An almost exact simulation method for the Heston model

Robert D. Smith

Abstract

ABSTRACT

The principal difficulty in pricing derivative payoffs on underlyings with stochastic volatility using Monte Carlo simulation is that many small time steps are needed in order to reduce the bias in the simulation error to an acceptable level. Many researchers have come up with inventive discretization methods that are more sophisticated than the Euler discretization. Several papers (Glasserman 2003; Jäckel and Kahl 2006; Klaus and Schmitz 2004) compare many of these alternative methods and propose a favorite. Nevertheless, none of these methods (which are generally based on Milstein methods) obtain a change in order of convergence; the best they can do is reduce the magnitude of the leading error term. One paper by Broadie and Kaya (2006) stands out in that it proposes what the authors call “an exact simulation method” by which they mean that there it has no bias and therefore no additional time steps are required. In this paper we address the major drawback of their method: that it is only effectively applicable to derivative payoffs which depend only on observations of the underlying at very few points in time.