An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options

Handling discontinuities in financial engineering is a challenging task when using quasi-Monte Carlo (QMC) method. This paper develops a so-called sequential importance sampling (SIS) method to remove multiple discontinuity structures sequentially for pricing discrete barrier options. The SIS method is a smoothing approach based on importance sampling, which yields an unbiased estimate with reduced variance. However, removing discontinuities still may not recover the superiority of QMC when the dimensionality of the problem is high. In order to handle the impact of high dimensionality on QMC, one promising strategy is to reduce the effective dimension of the problem. To this end, we develop a good path generation method with the smoothed estimator under the Black–Scholes model and models based on subordinated Brownian motion (e.g., Variance Gamma process). We find that the order of path generation influences the variance of the SIS estimator, and show how to choose optimally the first generation step. As confirmed by numerical experiments, the SIS method combined with a carefully chosen path generation method can significantly reduce the variance with improved rate of convergence. In addition, we show that the effective dimension is greatly reduced by the combined method, explaining the superiority of the proposed procedure from another perspective. The SIS method is also applicable for general models (with the Euler discretization). The smoothing effect of the SIS method facilitates the use of general dimension reduction techniques in reclaiming the efficiency of QMC.