Option pricing is one of the most important application fields for Monte Carlo methods, since option prices may be expressed as expected values under a suitable probability measure. Monte Carlo sampling is a quite flexible strategy that can be adopted when alternative numerical methods, like binomial/trinomial lattices or finite difference methods for solving partial differential equations, are inefficient or impossible to apply. For low-dimensional and path-independent options, Monte Carlo methods are not quite competitive with these alternatives, but for multidimensional or path-dependent options they are often the only viable computational strategy. Even more so when one has to deal with more complicated processes than an innocent geometric Brownian motion (GBM), or even a full-fledged term structure of interest rates. In the past, it was claimed that a significant limitation of Monte Carlo methods was their inability to cope with early exercise features. More recently, ideas from stochastic optimization and stochastic dynamic programming have been adapted to option pricing, paving the way to some Monte Carlo methods for American- and Bermudan-style options.
In this chapter we rely on the path generation methods of Chapter 6, where we have first seen how to price a vanilla call option by random sampling in Section 6.2.1. We will also make good use of the variance reduction strategies of Chapter 8, which were also illustrated in the simple case of ...