Importance Sampling for Option Greeks with Discontinuous Payoffs

The Greeks are derivatives of option price with respect to market parameters and play an important role in financial risk management. Among various simulation methods for estimating the Greeks, the pathwise method typically has a low variance. However, when the option payoff is discontinuous, the pathwise method is not applicable and the Greek involves a conditional expectation taken over a hypersurface that is a probability-zero set. In this paper, we propose an importance sampling (IS) method to estimate this conditional expectation. More specifically, IS is applied in a way that all simulated observations fall into a set constructed by thickening the hypersurface. Allowing the thickness of the set to go to zero then leads to a new representation of the Greek as an ordinary expectation, thus leading to an unbiased estimator. The resulting estimator makes use of the pathwise derivatives and can be viewed as an extension of the pathwise method to cases with discontinuous payoffs. Numerical results show that the proposed IS method works well.