Optimal importance sampling in securities pricing

Abstract

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 infinitesimal perturbation analysis (IPA) and stochastic approximation (SA). Compared with existing methods, the IPA estimator we derive is more universally applicable and more computationally efficient. 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.

Monte Carlo Computation in Finance

“…In the variance-minimization approach, [21] uses SAA while [4,93] use SA. The procedures using SA can have multiple stages: at stage n, variance reduction is performed using the parameter ϑ n−1 , and then the parameter is updated to ϑ n based on the new simulation output.…”

“…The procedures using SA can have multiple stages: at stage n, variance reduction is performed using the parameter ϑ n−1 , and then the parameter is updated to ϑ n based on the new simulation output. The estimator is computed by [93] as an average of fresh replications in the last stage, which were never used to choose the variance reduction parameter; it is an average of all replications in [4] and papers that follow it. Under suitable conditions, the variance reduction parameter ϑ n converges to an optimal choice, and the average over all replications is a consistent, asymptotically normal estimator.…”

Adaptive Monte Carlo Variance Reduction with Two-time-scale Stochastic Approximation

“…(See also [9].) Let {W n,t : t ∈ [0, T ]} n∈N be a sequence of iid replications of the stochastic process…”

“…In financial engineering settings, for example, a systematic way to derive nearly optimal parameter is proposed in [6], while the application of the Robbins-Monro algorithm, which is a single-time-scale stochastic approximation algorithm, in a fairly general formulation is studied in Su and Fu [9] and Arouna [1], and a pure-jump Lévy process framework in Kawai [7]. In order to search two-fold parameters of CV and of IS at the same time, we apply the two-time-scale stochastic approximation algorithm, which is a stochastic recursive algorithm in which some of the components are updated using step-sizes that are very small compared to those of the remaining components and whose almost sure convergence is first rigorously proved in Borkar [3].…”

Adaptive Importance Sampling Technique for Markov Chains Using Stochastic Approximation

“…These works, however, use a fixed or static change of measure, whereas our focus is on adaptively learning the best zero variance change of measure. Stochastic approximation approaches to importance sampling in the specific context of option pricing have been developed in Vázquez-Abad and Dufresne (1998), Su and Fu (2000) and Su and Fu (2002). These take a 'stochastic gradient'-based approach using an approximate gradient search.…”