Variance reduction of Monte Carlo and randomized quasi-Monte Carlo estimators for stochastic volatility models in finance

Abstract

We illustrate by numerical examples how certain variance reduction methods dramatically improve the efficiency of Monte Carlo simulation for option pricing and other estimation problems in finance, in the context of a geometric Brownian motion model with stochastic volatility. We consider lookback options and partial hedging strategies, with different models for the volatility process. For variance reduction, we use control variates, antithetic variates, conditional Monte Carlo, and randomized lattice rules coupled with a Brownian bridge technique that reduces the effective dimension of the problem. In some of our examples, the variance is reduced by a factor of more than 100 millions without increasing the work. The examples also illustrate how randomized quasi-Monte Carlo can be effective even if the problems considered involve a large number of dimensions.