### Chapter 5

#### Monte Carlo Method

# 5.1 Motivation

Most financial professionals have some familiarity with the Monte Carlo method. Many learn of it as a tool financial engineers use for pricing derivatives. We have already given, in Chapter 1, an intuitive description of how it is employed with value-at-risk measures. Such intuitive familiarity is an inadequate foundation if we are to implement practical value-at-risk measures based on the Monte Carlo method.

Many value-at-risk measures that employ the Monte Carlo method take hours to run, even with parallel processing. Run times are dramatically improved with variance reduction techniques, which we apply to value-at-risk measures in Chapter 10. To understand how these work, we need a formal understanding of the Monte Carlo method.

The purpose of this chapter is to replace intuitive familiarity with formal understanding. We place the Monte Carlo method in its (nonfinancial) historical context. We then consider two applications of the Monte Carlo method. Both applications can be expressed as definite integrals, which leads us to consider the Monte Carlo method as an alternative technique of numerical integration. This perception sets the stage for variance reduction techniques, which we introduce at the end of the chapter.

A secondary purpose of the chapter is to describe standard techniques for generating pseudorandom numbers. Many generators that come packaged with software are unsuitable for the high-dimensional analyses that some value-at-risk measures entail. We clarify this problem and offer guidance in selecting appropriate generators.