On Asymptotical Efficiency of Importance Sampling with p-order Relative Moment Minimization

The optimal parameters of candidate measures established by importance sampling are usually obtained by minimizing the quadratic criterion. To begin with, in this paper a scheme is developed for finding the alternative measure that is optimal in the sense that the p-order relative moment is minimized. Secondly, a necessary and sufficient condition for asymptotic efficiency of the optimal change of drift is demonstrated. Lastly, the change of drift is selected through Robbins-Monro type algorithm. When pricing options via Monte Carlo simulations, several numerical examples contrasting different p values that illustrate the efficiency of the proposed method are also included, as well as an interpretation of their different performances in terms of strike prices.