Numerical Valuation of Cross-Currency Swaps and Swaptions

Michael A. H. Dempster and J. P. Hutton
Department of Mathematics, University of Essex
mahd@essex.ac.uk
hutto@essex.ac.uk

Abstract

We investigate numerical valuation of cross-currency interest-based derivatives under Babbs' extended Vasicek-style model by numerical solution of the associated partial differential equations (PDE) - in particular, we consider the terminable differencial (diff) swap.

Firstly we precisely formulate, in terms of their cash flows, various types of single and cross-currency swaps and swaptions. We describe Babbs' model for the domestic and foreign term structures and the exchange rate, its formulation in terms of three correlated driftless Gaussian process and the associated four state variable parabolic PDE. We then formulate finite difference approximations to the PDE and discuss explicit and implicit methods. With this discrete approxiation to the valuation problem in a period, we proceed to value the terminable diff swap and other deals numerically by backwards recursion through the LIBOR dates, and investigate the solution found graphically.

We conclude that it is certainly practical, on a fast workstation, to solve for the value function of a very wide range of cross-currency derivative securities by solution of explicit finite approximations of the PDE.