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Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models

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Abstract

Option prices in exponential Lévy models solve certain partial integro-differential equations. This work focuses on developing novel, computable error approximations for a finite difference scheme that is suitable for solving such PIDEs. The scheme was introduced in (Cont and Voltchkova, SIAM J. Numer. Anal. 43(4):1596–1626, 2005). The main results of this work are new estimates of the dominating error terms, namely the time and space discretisation errors. In addition, the leading order terms of the error estimates are determined in a form that is more amenable to computations. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschitz continuous as in previous works. If the underlying Lévy process has infinite jump activity, then the jumps smaller than some \(\epsilon > 0\) are approximated by diffusion. The resulting diffusion approximation error is also estimated, with leading order term in computable form, as well as the dependence of the time and space discretisation errors on this approximation. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the cut off parameter \(\epsilon \).

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Acknowledgments

The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project “Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar” and King Abdullah University of Science and Technology (KAUST) is also acknowledged. The research has also been supported by the Swedish Foundation for Strategic Research (SSF) via the Center for Industrial and Applied Mathematics (CIAM) at KTH. The second author is a member of the KAUST SRI center for Uncertainty Quantification.

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Correspondence to Jonas Kiessling.

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Communicated by Desmond Higham.

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Kiessling, J., Tempone, R. Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models. Bit Numer Math 54, 1023–1065 (2014). https://doi.org/10.1007/s10543-014-0490-4

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