An Efficient Numerical PDE Approach for Pricing Foreign Exchange Interest Rate Hybrid Derivatives

Abstract

We discuss efficient pricing methods via a Partial Differential Equation (PDE) approach for long dated foreign exchange (FX) interest rate hybrids under a three-factor multi-currency pricing model with FX volatility skew. The emphasis of the paper is on Power-Reverse Dual-Currency (PRDC) swaps with popular exotic features, namely knockout and FX Target Redemption (FX-TARN). Challenges in pricing these derivatives via a PDE approach arise from the high-dimensionality of the model PDE, as well as from the complexities in handling the exotic features, especially in the case of the FX-TARN provision, due to its path-dependency. Our proposed PDE pricing framework for FX-TARN PRDC swaps is based on partitioning the pricing problem into several independent pricing sub-problems over each time period of the swap's tenor structure, with possible communication at the end of the time period. Each of these pricing sub-problems can be viewed as equivalent to a knockout PRDC swap with a known time-dependent barrier, and requires a solution of the model PDE, which, in our case, is a time-dependent parabolic PDE in three space dimensions. Finite difference schemes on non-uniform grids are used for the spatial discretization of the model PDE, and the Alternating Direction Implicit (ADI) timestepping methods are employed for its time discretization. Numerical examples illustrating the convergence properties and efficiency of the numerical methods are provided.

Keywords: Power-Reverse Dual-Currency (PRDC) swaps, Target Redemption (TARN), knockout, Partial Differential Equation (PDE), finite differences,non-uniform grids

JEL Classification: E40, E43, G12, G13, C61, C63