Variance Reduction for Monte Carlo Simulation of European, American or Barrier Options in a Stochastic Volatility Environment

Abstract

In this work we develop a methodology to reduce the variance when applying Monte Carlo simulation to the pricing of a European, American or Barrier option in a stochastic volatility environment. We begin by presenting some applicable concepts in the theory of stochastic differential equations. Secondly, we develop the model for the evolution of an asset price under constant volatility. We next present the replicating portfolio and equivalent martingale measure approaches to the pricing of a European style option. Modeling an asset price utilizing constant volatility has been shown to be an inefficient model[8,16]. One way to compensate for this inefficiency is the use of stochastic volatility models, which involves modeling the volatility as a function of a stochastic process[26]. A class of these models is presented and a discussion is given on how to price European options in this framework.

After developing the methods of how to price, we begin our discussion on Monte Carlo simulation of European options in a stochastic volatility environment. We start by describing how to simulate Monte Carlo for a diffusion process modeled as a stochastic differential equation. The essential element to our variance reduction technique, which is known as importance sampling, is hereafter presented. Importance sampling requires a preliminary approximation to the expectation of interest, which we obtain by a fast mean-reversion expansion of the pricing partial differential equation[22,6].

A detailed discussion is given on this fast mean-reversion expansion technique, which was first presented in [10]. We shall compare utilizing this method of expansion with that developed in [11], which is know as small noise expansion, and demonstrate numerically the efficiency of the fast mean-reversion expansion, in particular in the presence of a skew. We next wish to apply our variance reduction technique to the pricing of an American and barrier option. A discussion is given on how to price these options under constant volatility and in the presence of stochastic volatililty. Applying the importance sampling variance reduction method to a barrier option is similar to that of a European option since there exists a closed form solution to the price of this option in the context of constant volatility[4,15]. However, in the case of an American option Monte Carlo simulation and applying importance sampling are more complex. We present an algorithm to compute an American option price via Monte Carlo and describe an approximation technique to obtain a preliminary estimate to the pricing function under constant volatility. Hence, we are able to apply our variance reduction methodology to pricing of an American option. We subsequently present numerical results for both of these options.