A Comparison of Markov–Functional and Market Models

Abstract

The LIBOR Markov-functional model is an efficient arbitrage-free pricing model suitable for callable interest rate derivatives. We demonstrate that the one-dimensional LIBOR Markov-functional model and the separable onefactor LIBOR market model are very similar. Consequently, the intuition behind the familiar SDE formulation of the LIBOR market model may be applied to the LIBOR Markov-functional model.The application of a drift approximation to a separable one-factor LIBOR market model results in an approximating model driven by a one-dimensional Markov process, permitting efficient implementation. For a given parameterisation of the driving process, we find the distributional structure of this model and the corresponding Markov-functional model are numerically virtually indistinguishable for short maturity tenor structures over a wide variety of market conditions, and both are very similar to the market model. A theoretical uniqueness result shows that any accurate approximation to a separable market model that reduces to a function of the driving process is effectively an approximation to the analogous Markov-functional model. Therefore, our conclusions are not restricted to our particular choice of driving process. Minor differences are observed for longer maturities, nevertheless the models remain qualitatively similar. These differences do not have a large impact on Bermudan swaption prices.Under stress-testing, the LIBOR Markov-functional and separable LI-BOR market models continue to exhibit similar behaviour and Bermudan prices under these models remain comparable. However, the drift approximation model now appears to admit arbitrage that is practically significant. In this situation, we argue the Markov-functional model is a more appropriate choice for pricing.

Implications for Hedging of the Choice of Driving Process for One-Factor Markov-Functional Models

“…An important result that was observed in [2] is the approximate linear relationship between the logarithms of the co-terminal forward swap rates and x…”

“…The contribution we make here is to study the implications for hedging of the choice of the instantaneous volatility for the driving Markov process. This is a topic which seems to have received little attention in the literature for one factor Markov-functional models or equivalently for one factor separable market models (see [2,10]). One popular choice is to take a Gaussian process with exponential instantaneous volatility, referred to as the mean reversion process (MR), as is done in [9].…”

“…One popular choice is to take a Gaussian process with exponential instantaneous volatility, referred to as the mean reversion process (MR), as is done in [9]. We begin our investigation by comparing this candidate with one based on the Hull-White short-rate model, referred to as the Hull-White process (HW) which was first introduced in [2].…”

A comparison of single factor Markov-functional and multi factor market models

“…The LIBOR Markov-functional model has been compared with the LIBOR market model before by Bennett and Kennedy (2005). These authors show that the one factor LIBOR Markov-functional model with mean reversion and the one factor separable LIBOR market model are largely similar in terms of dynamics and pricing.…”

“…They also show this for an approximated version of the LIBOR market model by drift approximations, as introduced by and Hunter et al (2001). Relative to Bennett and Kennedy (2005) this paper makes the contribution of also comparing multi factor models with the Markov-functional model. Moreover, we show how multi factor models can a priori be compared to the Markov-functional model which is not a straightforward extension from the one-dimensional case.…”