Asymmetric Variance Reduction for Pricing American Options

Based on the dual formulation by Rogers (2002), Monte Carlo algorithms to estimate the high-biased and low-biased estimates for American option prices are proposed. Bounds for pricing errors and the variance of biased estimators are shown to be dependent on hedging martingales. These martingales are applied to (1) simultaneously reduce the error bound and the variance of the high-biased estimator, and (2) reduce the variance of the low-biased estimator while preserving its biased level. For a class of stochastic volatility models, projected hedging martingales are constructed based on an application of asymptotic expansion of option prices. These martingales are easy to compute. Numerical results demonstrate the robustness and effectiveness of these projected hedging martingales.