Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012
Outline Introduction 1 Introduction 2 Generating Random Numbers Generating Random Variables Generating Sample Path 3 Variance Reduction Techniques Quasi Monte Carlo Method 4 Black-Scholes Equations Monte Carlo Simulations for Option Pricing
Introduction Background History: John von Neumann, Stainslaw Ulam and Nicholas Metroplis Manhattan Project in Los Alamous National Laboratory Monte Carlo Casino, Monaco Monte Carlo methods: experimental mathematics large number of random variable simulation strong law of large numbers the sample average converges almost surely to the expected value X1 + + Xn a.s. X n = µ n i.e. n ( ) P lim X n = µ = 1. n
Introduction (Cont.) Two broad classes of Monte Carlo methods: Direct simulation of a naturally random system Operations research (inventory control, hospital management) Statistics: properties of complicated distribution Finance: models for stock prices, credit risk Physical, biology and social science: models with complex nondeterministic time evolution Adding artificial randomness to a system, then simulating the new system Solving some partial differential equations Markov chain Monte Carlo methods: for problems in statistical physics and in Bayesian statistics Optimization: travelling salesman, genetic algorithms
Example Introduction Objective: integral ˆ 1 α = f (x) dx = E [f (U)] 0 U uniformly distributed between 0 and 1 Get points U 1, U 2, independently and uniformly from [0, 1] The Monte carlo estimate ˆα n = 1 n f (U i ) n i=1 If f is integrable over [0, 1], by strong law of large numbers ˆα n α with probability 1 as n The error α n α is approximately normally distributed with mean 0 and standard deviation σ f/ n, where σf 2 = 1 0 (f (x) α)2 dx and can be approximated by smaple standard deviation s f = n i=1 (f (U i ) ˆα n) 2 1 n 1
Principles of Derivative Pricing Principles of theory for Monte Carlo If a derivative security can be perfectly replicated through trading in other assets, then the price of the derivative security is the cost of the replicating trading strategy. Discounted asset prices are martingales under a probability measure associated with the choice of discount factor. Prices are expectation of discounted payoffs under such martingale measure. In a complete market, any payoff can be realized through a trading strategy and the martingale measure associated with the discount rate is unique.
Random Number Generator Generating Random Numbers Generating Random Variables Generating Sample Path A generator of genuinely random numbers has the mechanism for producing random variables U 1, U 2, such that each U i is uniformly distirbuted between 0 and 1 the U i are mutually indepedent A random number generator produces a finite sequence of numbers u 1, u 2,, u K in the unit interval pseudorandom number generator not real random number, only mimics randomness
Linear Congruential Gnerators Generating Random Numbers Generating Random Variables Generating Sample Path Definition The general linear congruential generator proposed by Lehmer takes the form x i+1 = (ax i + c) mod m, u i+1 = x i+1 m. a, c, m are integer constants that determine the value generated. Initial value x 0 is called seed.
Inverse Transform Method Generating Random Numbers Generating Random Variables Generating Sample Path Definition In order to sample from a cumulative distribution function F, i.e. generate a random variable X with the property that P (X x) = F (x) for all x. The inverse transform method sets X = F 1 (U), U Unif [0, 1] where F 1 is the inverse of F and Unif [0, 1] denotes the uniform distribution on [0, 1]. Proof sketch: P (X x) = P ( F 1 (U) x ) = P (U F (x)) = F (x) Example: The exponential distribution with mean θ has distribution F (x) = 1 e x/θ, x 0. Inverting the exponential distirbution yields X = θ log (1 U) and can be implemented as X = θ log (U).
Acceptance-Rejection Method Definition Generating Random Numbers Generating Random Variables Generating Sample Path In order to generate random variable X with density function f (x), we first generate X from distribution g (x). Then generate U from Unif [0, 1]. If U f (X ) /cg(x ), this is the expected X ; else, repeat above steps. Proof sketch: Let Y be a sample returned by the algorithm and observe that Y has the distribution of X conditional on U f (X ) /cg(x ). For any A R P (Y A) = P (X A U f (X )/cg(x ) ) = P (X A, U f (X )/cg(x )) P (U f (X )/cg(x )) P (X A, U f (X ) /cg(x )) = A f (x) g (x) dx = 1 f (x) dx cg(x) c A P (U f (X ) /cg(x )) = R f (x) g (x) dx = 1 cg(x) c P (Y A) = f (x) dx conclusion proved. A
Brownian Motion Introduction Generating Random Numbers Generating Random Variables Generating Sample Path Definition A standard one-dimensional Brownian motion on [0, T ] is a stochastic process {W (t), 0 t T } with following properties: W (0) = 0; The mapping t W (t) is a continuous function; The increments W (t 1) W (t 0), W (t 2) W (t 1),, W (t k ) W (t k 1 ) are independent for any k and any 0 < t 0 < t 1 < < t k T ; W (t) W (s) N (0, t s) for any 0 < s < t < T. Simulation of Brownian Motion Based on the stationary and independent increment properties, generate n independent and identically distributed random variable B 1,, B n such that B i N ( ) 0, T n, i = 1,, n. Define Ŵ ( ) kt n = k i=1 B i, then as n, Ŵ is an appropriate of W.
