A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives

Abstract

We propose a general framework for efficient pricing via a Partial Differential Equation (PDE) approach of cross-currency interest rate derivatives under the Hull-White model. In particular, we focus on pricing long-dated foreign exchange (FX) interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We formulate the problem in terms of three correlated processes that incorporate FX skew via a local volatility function. This formulation results in a time dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE. The Crank-Nicolson (CN) method and the Alternating Direction Implicit (ADI) method are considered for the time discretization. In the former case, the preconditioned Generalized Minimal Residual (GMRES) method is employed for the solution of the resulting block banded linear system at each time step, with the preconditioner solved by Fast Fourier Transform (FFT) techniques. Numerical results indicate that the numerical methods considered are second-order convergent, and, asymptotically, as the discretization granularity increases, almost optimal, with the ADI method being modestly more efficient than CN-GMRES-FFT. An analysis of the impact of the FX volatility skew on the PRDC swaps' prices is presented, showing that the FX volatility skew results in lower prices (i.e. profits) for the payer of PRDC coupons.