Abstract
As the only practical way to deal with most path-dependent instruments, Monte Carlo estimation is now one of the workhorses of modern derivatives valuation. It has the advantage of being relatively easy to implement in its basic form, and, given enough computer resources, it will converge asymptotically to the correct answer. Yet, once these general principles are acknowledged, one faces the fact that many problems have such high dimension that the basic Monte Carlo technique can require an enormous number of simulations before convergence to a reasonably accurate answer is achieved. Some variance reduction techniques, like the use of antithetic variables, are helpful and easy to apply, but limited in effectiveness. Others, like non-random sample selection through use of stratified sampling or importance sampling, may depart from the basic, fully random Monte Carlo technique but can produce enormous improvement in performance in some cases. This article presents a remarkably effective simulation technique for high-dimensional problems that combines stratification with importance sampling, so that the simulation sample closely matches the desired probability distribution and resolution is maximized in the regions that are most important for the value of the derivative instrument.