The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations. In this document, I replicate the numerical example comparing small noise expansion SNE and fast mean reversion FMR described in that paper using Matlab and NAG routines. The NAG s Mersenne Twister random number generator produced more stable results than many other random number generators. 1 Importance Sampling Importance sampling is a popular variance reduction technique. Suppose we wish to estimate θ = E f [φ X], where X has probability density f, and φ is a payoff function. Let g be another probability density such that fx > gx > ˆ θ = E f [φ X] = ˆ φxfxdx = φ x fx [ gx gxdx = E g φ X f X ] g X Here, f and g are referred to as the original and importance sampling densities, and fx gx represents the likelihood ratio. It is the Radon-Nikodym derivative of the original measure with respect to the importance sampling measure. If density g can be chosen so that the random variable φ X fx gx an efficient estimation of θ. To make the variance of φ X fx gx inversely related to φx. has small variance, then importance sampling can result in small, fx gx should be In the context of the Black-Scholes European call option, the importance sampling estimator is given by { θ = exp rt E g [max S exp r σ T +σ T X K, } fx gx Stan Stilger przemyslaw.stilger@postgrad.mbs.ac.uk is at Manchester Business School. I would like to thank Mike Croucher, Jean-Pierre Fouque, Eberhard Mayerhofer and Ser-Huang Poon for their helpful comments and suggestions. ] 1
{ where φx = max S exp r σ T +σ } T X K, and X N, 1. Thus, θ is the price of European call option, φx is the payoff function and fx gx ratio. The original density is ln S T d = N ln S + r σ T, σ T whereas the importance sampling density is given by and the likelihood ratio is with α = ln S T d = N ln K + r σ T, σ T fx gx = exp α βx + α β /σ T T and β = ln K S + r σ represents the likelihood T. The change of measure is done by r σ changing the drift of the stochastic process. Here, the drift of the stochastic process under K the importance sampling measure has been shifted by ln S, and all simulated paths end up at-the-money or in-the-money which is more efficient as some simulated paths under the original measure end up out-of-the-money. Stochastic Volatility Application Fouque and Tullie present a variance reduction scheme stochastic volatility simulations. In their setup, the price of a risky asset X t, evolves according to the following SDE dx t = µx t dt + σy t X t dw t where µ represents a constant mean return rate, σy t represents the volatility which is driven by another stochastic process Y t which takes the following form The invariant distribution of Y t is N dy t = αm Y t dt + βρdw t + 1 ρ dz t. 1 m, β α. Let v = β α and dv t = bt, V t dt + at, V t dη t
where dv t = and dη t = W t Z t rx σy t x, bt, V t =, at, V t = αm Y t ν αρ ν, α 1 ρ X t Y t. The price P t, V t of an European option at time t is then given by P t, v = E [ e rt t φv T V t = v ] Next, Fouque and Tullie consider the following process ˆ T Q t = exp hs, V s dη s + 1 ˆ T hs, V s ds which is a positive martingale if E [ ] Q 1 t = 1. Then they define a new probability measure P equivalent to P dp dp = Q T 1 By Girsanov Theorem, the process under the new measure, P, η t = η t + ˆ t hs, V s dη s is a standard Brownian motion. Processes V t and Q t can be rewritten in terms of η t dv t = bt, V t at, V t ht, V t dt + at, V t d η t ˆ T Q t = exp hs, V s d η s 1 ˆ T hs, V s ds With respect to this new measure, P t, V t can be written as P t, v = Ẽ [ e rt t φv T Q T V t = v ] Applying Ito s lemma to P t, V t Q t and using Kolmogorov backward equation for P t, V t yields d P t, V t Q t = Q t P t, Vt ht, V t + at, V t T P t, V t d η t Integrating d P t, V t Q t from to T gives P T, V T Q T = P, V Q + This reduces to φv T Q T = P, v + ˆ T ˆ T Q t P t, Vt ht, V t + at, V t T P t, V t d η t Q t P t, Vt ht, V t + at, V t T P t, V t d η t 3
Therefore, the variances of Monte Carlo estimators under P and P are given by [ˆ T V ar P φv T Q T = Ẽ [ˆ T V ar P φv T = Ẽ ] Q t P t, V t ht, V t + at, V t T P t, V t dt ] at, V t T P t, V t dt Hence, the optimal choice of h for which the variance of φv T Q T under P is ht, V t = 1 P t, V t at, V t T P t, V t 3 In Fouque s and Tullie s work importance sampling is associated with ht, v as it dictates the change of drift of the stochastic process given by equation. An appropriate choice of ht, v leads to smaller variance for Monte Carlo estimator under P than under P. Unfortunately, ht, v can not be calculated directly from equation 3 as it depends on P t, v which is an unknown. However, this equation gives some intuition about the possible choice of ht, v. To determine ht, v, Fouque and Tullie approximate P t, v by its fast mean reversion expansion. Then they compare fast mean reversion with small noise expansion. P SNE is a small noise expansion of P t, v given by the Black-Scholes formula with d 1 = ln x K +r+ 1 σ yt t σy T t P SNE = xnd 1 Ke rt t Nd and d = d 1 σy T t. Here h takes the following form h = 1 P SNE xσy P SNE x P F MR is a fast mean reversion expansion of P t, v derived from the following equation where P BS σ is given by P F MR = P BS σ T t V x P BS σ + V x 3 x 3 3 P BS σ x 3 xnd 1 Ke rt t Nd with d 1 = ln x K +r+ 1 σ T t σy, d T t = d 1 σ T t and V, V 3 are parameters calibrated from the implied volatility skew. In this case, h takes the following form h = 1 P F MR xσy P F MR x In general, Small Noise Expansion works best when the mean reversion rate is low 4 5 6 4
α, whereas Fast Mean Reversion works best when the mean reversion rate is high α. 3 Numerical Implementation Following the importance sampling schemes described in the previous section, I employ small noise expansion and fast mean reversion to price a European call using the following parameters: r =.1, σ y = exp y m =.6, ν = 1, ρ =.3 X = 11, exp y =.983 K = 1, T = 1, α = 1 where r is the risk-free rate, ρ is the correlation in equation 1. As for the fast mean reversion, I use order zero expansion, where 5 reduces to P BS σ. In order to apply importance sampling, I calculate h according to equation 4. For FMR it is given by equation 6. Implementation is done in Matlab and using the following NAG routines: s3ab - Black-Scholes-Merton option pricing formula with greeks g5kf - initialization function for pseudo random number generator using g5sk - pseudo random number generator Figures 1 and plot the results of basic Monte Carlo MC, small noise expansion SNE and fast mean reversion FMR. The numerical example shows that both schemes converge well compared to the plain Monte Carlo simulations, while the Fast Mean Reversion performs slightly better the Small Noise Expansion based on the speed of price convergence and with a small variance. According to Fouque and Tullie Fast Mean Reversion outperforms Small Noise Expansion even when the mean reversion rate is 1. References [1] Fouque J.P. and T. Tullie Variance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment. Quantitative Finance, 1:4-3. 5
Figure 1: Price Figure : Variance 6