GPU Pricing of Exotic Cross-Currency Interest Rate Derivatives with a Foreign Exchange Volatility Skew Model

Ortiz-Gracia

2017

J Sci Comput

We present a robust and highly efficient dimension reduction Shannon-wavelet method for computing European option prices and hedging parameters under a general jump-diffusion model with square-root stochastic variance and multi-factor Gaussian interest rates. Within a dimension reduction framework, the option price can be expressed as a two-dimensional integral that involves only (i) the value of the variance at the terminal time, and (ii) the time-integrated variance process conditional on this value. A Shannon wavelet inverse Fourier technique is developed to approximate the conditional density of the time-integrated variance process. Furthermore, thanks to the excellent approximation properties of Shannon wavelets, the overall pricing procedure is reduced to the evaluation of just a single integral that involves only the density of the terminal variance value. This single integral can be accurately evaluated, since the density of the variance at the terminal time is known in closed-form. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.

Section: A General Jump-diffusion Modelmentioning

confidence: 99%

A Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

Ortiz-Gracia

2017

J Sci Comput

“…, l, come from changing from the foreign risk-neutral measure to the domestic risk-neutral one. We emphasize the generality of the cross-currency model (1a)-(1d), as well as its strong suitability for modeling FX products, especially long-dated (maturities of 30 years or more) hybrid FX products, such as Power-Reverse Dual-Currency (PRDC) swaps [2,4].…”

Section: Cross-currency Modelmentioning

confidence: 99%

“…via the very popular square-root CoxIngersoll-Ross (CIR) model [1], and/or a stochastic interest rate, e.g. via a one/two-factor Hull-White model [2,4,3]. In addition, the financial markets have become more diverse, with trading not only of stocks, but also of many types of financial derivatives.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Dimension Reduction Monte-Carlo Simulation for High-dimensional Stochastic Models in Finance

Procedia Computer Science

2015

One-way coupling often occurs in multi-dimensional stochastic models in finance. In this paper, we develop a highly efficient Monte Carlo (MC) method for pricing European options under a N -dimensional one-way coupled model, where N is arbitrary. The method is based on a combination of (i) the powerful dimension and variance reduction technique, referred to as drMC, developed in Dang et. al (2014), that exploits this structure, and (ii) the highly effective multilevel MC (mlMC) approach developed by Giles (2008). By first applying Step (i), the dimension of the problem is reduced from N to 1, and as a result, Step (ii) is essentially an application of mlMC on a 1-dimensional problem. Numerical results show that, through a careful construction of the ml-dr estimator, improved efficiency expected from the Milstein timestepping with first order strong convergence can be achieved. Moreover, our numerical results show that the proposed ml-drMC method is significantly more efficient than the mlMC methods currently available for multi-dimensional stochastic problems.

“…non-perfect correlations, between rates for different maturities. This issue is particularly crucial in modelling of (long-dated) FX interest rate derivatives, such as Power-Reverse Dual-Currency (PRDC) swaps and FX Target Redemption Notes, due to their strong dependence on movements in both domestic and foreign interest rates (Caps, 2007;Col et al, 2013;Dang et al, 2014Dang et al, , 2010Dang et al, , 2015aMallo, 2010;Piterbarg, 2006;Sippel and Ohkoshi, 2002). These derivatives have become increasingly important and are traded in large quantities in Over-the-Counter markets.…”

Section: Introductionmentioning

confidence: 99%

A Shannon wavelet method for pricing foreign exchange options under the Heston multi-factor CIR model

Berthe

Applied Numerical Mathematics

Ortiz-Gracia

2019

We present a robust and highly efficient Shannon wavelet pricing method for plain-vanilla foreign exchange European options under the jump-extended Heston model with multi-factor CIR interest rate dynamics. Under a Monte Carlo and partial differential equation hybrid computational framework, the option price can be expressed as an expectation, conditional on the variance factor, of a convolution product that involves the densities of the time-integrated domestic and foreign multi-factor CIR interest rate processes. We propose an efficient treatment to this convolution product that effectively results in a significant dimension reduction, from two multi-factor interest rate processes to only a single-factor process. By means of a state-of-the-art Shannon wavelet inverse Fourier technique, the resulting convolution product is approximated analytically and the conditional expectation can be computed very efficiently. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and efficiency of the method.