Outline  Describe a commonly used mathematical model for portfolio credit risk  Incorporate extremal dependence in this framework Asymptotic regime to analyze probability of large losses and expected shortfall Sharp asymptotics for these measures and their implications Provably efficient importance sampling techniques to estimate these performance measures

The Portfolio Credit Risk Problem  Consider a portfolio with n obligors The obligor i has exposure e i.  If it defaults, a loss of amount e i is incurred  This amount may be random to incorporate credit quality changes, recovery variation etc. The default probability of obligor is p i.  May be measured using historical default data based on its ratings  KMV modifies Merton’s seminal ideas combined with empirical data to come up with Expected Default Frequency

Historical Credit Migration Data to Compute Default Probabilities This data may be adjusted for prevalent conditions. It may be used to compute losses due to change in credit quality

Latent Variable Approach based on Merton’s Model 1 Yr Distribution of asset value at horizon Asset Value Today Time Value Debt face value Default Path

KMV’s Approach to Finding Expected Default Frequency 1 Yr Distribution of asset value at horizon Asset Value Today EDF Time Value Default Point Distance-to-Default = 3 Standard deviations Asset Volatility (1 Std Dev) Courtesy: KMV website

Modeling Dependence through Multi-Variate Latent Variables  Latent random variable X i models the value of obligor i If X i goes below a threshold x i the obligor i defaults resulting in loss e i  Total Loss L= e 1 I(X 1 <x 1 ) + e 2 I(X 2 <x 2 ) …. + e n I(X n <x n )  We focus on developing sharp asymptotics and Monte Carlo importance sampling techniques to estimate P(L>x) and E(L-x|L>x) for large x when latent variables (X 1, X 2,…, X n ) have extremal dependence

Typically Latent Variables are assumed to have Normal Distributions  J. P. Morgan’s CreditMetrics and Moody’s KMV system assume that the latent variables (X 1, X 2,…, X n ) follow a multi- variate normal distribution. Correlations captured through dependence on factors

Normal Variables often Inadequate to Capture Extremal Dependence Empirical evidence suggests that financial variables often exhibit stronger dependence than that captured by correlation based multi-variate normal model. Example: P(X 1 >x | X 2 >x)  0 as x  infinity, in normal setting If instead latent variables have a multivariate t-distribution, extremal dependence is captured, i.e., random variables may take large values together with non-negligible probability T-distributions often show better fit to financial data

Modeling Extremal Dependence

Asymptotic Regime to Analyze Loss Distribution

Sharp Asymptotic for Loss Probability

Comparison with Normal Copula

Monte Carlo Simulation  Accurate estimation via Monte Carlo Simulation  Naïve implementation Generate samples of Z, W and the Bernoulli variables with probability of success P(X i < -a i n | Z, W) for each i. Then a sample of I{ L n >nb } is seen. Average of many samples provides an estimator for P(L n >nb) Central limit theorem may be used to construct confidence intervals  Computational problem of estimating rare event probabilities

Importance Sampling in Our Setting

Evaluating Importance Sampling Estimator

Performance of Importance Sampling Algorithms  In the range of practical importance, P(L n >nb) approximately 1 in 1000, algorithm 1 reduces variance by about 150 times.  All else being equal, greater the impact of W in causing the rare event, better the performance  The results extend easily to multi-factor models

Monte Carlo Methods for Pricing American Options

Multi-Period Binomial Model: American Options  The decision to exercise can be made at any point in the lattice S0S0 S 1 (H) S 1 (T) S 2 (HH) S 2 (HT,TH) S 2 (TT) S 3 (HHH) S 3 (HHT,HTH,THH) S 3 (TTH,HTT,THT) S 3 (TTT)

American Options and Stopping Times

Pricing American Options

Dynamic Programming Recursion satisfied by the Price Process

General Models  We assume that the option can be exercised at times 0,1,2,...,N (Bermudan option)  The discounted value of the option at time m if exercised at time m equals G m (S m ) > 0  Let T m denote the set of stopping times taking values in (m, m+1,...,N) Then  Where the expectation is under risk neutral measure  If s 0 denotes the initial price then our interest is in finding J 0 (s 0 )

Dynamic Programming Formulation  Let C m (s) = E(V m+1 (S m+1 )|S m =s)=  V m+1 (y) f m (s,y)dy = P m (V m+1 )(s) denote the continuation value. V N (s) = G N (s) V m (s) = max(G m (s), C m (s)) for m=0,1,...,N-1 Alternatively, C n-1 (s) = P n (G n )(s) C m (s) =P m (max(G m+1,C m+1 ))(s) for m=0,1,2,…N-2 Even if the state space is discretized, the DP formulation suffers from the curse of dimensionality

Monte Carlo Methods for American Options  Random Tree Method  Regression based Function Approximation method

The Random Tree Method (Broadie and Glasserman 1997) t=0 t=1 Generate a sample tree With b branches From every state visited s0s0 s11s11 s12s12 s13s13 s 2 11 s 2 31

Random Tree Method  Does not depend upon the number of underlying securities  The effort increases exponentially with the number of exercise opportunities.

Regression Based Function Approximations

The Broad Approach

Simulation Methodology  Generate n sample paths (s m,j : m=0,...,N and j=1,...,n) of the process (S m : m=0,1,...,N)  Set N m

Regression based Methodology  Using this methodology the optimal exercise policy  * is learnt quickly  The expectation corresponding to this stopping policy is evaluated using the usual Monte-Carlo to generate samples of G * (S * )  The first phase is empirically seen to be quick. Mistakes here are not crucial.  The second phase requires significant effort...hence a need to speed-up through variance reduction