Numerical pricing of American options under infinite activity Lévy processes

Abstract

Under infinite activity Lévy models, American option prices can be obtained by solving a partial integro-differential equation (PIDE), which has a singular kernel. With increasing degree of singularity, standard time-stepping techniques may encounter difficulties. This study examines exponential time integration (ETI) for solving this problem and the performance of this scheme is compared with the Crank-Nicolson (CN) method and an implicit-explicit method in conjunction with an extrapolation (IMEX-Extrap), in terms of computational speed and convergence orders. These findings indicate that ETI is faster and more accurate among PIDE-based methods for solving the system of ordinary differential equations resulting from spatial discretization of the PIDE. For very singular problems, it is shown that the IMEX-Extrap scheme becomes unfavorable compared with the other schemes as it is relatively more time consuming and the global convergence deteriorates from quadratic to linear, whereas the ETI scheme yields both point-wise and global quadratic convergence. For illustration, under the infinite variation process, the IMEX-Extrap achieves a precision of the order of 10 Ϫ4 in 663.016 s, whereas for the same set of parameters, the CN method and the ETI scheme reach an accuracy of the order of 10 Ϫ5 in 237.891 s and 22.772 s, respectively.

Fast approximations of bond option prices under CKLS models

“…The above scheme is known as ETI and has been previously used for pricing options on assets in Tangman et al (2008Tangman et al ( , 2010 and Rambeerich et al (2009Rambeerich et al ( , 2010. However the application to the pricing of interest rate derivatives is new and is used for the first time in this paper for pricing bond options.…”

Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

“…Therefore, more general models for stochastic dynamics of the risky assets have been developed. We can mention the deterministic local volatility functions [8,15], the stochastic volatility [22,24], jump-diffusion [28,33], Lévy [30], and infinite Lévy models [16,41]. Note that infinite activity Lévy models incorporate jumps whose intensity is not a finite measure.…”

Pricing American Options in an Infinite Activity Lévy Market: Monte Carlo and Deterministic Approaches Using a Diffusion Approximation

“…Given a Lévy process X t which satisfies [FK1], [FK2], and [FK3], the localized problem admits a unique solution u 2 L 2 .0; T I V \ K/ where V is given by(23).…”