Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics

Chapter Six

Sample Path Generation for Continuous-Time Models

Many models of interest in financial engineering can be represented by continuous-time stochastic differential equations. The application of Monte Carlo methods to this class of models requires sample path generation, which in turn requires a way to discretize them. Numerical analysis offers several ways to discretize deterministic differential equations for simulation purposes, including some very ingenious ones; here we will consider rather simple strategies, but we note that, due to the stochastic character of our applications, we have to tackle both numerical and statistical issues. To motivate the content of the chapter practically, we will mostly consider simple asset pricing problems, leaving more complex examples to Chapter 11 on option pricing, where we also apply variance reduction techniques, and Chapter 13 on risk management and hedging. The risk-neutral pricing approach that we discussed in Section 3.9 allows to price financial instruments by taking suitable expectations, which in turn requires Monte Carlo simulation, when the model is complicated enough to preclude an analytical solution. The following cases, listed in increasing order of complexity, may occur, depending on the nature of the underlying stochastic process and the kind of derivative contract we are evaluating:

1. We just need the value of a random variable at a single point in time and we are able to sample paths exactly. This is the lucky ...