Fast exponential time integration scheme for option pricing with jumps

Abstract

SUMMARYA fast exponential time integration scheme is considered for pricing European and double barrier options in jump-diffusion models. After spatial discretization, the option pricing problem is transformed into the product of a matrix exponential and a vector, while the matrix bears a Toeplitz structure. The shift-andinvert Arnoldi method is then employed for fast approximation to such operation. Owing to the Toeplitz structure, the computational cost can be reduced by the fast Fourier transform. Furthermore, the discretized form of option pricing problem satisfies a given condition such that the error bound of the shift-and-invert Arnoldi approximation is unrelated to the norm of the matrix. Numerical tests are carried out to compare the proposed method with other option pricing methods.

Fast numerical solution for fractional diffusion equations by exponential quadrature rule

“…Recently, the Krylov subspace methods have been applied to approximate the matrix exponential [20,21,29,31], especially when the matrix is with a very large size and special structure (sparse or Toeplitz). The main idea is to approximately project the exponential of a large matrix onto a small Krylov subspace.…”

“…Let γ > 0 be the shift parameter. The shift-invert Arnoldi method for approximating exp(−tA)v is described as follows [20,21,29].…”

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

“…Then, the Kolmogorov backward equations for these processes are considered as one of the useful methods for option pricing. See Cont & Voltchkova (2005); Garreau & Kopriva (2013); Lee et al (2012) and the references therein.…”

“…Popular examples of such methods are finite difference methods (Gao et al, 2013;Huang & Oberman, 2013;Li et al, 2012;Meerschaert, 2004), finite element methods (Zhao & Lib, 2012), spectral methods (Bueno-Orovio et al, 2014;Huang et al, 2014), and other methods (Yan, 2013) for the forward equations describing anomalous diffusion etc. Further, such methods have been proposed for the backward equations in finance (Duquesne et al, 2010;Garreau & Kopriva, 2013;Kwok et al, 2012;Lee et al, 2012). In many of these methods, first, a time-evolution system of ordinary differential equations is derived from the given PIDE by the discretization of the spatial variable with finite differences, finite elements, polynomial expansions, etc., and some quadrature formulas.…”

Numerical pricing of American options under infinite activity Lévy processes

“…ETI (Cox & Matthews, 2002) is exact in time integration and therefore L 0 -stable. In addition, with efficient techniques for matrix function evaluations (Lee, Liu, & Sun, 2010;Schmelzer & Trefethen, 2007), ETI is becoming a promising method for solving systems of equations. Since only spatial discretization is incurred, this scheme combined with quadratic finite Journal of Futures Markets DOI: 10.1002/fut element methods has led to fourth order convergent numerical solutions for American options (Rambeerich, Tangman, Gopaul, & Bhuruth, 2009) under Merton's jump diffusion model (Merton, 1976).…”