I develop and analyze variance reduction techniques for Monte Carlo simulation, motivated by problems of pricing financial derivatives. There are two new approaches, both of which deal with the possibility of simulated paths stopping early.
In simulations for pricing barrier options, simulated paths may naturally stop early when a barrier is crossed. The first approach uses a change of measure at each step of the simulation in order to condition on one-step survival. When the probability of one-step survival is not available, I combine estimation of that probability with the change of measure. This method is more generally applicable to terminal reward problems on Markov processes with absorbing states.
In some applications, simulation effort is of greater value when applied to early time steps rather than shared equally among all time steps. The second approach stops paths when they would naturally continue, rather than forcing them to continue when they would naturally stop. I formulate and solve the problem of optimal allocation of computational resources to simulation steps with the objective of minimizing the variance of an estimator of a sum of sequentially simulated random variables. I also develop two ways to enhance variance reduction through early stopping: one takes advantage of the statistical theory of missing data, while the other redistributes the terms of the sum to make optimal use of optimal early stopping.