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

2. 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

3. 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).

4. Asset Price Models

5. Equivalent Martingale Measure, Risk Neutral Pricing

6. Single Dimension Setting

7. 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

8. Motivating Monte Carlo Method

9. 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).

10. 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

11. 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

12. 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

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

14. 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

15. Properties of a Good Random Number Generator

16. 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

17. 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= (L’Ecuyer)

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

19. 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)

20. Inverse Transform Method1
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

21. 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.

22. 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

23. Generating Normally Distributed Random Variates

24. Generating Multivariate Normal Random Variates

25. 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

26. 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

27. 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

28. 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)

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

30. 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)

31. Importance Sampling Assuming Independence

32. Rare Event Simulation problem through an Example

33. Importance Sampling for this