A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models

Abstract

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Lévy process or, more generally, a time-inhomogeneous jumpdiffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Lévy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed with smooth and non-smooth initial conditions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.

Exponential time integration and Chebychev discretisation schemes for fast pricing of options

“…Many works [1,2,6,[8][9][10]14] have studied this problem. While fully explicit schemes require very restrictive conditions to remain stable [6], the implicit-explicit (IMEX) methods used in [8] has slow first order convergence.…”

Approximation of jump diffusions in finance and economics

“…Under suitable conditions, the pricing function u(x, t) = E(g(X T )|X t = x) can be expressed as a solution of a partial integro-differential equation (PIDE). Therefore an approximation of the pricing function u(x, t) can be obtained by solving the corresponding PIDE via finite difference or finite element methods (see, for instance, Cont & Voltchkova, 2005;D'Halluin, Forsyth, & Vetzal, 2005). These methods are computationally efficient when we have a low-dimensional underlying process X.…”

Properties of Multinomial Lattices with Cumulants for Option Pricing and Hedging

“…This type of research may be categorized into two approaches: The first approach is to formulate a general continuous time model and discretize it to solve pricing problems (Amin, 1993;Cont and Voltchkova;Cox and Ross, 1976;Karandikar and Rachev, 1995). The second approach is to generalize the standard discrete model (such as the standard binomial tree or trinomial tree models (Cox et al, 1979;Hull and White, 1990)), directly, to incorporate real market behavior characterized as smiles or heavy tails (Derman and Kani, 1994;Rubinstein, 1994Rubinstein, , 1998.…”

“…Note that the ideas of a multinomial lattice and moment matching originate from the pioneering work of Cox and Ross (1976) using the binomial lattice model, and a number of extensions have been proposed since then for modeling stock (or option) dynamics (Amin, 1993;Cont and Voltchkova;Cox et al, 1979;Karandikar and Rachev, 1995;Rubinstein, 1994Rubinstein, , 1998 or interest rate dynamics (Hull and White, 1994;Li et al, 1995). Therefore, providing a multinomial lattice parameterization that matches moments itself might not be innovative.…”