Monte Carlo Methods
Out of intense complexities intense simplicities emerge.
Many problems in finance do not have deterministic solutions because of the inherently stochastic nature of the problems’ underlying behaviors. In these cases, solutions are interpreted probabilistically and our interest centers on the statistical properties of the conclusions we draw from studying certain statistics of interest. Monte Carlo methods are algorithms that perform repeated random samplings in which sample properties provide us insight into the sampling distributions of these statistics of interest. One particular application, for example, relates to credit risk and the sensitivity of joint default on bonds to changes in the correlation across bonds—the probability, say, that bonds A, B, and C default at the same time. The relationship of interest in this case is the dependence of joint default risk on the correlation of returns across the bonds. We study this problem in detail further on.
Monte Carlo methods are the topic of a voluminous literature in the sciences and my intent here is not to cover that literature but to introduce you to this important class of models as they relate to our study of risk. Although there is no hard consensus on what exactly constitutes a Monte Carlo study, they all share some common elements: Given a set of known parameters—means, volatilities, and correlations, for example—we generate samples of new observations consistent with ...