Financial Applications

Financial problems such as option pricing form a rich class of applications for simulation, variance reduction techniques, and quasi–Monte Carlo sampling. They provide a unique opportunity to present these topics in an applied setting and therefore represent a valuable learning tool that we believe will be useful to the reader. Readers interested in a more extensive treatment of Monte Carlo simulation in finance are referred to [145, 202, 314].

The problems studied in this chapter all fit in the following framework. We start with a market model where we have q underlying assets and denote by S _j (t) the value of the jth asset at time t for j = 1, . . . , q. We also have a bank account, which pays interest at a rate r _t ≥ 0 at time t. Most of the time, we assume that r _t = r is constant, and the corresponding value of r is called the risk-free rate. We think of an option in a loose sense as a security that entitles its holder to a certain payoff whose value depends on one or more of the q underlying assets. We are interested in determining different quantities related to the option, the most important one being its value at a given time, for a given model of the underlying assets.