Numerical solution of two asset jump diffusion models for option valuation

Authors:
Simon S. Clift;Peter A. Forsyth
Affiliations:
University of Waterloo, School of Computer Science, 200 University Avenue, Waterloo, Ontario, Canada;University of Waterloo, School of Computer Science, 200 University Avenue, Waterloo, Ontario, Canada
Venue:
Applied Numerical Mathematics
Year:
2008

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Abstract

Under the assumption that two financial assets evolve by correlated finite activity jumps superimposed on correlated Brownian motion, the value of a contingent claim written on these two assets is given by a two-dimensional parabolic partial integro-differential equation (PIDE). An implicit, finite difference method is derived in this paper. This approach avoids a dense linear system solution by combining a fixed point iteration scheme with an FFT. The method prices both American and European style contracts independent (under some simple restrictions) of the option payoff and distribution of jumps. Convergence under the localization from the infinite to a finite domain is investigated, as are the theoretical conditions for the stability of the discrete approximation under maximum and von Neumann analysis. The analysis shows that the fixed point iteration is rapidly convergent under typical market parameters. The rapid convergence of the fixed point iteration is verified in some numerical tests. These tests also indicate that the method used to localize the PIDE is inexpensive and easily implemented.