Longstaff-Schwartz, Effective Model Dimensionality and Reducible Markov-Functional Models

Abstract

The popularity of the so-called Market Models has led researchers and practitioners to ask two important questions about modelling interest-rate derivatives. The first (and highly contentious) question is, how many stochastic drivers are needed to value accurately any given derivative? The second, which arises because of the high dimensionality of Market Models, even those with a small number of stochastic drivers, is how can callable products be valued using Monte Carlo simulation? In this paper we consider the Longstaff-Schwartz algorithm, an effective algorithm developed to answer the second of these questions, and in so doing we shed light on the first of these questions. We show that the success of the Longstaff-Schwartz algorithm for high-dimensional models demonstrates that, in a way we make precise, low-dimensional models are sufficient, but that in another sense the higher dimensionality still plays a part. Using the insight gained from this analysis we go on to develop models which have these desirable properties - high dimensionality and accurate calibration properties on the one hand, but the ability to collapse the models onto lower-dimensional 'core' models for the purposes of valuing callable derivatives. The core models that we develop are Markovian and can thus be implemented efficiently using lattice methods, avoiding the need for more costly Monte Carlo simulation.