Adaptive optimal allocation in stratified sampling methods

In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction and our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm. For the pricing of arithmetic average Asian options in the Black and Scholes model, the variance is divided by a factor going from 1.1 to 50.4 (depending on the option type and the moneyness) in comparison with the standard allocation procedure, while the increase in computation time does not overcome 1%.