2.1 Mathematical Basis: Risk-Neutral Pricing

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1 Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t t Ê t F T ], (.1 Ê is the risk-neutral expectation. Consider an example that a European vanilla call has a payoff (S T K +, and evolves according to = µdt + σdb t. Then the option price at time t can be written as V t = e r(t t Ê t (ST K +], (. = (r qdt + σd B t. (.3 It is worthwhile pointing out that the Brownian motion B t is different from B t. On generating M different scenarios {S T,m } M m=1 of the underlying asset(s according to (.3, the expectation can be approximated by the averaging the payoff over the M scenarios: f t e r(t t M M (S T,m K +. m=1 Remark 8 The PDE associated with (. is the well known Black-Scholes equation: V t + 1 σ S V + (r q S V rv = 0, in t 0, T, S > 0. S S If you compute the expectation in a real world, U(, t = e r(t t E t (ST K +], 13

2 14 CHAPTER. MONTE-CARLO SIMULATION = µdt + σdb t. The corresponding equation that U(S, t satisfies is given by which is not what we desire. V t + 1 σ S V S + µs V S rv = 0, in t 0, T, S > 0, Remark 9 (.1 is correct for any European-style derivatives, including path-dependent options. Note that the price function of a (strongly path-dependent option may involve a pathdependent variable and the resulting governing PDE is likely different from the Black-Scholes equation. I refer you to Chapter 4 or MA457.. Basics of Simulation..1 Generating Stock Prices Suppose we wish to generate a GBM (.3. We can do it as follows: 1 Consider a partition {t i = i t} n i=0, t 0 = 0, t n = T, t = T/n. Increments can be generated using standard normal distributed random numbers ε i : i+1 i = (r q i t + σi ε i t. Note that we need to take t small so as to obtain a good approximation. If we insead look at x = log S, then dx = (r q σ dt + σdb. Increments for x can be generated in a similar way: x ti+1 x ti = (r q σ t + σε i t. Here there is no approximation. For European vailla options, we can just take t = T since prices at intermediate times are unnecessary. For path dependent options like Asian options, it is necessary to generate prices at each t i... Generating Normal Variables Any program language offers the generation of the random variables drawn from a uniform distribution over 0, 1]. To generate normal variable, a particularly useful distribution that is fast and easy to implement is the following approximation to the Normal distribution: ( 1 ψ i 6, i=1

3 .. BASICS OF SIMULATION 15 the ψ i are independent random variables, drawn from a uniform distribution over 0, 1]. A better method is the Box-Muller method. Let x 1 and x are two independent uniform random variables in (0, 1. We combine them to give two numbers y 1 and y that are both Normally distributed: y 1 = log x 1 cos (πx and y = log x 1 sin (πx. The polar form of the Box-Muller method can be found at method. In Matlab, there is a build-in function randn which generates normal random variables...3 Generation of Correlated Samples When pricing options on multi-assets by using Monte-Carlo simulation, we require n correlated samples from normal distributions. Assume we require sample i and sample j to be correlated with correlation coefficient ρ ij, i, j = 1,...n. We first sample n independent variables x i (1 i n, from univariate standardized normal distributions. The required samples are ǫ i (1 i n, i ǫ i = α ik x k. k=1 For ǫ i to have the correct variance and the correct correlation with the ǫ j (1 j n, we must have i αik = 1 and, for all j i, In matrix form, we denote k=1 j α ik α jk = ρ ij. k=1 C = (ρ ij n n and M = (α ij n n and Then, the matrix M must satisfy ǫ = Mx MM T = C, which leads to E ǫǫ T] = ME xx T] M T = MM T = C. This decomposition of the correlation matrix into the product of two matrices is not unique. The Cholesky factorization gives one way of choosing this decomposition. It results