Brownian Motion (Cont.) Generating Random Numbers Generating Random Variables Generating Sample Path 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0 20 40 60 80 100 120
Geometric Brownian Motion Generating Random Numbers Generating Random Variables Generating Sample Path Definition A stochastic process S (t) is a geometric Brownian motion if log S (t) is a Brownian motion with initial value log S (0). Geometric Brownian motion is the most fundamental model of the value of a financial asset. Suppose W is a standard Brownian motion and a geometric Brownian motion process is often specified by an SDE By Itô formula, we have ds (t) S (t) d (log S (t)) = = µdt + σdw (t) (µ 12 σ2 ) dt + σdw (t) and if S has initial value S (0) then S (t) = S (0) exp [(µ 12 ) ] σ2 t + σw (t).
Geometric Brownian Motion (Cont.) Generating Random Numbers Generating Random Variables Generating Sample Path 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0 20 40 60 80 100 120
The Stratified Sampling Variance Reduction Techniques Quasi Monte Carlo Method Definition Stratified sampling refers broadly to any sampling mechanism that constrains the fraction of observations drawn from specific subsets of the sample space. Let A 1,, A k be disjoint subsets of the real line for which P (Y i A i ) = 1, then K K E [Y ] = P (Y A i ) E [Y Y A i ] = p i E [Y Y A i ] i=1 Decide in advance what fraction of the sample should be drawn from each stratum A i and the theoretical probability p i = P (Y A i ) An unbiased estimator of E [Y ] is provided by i=1 Ŷ = K i=1 p i 1 n i n i j=1 Y ij = 1 n n K i i=1 j=1 Y ij
Stratified Sampling (Cont.) Variance Reduction Techniques Quasi Monte Carlo Method comparison of stratified sample (left) and random sample (right)
Antithetic Variates Introduction Variance Reduction Techniques Quasi Monte Carlo Method Definition The method of antithetic variates attempts to reduce variance by introducing negative dependence between pairs of replications. U and 1 U are both uniformly distributed over [0, 1] F 1 (U) and F 1 (1 U) both have distribution F and are mutually antithetic. Implement of the antithetic variates method: ) ) ) the pairs (Y 1, Ỹ 1, (Y 2, Ỹ 2,, (Y n, Ỹ n are i.i.d. ) each Y i and Ỹ i have same distribution and Cov (Y i, Ỹ i < 0 ) Monte Carlo estimate is Ŷ = 1 n 2n i=1 (Y i + Ỹ i ) Var (Ŷ = 1 n ( ) n Var Yi + Ỹ i = 1 ( )) Var (Y 1) + Cov (Y 1, Ỹ 1 2 2 i=1
Importance Sampling Variance Reduction Techniques Quasi Monte Carlo Method Definition Compute an expectation under a given probability measure Q of a random variable X, there is another measure Q equivalent to Q such that [ E Q [X ] = E Q X dq ]. d Q define Radon-Nikodym derivative L = dq d Q and measure Q is called an importance measure which give more weight to important outcomes. Theorem Let Q be define by dq dq = X E X the importance sampling estimator ZL under Q has a smaller variance than the estimator ZL under any other Q.
