Numerical valuation of options with jumps in the underlying

Authors:
Ariel Almendral;Cornelis W. Oosterlee
Affiliations:
Norwegian Computing Center, Gaustadalleen 23, Postbox 114, Blindern, N-0314, Oslo, Norway;Faculty of Information Technology Systems, Department of Applied Mathematical Analysis, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
Venue:
Applied Numerical Mathematics
Year:
2005

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Abstract

A jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integro-differential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in time by the second order backward differentiation formula (BDF2). The resulting system is solved by an iterative method based on a simple splitting of the matrix. Using the fast Fourier transform, the amount of work per iteration may be reduced to O(n log2 n) and only O(n) entries need to be stored for each time level. Numerical results showing the quadratic convergence of the methods are given for Merton's model and Kou's model.