Optimal Importance Sampling in Securities Pricing

To reduce variance in estimating security prices via Monte Carlo simulation, we formulate a parametric minimization problem for the optimal importance sampling measure, which is solved using in nitesimal perturbation analysis (IPA) and stochastic approximation (SA). Compared with existing methods, the IPA estimator we derive is more universally applicable and more computationally e cient. Under suitable conditions, we show that the objective function is a convex function, the IPA estimator is unbiased, and the stochastic approximation algorithm converges to the optimum. Lastly, we demonstrate how combining importance sampling with indirect estimation using put-call parity can lead to further substantial variance reduction. This research was supported in part by the National Science Foundation under Grants DMI-9713720 and DMI-9988867 and by the Semiconductor Research Corporation under Grant 97-FJ-491. An earlier version of this manuscript that accepted for presentation at the 2000 Winter Simulation Conference (Su and Fu 2000) did not include the proofs of the theoretical results nor the computational work on xed income securities and indirect estimation.