Cross-Currency and Hybrid Markov-Functional Models

Abstract

In this paper we consider cross-currency Markov-functional models and their calibration under the spot measure. Hunt, Kennedy and Pelsser introduced a single-currency Markov-functional interest rate model in the terminal measure and showed how to efficiently calibrate it to LIBOR or swaprate options. Building upon their work we will present a multi-factor cross-currency LIBOR model under different measures. We see zero correlated FX spot and LIBOR rates as a natural starting point. Under this assumption we don't need a change of numeraire drift correction and the functionals of the foreign currency rates under the domestic numeraire are identical to the functionals of the foreign currency model under its (foreign) spot measure. This provides the motivation for first deriving a spot measure version of single-currency Markov-functional model. We will show that in the spot measure it is possible to formulate and implement a very efficient calibration procedure comparable to that provided by Hunt, Kennedy and Pelsser for the terminal measure.

Combining single-currency Markov-functional interest rate models with a Markov-functional FX spot model we build two and three factor cross-currency models. Relaxing the zero correlation assumption is technically quite simple, but it entails considerable additional computational costs, mainly for the calibration of the model to FX options. To circumvent this problem we suggest a more efficient approximate procedure, which seems to work quite well for low correlations.

Keywords: Interest Rate Model, Markov Functional Model, Cross Currency Model, Spot Measure in Time Discrete Lattice Model

JEL Classification: G13, C63