Monte Carlo Methods

Abstract

The term “Monte Carlo” for computational methods involving simulated random numbers was invented by Metropolis and Ulam when they were working at the Las Alamos Laboratory (didn’t some people say that quants were rocket scientists?). Like deterministic pricing schemes, simulation pricing schemes can be used in any Markovian (or, of course, static one-period) setup. In the case of European claims, simulation pricing schemes reduce to the well known Monte Carlo loops. For products with early exercise features, or for more general control problems and the related BSDEs, numerical schemes by simulation are available too, yet these are more sophisticated and will be dealt with separately in Chaps. 10 and 11.

Monte Carlo methods are attractive by their genericity: genericity of their theoretical properties (such as the confidence interval they provide for the solution, at least for genuine pseudo Monte Carlo methods, as opposed to the quasi Monte Carlo methods also reviewed in this chapter); and also genericity of implementation. But (pseudo) Monte Carlo methods are slow, only converging at the rate $\sigma/ \sqrt{m}$, where m is the number of simulation runs and σ is the standard deviation of the sampled payoff. So ultimately with Monte Carlo it’s all about how to make it faster. To accelerate the convergence, various variance reduction techniques (e.g. control variate and importance sampling) can be used to transform a given payoff into another one with less variance. An alternative to variance reduction is quasi Monte Carlo, which converges faster in practice than pseudo Monte Carlo (but beware of the dimension; moreover quasi Monte Carlo estimates do not come with confidence intervals). A last “acceleration” technique is of course to resort to a parallel implementation, which with Monte Carlo is an easy thing to do, but unfortunately parallelization techniques are not dealt with in this book!