Graphics processing unit pricing of exotic cross‐currency interest rate derivatives with a foreign exchange volatility skew model

Abstract

We present a graphics processing unit (GPU) parallelization of the computation of the price of exotic crosscurrency interest rate derivatives via a partial differential equation (PDE) approach. In particular, we focus on the GPU-based parallel pricing of long-dated foreign exchange (FX) interest rate hybrids, namely power reverse dual currency (PRDC) swaps with Bermudan cancelable features. We consider a three-factor pricing model with FX volatility skew, which 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, and the alternating direction implicit (ADI) technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU via a parallelization of the ADI timestepping technique. Numerical results indicate that GPUs can provide significant increase in performance over CPUs when pricing PRDC swaps. An analysis of the impact of the FX skew on such derivatives is provided. PRDC swaps. The use of a local volatility model provides better modeling for the skewness of the FX rate and at the same time avoids introducing more stochastic factors into the model.The ever growing interest in cross-currency interest rate derivatives in general, and PRDC swaps in particular, has created a need for efficient pricing and hedging strategies for them. The popular choice for pricing PRDC swaps is Monte-Carlo (MC) simulation, but this approach has several major disadvantages, including slow convergence, and the limitation that the price is obtained at a single point only in the domain as opposed to the global character of the partial differential equation (PDE) approach. In addition, accurate hedging parameters, such as delta and gamma, are generally harder to compute via an MC approach than via a PDE approach. On the other hand, when pricing PRDC swaps by the PDE approach, each stochastic factor in the pricing model gives rise to a spatial variable in the PDE. Because of the 'curse of dimensionality' associated with highdimensional PDEs-not to mention additional complexity due to multiple cash flows and exotic features, such as Bermudan cancelability-the pricing of such derivatives via the PDE approach is not only mathematically challenging but also very computationally intensive.Over the last few years, the rapid evolution of graphics processing units (GPUs) into powerful, cost efficient, programmable computing architectures for general purpose computations has provided application potential beyond the primary purpose of graphics processing. In the area of computational finance, although there has been great interest in utilizing GPUs in developing eff...

Alternating direction implicit time integrations for finite difference acoustic wave propagation: Parallelization and convergence

“…However, hybrid CR-TA solvers have been also proposed in [18] and [23]. In addition, solution methods based on approximate matrix inversion have been alternatively used by [48], and Authors in [11] and [12] developed a problembased solution approach. Thus, last word on GPU tridiagonal solvers has not yet been said, and this motivates recent the reviews and exploratory studies in [20] and [27].…”

A dimension and variance reduction Monte-Carlo method for option pricing under jump-diffusion models

“…In addition to stochastic interest rates, due to long maturity, stochastic volatility is also crucial in modelling long-dated FX options. This is due to (1) the typical observed volatility skew in the FX markets, and (2) the products' payoff structures, which makes the prices of these options very sensitive to the skews (Dang, Christara, and Jackson 2014;Dang et al 2015a;Piterbarg 2006). A major challenge in modelling long-dated options in general, and FX options in particular, is that they are usually embedded with exotic features which provide possibilities of very early termination of the products, usually within only a couple of years of their inception.…”