Numerical Valuation of Cross-Currency Swaps and Swaptions

We investigate numerical valuation of cross-currency interest rate-based derivatives under Babbs' extended Vasicek-style model by numerical solution of the associated partial differential equation (PDE) --- in particular, we consider the terminable differential (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 processes and the associated three state variable parabolic PDE. We then formulate finite difference approximations to the PDE, and discuss explicit and implicit methods. With this discrete approximation to the valuation problem in a period, we proceed to value the terminable diff swap and other deals numerically by backwards recursion through the payment dates, and investigate the solutions found graphically. We conclude that it is certainly practical, on a fast workstation, to solve for the value function of a wide range of cross-currency derivative securities by solution of explicit finite difference approximations of the PDE.