Quasi Monte Carlo Method Variance Reduction Techniques Quasi Monte Carlo Method Definition Quasi Monte Carlo method is a method numerical integration and solving some problems using low-discrepancy sequences (or quasi-random sequence or sub-random sequences) Properties of Quasi Monte Carlo Method Quasi Monte Carlo method make no attempt to mimic randomness, but to generating points evenly distributed. ( ) 1 Accelerate convergence of ordinary Monte Carlo method from O n to Quasi Monte Carlo method O ( 1 n ). Example: Suppose the objective is to calculate ˆ E [f (U 1,, U d )] = f (x) dx 1 [0,1) d n n f (x i ) for carefully and deterministically chosen points x 1,, x n in [0, 1) d. i=1
European Option Introduction Black-Scholes Equations Monte Carlo Simulations for Option Pricing Definition An option is a derivative financial instrument that specifies a contract between two parties for future transaction on an asset at a reference price (the strike). The buyer of the option gains the right, but not the obligation, to engage in the transaction, while the seller incurs the corresponding obligation to fulfill the transaction. An option conveys the right to buy something is called a call option; an option conveys the right to sell something is called a put option. The reference price at which the underlying asset may be traded is called strike price or exercise price. Most options have an expiration date and if it is not exercised by the expiration date, it becomes worthless. A European option may be exercised only at the expiration date of the option.
Black-Scholes Equations Black-Scholes Equations Monte Carlo Simulations for Option Pricing Black-Scholes model of the market follows these assumptions: There is no arbitrage opportunity. The stock price follows a geometric Brownian motion with constant drift and volatility. It is possible to borrow and lend cash at a known constant risk free interest rate. It is possible to but and sell any amount, even fractional of stock. The transactions do not incur any fees or costs and underlying security does not pay a dividend. Definition The Black-Scholes equation is a partial differential equation which describes the price of the option over time: V t + 1 2 σ2 S 2 2 V S 2 V + rs S rv = 0.
Black-Scholes Solution Black-Scholes Equations Monte Carlo Simulations for Option Pricing The value of a call option for a non-dividend paying underlying stock in terms of Black-Scholes parameter is ( ln S K d 2 = r(t t) C (S, t) = N (d 1 ) S N (d 2 ) Ke ( ln S K d 1 = ) + ) + (r σ2 2 σ T t (r + σ2 2 σ T t ) (T t) ) (T t) = d 1 σ T t. The price of a corresponding put option based on put-call parity is P (S, t) = Ke r(t t) S + C (S, t) = N ( d 2 ) Ke r(t t) N ( d 1 ) S. N ( ) is the cumulative distribution function of the standard normal distribution. T t is the time to maturity. S is spot price of the underlying asset and K is strike price. r is the risk free rate and σ is the volatility of the returns.
Numerical Scheme Introduction Black-Scholes Equations Monte Carlo Simulations for Option Pricing With boundary conditions C (0, t) = 0 for all t, C (S, t) S as S, C (S, T ) = max {S K, 0} corresponding numerical schemes can be developed.
No-Arbitrage Pricing Formula Black-Scholes Equations Monte Carlo Simulations for Option Pricing Non-Arbitrage Pricing Formula Price of a European option can be obtained by the expectation of the present value of the payoff for the options under the equivalent martingale measure Q. That is, at time t < T, the non-arbitrage price of a European option V t with the payoff Π(T ) and the maturity T is obtained by V t = e r(t t) E Q [Π(T ) F t ]. Consider European call option, then Π(T ) = (S T K) +. Then we have C (S t, t) = e r(t t) E Q [ (ST K) + F t ] and the stock price follows geometric Brownian motion ds t S t = µdt + σdw t.
Monte Carlo Simulation Black-Scholes Equations Monte Carlo Simulations for Option Pricing Simulate N (about ten thousands scale) paths of S i, i = 1,, N Every path S i is genererated from time t to T step by step S k = S k 1 exp [(µ 12 ) ] σ2 t + σw t, W t N (0, t) Start from Si 0 T t t = S t, i = 1,, N we have N paths ended with Si, then the simulated Monte Carlo result of call option can be approximated as C (S t, t) 1 N N ( S i=1 i T t t ) + K
Monte Carlo Simulation (Cont.) Black-Scholes Equations Monte Carlo Simulations for Option Pricing 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 2 1.8 1.6 1.4 1.2 1
References Introduction Black-Scholes Equations Monte Carlo Simulations for Option Pricing Paul Glasserman, Monte Carlo Methods in Financial Engineering, Springer, ISBN-10: 0387004513, ISBN-13: 978-0387004518, August 2003 Phelim P. Boyle, Options: A Monte Carlo Approach, Journal of Financial Economics 4(1977) 323-338 Phelim Boyle, Mark Broadie, Paul Glasserman, Monte Carlo Methods for Security Pricing, Journal of Economic Dynamics and Control 21(1997) 1267-1321