Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

Abstract

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle

“…We focus on the fast computation of TME, where it exists in the linear solution operator. Recently, some investigators applied the Krylov subspace methods to the matrix exponential [22,23,30,43,44,42], here we use the shift-invert Lanczos method [33,35] to calculate the TME.…”

Fast numerical solution for fractional diffusion equations by exponential quadrature rule

“…In next section, we will exploit the shift-invert Arnoldi method [21,31,33] to approximate such a matrix exponential that only needs O(n log n) operations.…”

“…Note that the norm of such a Toeplitz-like matrix is very large. Therefore, the shift-invert Arnoldi method [21,31,33] is exploited to approximate the Toeplitz-like matrix exponential. Furthermore, the coefficient matrix is proved to be sectorial and hence the convergence of the approximation by the shift-invert Arnoldi method is independent of the size of the matrix norm.…”

European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme

“…It is noticed that what we ultimately require is the multiplication of a matrix exponential and a vector, but not the exact matrix exponential. For such case, the recently used Krylov subspace methods such as the one in [34] work well.…”