Variance-Reduction Methods for Monte Carlo Simulation of Radiation Transport

After a brief description of the essentials of Monte Carlo simulation methods and the definition of simulation efficiency, the rationale for variance-reduction techniques is presented. Popular variance-reduction techniques applicable to Monte Carlo simulations of radiation transport are described and motivated. The focus is on those techniques that can be used with any transport code, irrespective of the strategies used to track charged particles; they operate by manipulating either the number and weights of the transported particles or the mean free paths of the various interaction mechanisms. The considered techniques are 1) splitting and Russian roulette, with the ant colony method as builder of importance maps, 2) exponential transform and interaction-forcing biasing, 3) Woodcock or delta-scattering method, 4) interaction forcing, and 5) proper use of symmetries and combinations of different techniques. Illustrative results from analog simulations (without recourse to variance-reduction) and from variance-reduced simulations of various transport problems are presented.

Efficient Variance Reduction for American Call Options Using Symmetry Arguments

Recently it was shown that the estimated American call prices obtained with regression and simulation based methods can be significantly improved on by using put-call symmetry. This paper extends these results and demonstrates that it is also possible to significantly reduce the variance of the estimated call price by applying variance reduction techniques to corresponding symmetric put options. First, by comparing performance for pairs of call and (symmetric) put options for which the solution coincides, our results show that efficiency gains from variance reduction methods are different for calls and symmetric puts. Second, control variates should always be used and is the most efficient method. Furthermore, since control variates is more effective for puts than calls, and since symmetric pricing already offers some variance reduction, we demonstrate that drastic reductions in the standard deviation of the estimated call price is obtained by combining all three variance reduction techniques in a symmetric pricing approach. This reduces the standard deviation by a factor of over 20 for long maturity call options on highly volatile assets. Finally, we show that our findings are not particular to using in-sample pricing but also hold when using an out-of-sample pricing approach.

Optimising Poisson bridge constructions for variance reduction methods

In this paper we discuss different Monte Carlo (MC) approaches to generate unit-rate Poisson processes and provide an analysis of Poisson bridge constructions, which form the discrete analogue of the well-known Brownian bridge construction for a Wiener process. One of the main advantages of these Poisson bridge constructions is that they, like the Brownian bridge, can be effectively combined with variance reduction techniques. In particular, we show here, in practice and proof, how we can achieve orders of magnitude efficiency improvement over standard MC approaches when generating unit-rate Poisson processes via a synthesis of antithetic sampling and Poisson bridge constructions. At the same time we provide practical guidance as to how to implement and tune Poisson bridge methods to achieve, in a mean sense, (near) optimal performance.