There is a multifaceted interaction between optimization and Monte Carlo simulation, which is illustrated by the rich connections between this and other chapters of this book. Monte Carlo methods may be used when dealing with stochastic optimization models, i.e., when some of the parameters characterizing a decision problem are uncertain. This is clearly relevant in finance, and there is an array of approaches depending on the problem structure and on our purpose:
In stochastic programming (SP) models we generalize deterministic mathematical programming models in order to deal with uncertainty that is typically represented by a scenario tree. The scenario tree may be generated by crude Monte Carlo sampling, or we may adopt other approaches like Gaussian quadrature (see Chapter 2), variance reduction (see Chapter 8), or low-discrepancy sequences (see Chapter 9). The stochastic programming models that we consider here can be extended to account for risk aversion, as illustrated in Chapter 13.
Another approach to decision making under uncertainty relies on dynamic programming (DP). We will illustrate the advantages and the disadvantages of DP with respect to SP models. The principle behind DP is embodied in the Bellman recursive equation and is extremely powerful; ...