An efficient numerical partial differential equation approach for pricing foreign exchange interest rate hybrid derivatives

Duy-Minh Dang, Christina C. Christara, Kenneth R. Jackson and Asif Lakhany

Abstract

ABSTRACT

In this paper, 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 multicurrency pricing model with FX volatility skew. The emphasis of this 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 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 subproblems over each time period of the swap's tenor structure, with possible communication at the end of the time period. Each of these pricing subproblems 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 nonuniform grids are used for the spatial discretization of the model PDE, and the alternating direction implicit time-stepping methods are employed for its time discretization. Numerical examples illustrating the convergence properties and efficiency of the numerical methods are also provided.