Quantitative Finance

Chapter 7

Monte Carlo simulation

7.1 Background

In previous chapters (e.g. Chapters 3 and 4) we have seen that option prices can be represented as expected discounted payoffs under some “risk neutral” probability measure. Monte Carlo simulation provides a method to calculate such expected values numerically in the sense that the expectation of a random variable can be approximated by the mean of a sample from its distribution.1 Furthermore, Monte Carlo simulation can be useful to generate scenarios to evaluate the risk of a particular trading position.

Moving away from the purely probabilistic interpretation, Monte Carlo simulation can also be seen as simply a method of numerical integration. Suppose one wishes to calculate

$\begin{array}{cc}I={\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}& \left(7.1\right)\end{array}$