4 16 CHAPTER. MONTE-CARLO SIMULATION in a Matrix M that is lower triangular. For example, when n = 3, ǫ 1 = x 1 ǫ = ρ 1 x ρ 1x ǫ 3 = ρ 31 x 1 + ρ 3 ρ 31 ρ 1 1 ρ 1 x ρ 3 ρ 1 ρ 31 ρ 1 ρ 31 ρ 3 x 1 ρ Number of Trials and Error of Estimate The number of simulation trials carried out depends on the accuracy required. If M independent trials are carried out, it is usual to calculate the standard deviation as well as the mean of the discounted payoffs given by the simulation trials for the derivative. Denote the mean by b and the standard deviation by ω. Then variable b is the simulation s estimate for the true value of derivative f. The standard error of the estimate ω M. This shows that our uncertainty about the value of the derivative is inversely proportional to the square root of the number of trials. To double the accuracy of a simulation, we must quadruple the number of trials; to increase the accuracy by a factor of 10, the number of trials must increase by a factor of Sampling through a Tree Instead of implementing Monte Carlo simulation by randomly sampling from the stochastic process for an underlying variable, we can sample paths for the underlying variable using a binomial tree. Suppose we have a binomial tree the probability of an up movement p = e(r q t d. u d The procedure for sampling a random path through the tree is as follows. At each node, we sample a random number between 0, 1]. If the number is less than or equal to 1 p, we take the down path. If it is greater than p, we take the up path. Once we have a complete path from the initial node to the end of the tree we can calculate a payoff. This completes the first trial. A similar procedure is used to complete more trials. The mean of the payoffs is discounted at the risk-free rate to get an estimate of the value of the derivative..3 Calculating the Greek Letters Suppose we are interested in the partial derivative of f with q, f is the value of the derivative and q is the value of an underlying variable or a parameter. First, Monte-Carlo simulation is used in the usual way to calculate an estimate, f, for the value of the derivative, we suppose that the number of time intervals is N, the random number streams used

5 .3. CALCULATING THE GREEK LETTERS 17 {ε i, i = 1,..., N}, and the number of trials M. Then we will use the same N, ε i, M to estimate the value of the derivative f with q + q. Then f q =: f f. q Alternatively, we can use the central difference that has a higher order of accuracy. Another way to calculate the Greek letters is to exploit the differential equation satisfied by the Greek letters. For example, differentiate the B-S equation with respect to S, then the hedging ratio for call satisfies with t + 1 σ S S + ( r q + σ S q = 0, S > 0, t 0, T S (S, T = { 1, if S > X 0, if S < X. Then, we can write = : I {S>X} (S, t = Ẽt e q(t t I {ST >X}], (.4 = ( r q + σ dt + σd B t. We can estimate the value of the by a Monte Carlo simulation. Let us revisit the above idea by probabilistic approach. or equivalently, S T = Se (r q σ = V S = Êt = Êt V (S, t = Êt e r(t t (S T X +], = (r q dt + σd S B t t (T t+σ( B T B t. So e r(t t σ (r q I {ST >X}e (T t+σ( ] B T B t σ ( q e (T t+σ( B T B t I{ST >X} ]. (.5 We emphasize that two expectations in (.4 and (.5 are taken in the different world. However, we can show ] ] Ẽ t e q(t ( q t I {ST >X} = Ê t e σ (T t+σ( B T B t I{ST >X} in terms of the Girsanov transformation.

6 18 CHAPTER. MONTE-CARLO SIMULATION.4 Other Issues.4.1 Variance Reduction Techniques Variance reduction techniques are any method that reduces the variance of prices given by an MC method using a pseudo random number generator. The easiest technique is antithetic variance reduction. Suppose that we need to generate random numbers {ε i } M i=1 that should have mean 0. Normally this set won t actually have mean 0, but {ε i, ε i } M i=1 will. Few extra computations are needed for this (no new random numbers need to be generated, but the effect on convergence is limited..4. Control Variates The method of control variates is used when we are trying to use MC to price a derivative A, and there is a derivative B with payoff correlated to A and with a price available in analytically closed form. Suppose the price F A of derivative A can be written as F A = (F A F B + F B. We can obtain F A by computing F A F B via MC and adding the exact price F B to the difference. Let σ (F be the variance on a set of MC generated prices for a derivative. Then, the control variates method will be effective if Since σ (F A F B < σ (F A. σ (F A F B = σ (F A ρ A,B σ(f A σ(f B + σ (F B, we infer that the method is effective if ρ A,B > σ(f B σ(f A. An example would be to use a portfolio of plain vanilla calls (which has an analytic price as a control variate for pricing a rainbow call (an option on two underlying assets: (max (S 1, S K + 1 (S1 K + + (S K +] = { 1 (S 1 K + (S K + ], if S 1 S 1 (S K + (S 1 K + ], if S 1 < S.4.3 *Simulation with Least-squares Approach for American Options American options are contracts that may be exercised early, prior to expiry. These options are contrasted with European options for which exercise is only permitted at expiry. Most traded stock and futures options are American style, while most index options are European.

7 .4. OTHER ISSUES 19 From the view point of probabilistic theory, the American option pricing is an optimal stopping problem. The price function of an American put can be rewritten as ] V (S, t = max Ê e r(t t (K + = S, (.6 t t is a stopping time. Intuitively t (. can be thought of as a strategy to exercise the option and the option s value corresponds to the optimal exercise strategy. Longstaff and Schwartz (001: Valuing American options by simulation: a simple least squares approach, Review of Financial Studies, 14(1, *Low-discrepancy Sequences and Quasi Monte Carlo Reference: 1 mc.html; P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer