Monte Carlo Methods in Finance IIM Ahmedabad, Nov 6, 2005 Sandeep Juneja

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

2. 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

3. European Call Option 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(ST)= (ST-K)+ BUY CALL WRITE CALL Option Payoff Option Payoff K 0 profit 0 Underlying price K Payoff on the Maturity Date

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

5. 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)

6. Examples of Options on Multiple Assets • Basket Option ([c1S1(T) + c2S2(T) +...+ cdSd(T)] - K)+ • Out-performance Option (max{c1S1(T), c2S2(T),...,cdSd(T)} - K)+ • Barrier Option I(mini=1,..,n{S2(ti) <b}(K - S1(T))+ • Quantos S2(T)(S1(T) - K)+ They all have an associated American version

7. 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

8. Simple One Period Binomial Model to Price Options Two securities exist in this world S1(H)= uS0 1+r 1 S0 1+r S1(T)= dS0 d < 1+r < u from no-arbitrageconsiderations Consider an option V1(H), e.g., S1(H)-K (If S1(T) <K<S1(H)) V0 ? V1(T), e.g., 0

9. Create a Replicating Portfolio

10. The Risk Neutral Measure

11. S3(HHH) S2(HH) S3(HHT,HTH,THH) S1(H) S2(HT,TH) S3(TTH,HTT,THT) S0 S1(T) S2(TT) S3(TTT) Multi-Period Binomial Model • The analysis extends to multiple periods to more realistic models.

12. Solving for Option Price through Backward Recursion

13. A Numerical Example:Pricing a Lookback Option S3(HHH)=32 S2(HH)=16 S3(HHT,HTH,THH)=8 S1(H)=8 S2(HT,TH)=4 S0=4 S3(TTH,HTT,THT) =2 S1(T)=2 S2(TT)=1 S3(TTT)=0.5

14. The Discounted Price Process is a Martingale

15. Binomial Tree Model is Complete • Every security VN 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

16. Fundamental Theorem in Option Pricing

17. 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).

18. Single Dimension Asset Pricing Model

19. Asset Price an Expectation under Equivalent Martingale Measure

20. 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 dSt = r Stdt + s(t) StdW(t)

21. 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 Yk denote the loss from obligor k. • Our interest is in estimating P(Y1+...+Ym>u) for large u. • Note that P(Y1+...+Ym>u)= E[I(Y1+...+Ym>u)] • Loss given default E [Y1+...+Ym|Y1+...+Ym>u]=E[Y1+...+Ym I(Y1+...+Ym>u)]/P(Y1+...+Ym>u)

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

23. 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).

24. 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

25. Key Statistical Ideas • Law of large numbers: If X1, X2, … are independent identically distributed random variables with mean m = EX, then For dice m =1*1/6 +2*1/6 +3*1/6 +4*1/6+5*1/6 +6*1/6 = 7/2 • Central limit theorem s2 is the variance of each Xi determines the convergence rate

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

27. Constructing Estimators…

28. 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

29. 0 1/2 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’s are mutually independent

30. 0 1/2 Linear Congruential Generators 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

31. Properties of a Good Random Number Generator

32. Periodicity of 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

33. 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 Example of good LCG a=40014, m=214748563

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

35. 1 F(x) U F-1(U) 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. 1 F(x) 2/3 1/3 2 3 1 x F-1(U) has distribution function F(x)

36. 1 F(x) U F-1(U) Inverse Transform Method P(F-1(U) < y) = P(U < F(y))=F(y) Also F(X) has U(0,1) distribution 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

37. 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. Accept the sample if f(Y) < c g(Y). Otherwise, reject and repeat.

38. 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

39. Generating Normally Distributed Random Numbers

40. Algorithm for Normal Distribution

41. Generating Multivariate Normal Random Numbers

42. Algorithm for Multivariate Normal Random Numbers

43. 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

44. 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

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

46. 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 Xn • estimate EY by its sample mean Yn • 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

47. Common Random Numbers to Estimate Sensitivity

48. 1 U 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

49. Example of Antithetic Technique • Example: Asian Option Antithetic

50. 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)