Calculating Prices and Sensitivities for Path-Independent Derivatives Securities in Multifactor Models

Abstract

Pricing derivatives written on assets whose prices depend on multiple factors can be difficult, even if payoffs depend only on prices at maturity. In complicated problems, Monte Carlo simulation is a popular technique, but it can be time-consuming when accuracy is needed. Conditional Monte Carlo is a known variance reduction method, but its use has been limited by a presumption in the literature that it can be used only when factors are instantaneously uncorrelated. I show that this presumption is not necessarily true. For example, we can express the prices and sensitivities of an European call option written on an asset that has stochastic volatility as an expectation over its Black-Scholes counterparts, even when the volatility and price processes exhibit instantaneous correlation. Combining this conditional Monte Carlo technique with low-discrepancy ("quasi-random") techniques provides price and delta estimators that require a fraction of a second to compute and yet have exact errors that are accurate to a few one-hundredths of a percent of the option's price. By comparison, conventional Monte Carlo simulation requires at least several minutes of computation time to achieve the same accuracy.