In this chapter, we provide a detailed survey of simulation methods applied to numerical pricing of European, and, more recently, American options. Since European methods option prices can be calculated as expected values, it is natural to use Monte Carlo for computing them. However, this can often be quite slow, and this chapter reviews and compares different methods used to improve the efficiency of Monte Carlo methods. So-called "variance reduction" techniques are surveyed, including control variates, antithetic variates, moment matching, importance sampling and conditional Monte Carlo methods. Next, the quasi-Monte Carlo approach is reviewed, in which, instead of random numbers, deterministic sequences are generated — so-called quasi-random numbers or low-discrepancy sequences. These are more evenly dispersed than random sequences. It is interesting that these procedures are typically based on number-theoretic methods. The chapter also discusses the use of Monte Carlo methods for computing sensitivities ("Greeks") of the option price with respect to different parameters, and the difficult problem of computing American option prices using simulation. The difficulty stems from the fact that the price of an American option is a maximum of expected values, rather than a single expected value.
Copyright 2001 Cambridge University Press. Reprinted with permission.