Just-In-Time Monte Carlo for Path-Dependent American Options

Abstract

The hardest computational problems in derivatives are those that involve American exercise and path-dependent payoffs. Standard Monte Carlo simulation generates paths in a forward direction, so path-dependent payoffs can be computed easily. But optimizing the early exercise decision requires generating separate families of paths going forward from each possible exercise point. Achieving an accurate solution for a complex problem can easily require more paths than can be held in computer memory. The popular Longstaff-Schwartz approximation technique for these problems works well but suffers from the same difficulty when the number of early exercise decisions is large, or in general, when problems become too complex. As shown here, Longstaff-Schwartz with too few secondary paths induces bias in addition to inaccuracy in the valuation. In this article, Dutt and Welke introduce an innovative approach that amounts to generating Monte Carlo paths in the reverse direction. Starting from the desired probability distribution of terminal asset prices, it simulates paths backward in time in such a way as to arrive with probability 1 at the initial asset value, with the resulting paths being fully consistent with the underlying returns process. The great advantage is that the simulated prices for later dates can be dropped from memory once they have been used. The authors illustrate the power of their “Just-in-time” approach on several examples, including valuing a 30 year mortgage under CIR interest rates taking into account both the prepayment and the default options.