Markov-Functional Models

Abstract

In this chapter we propose models that can fit the observed prices of liquid instruments in a similar fashion to the market models, but which also have the advantage that prices can be calculated just as efficiently as in the spot rate models of the first part of this book. To achieve this we consider the general class of Markov-Functional interest rate models (MF models), first introduced by Hunt, Kennedy and Pelsser (2000). The defining characteristic of MF models is that pure discount bond prices are assumed at any time to be a function of some low-dimensional process which is Markovian in some martingale measure. This ensures that implementation is efficient since it is only necessary to track the driving Markov process. Market models do not possess this property (for a low-dimensional Markov process) and this is the impediment to their efficient implementation. The freedom to choose the functional form is what permits accurate calibration of Markov-Functional models to relevant market prices, a property not possessed by spot rate models. The remaining freedom to specify the law of the driving Markov process is what allows us to make the model realistic. As we shall see, given a Markov process, it is possible to fit the marginal distributions of market interest rates implied by the market option prices. However, for successfully pricing exotic interest rate products it is important to capture the joint distribution of the interest rates under consideration. We will discuss this important point further in Section 9.5.