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

Jonas Kiessling; Raul Tempone

doi:10.1007/s10543-014-0490-4

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

Research output: Contribution to journal › Article › peer-review

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 (Formula presented.) 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 (Formula presented.). © 2014 Springer Science+Business Media Dordrecht.

Original language	English (US)
Pages (from-to)	1023-1065
Number of pages	43
Journal	BIT Numerical Mathematics
Volume	54
Issue number	4
DOIs	https://doi.org/10.1007/s10543-014-0490-4
State	Published - May 6 2014

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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 for stokastiska differentialekvationer med tillampningar" 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.

ASJC Scopus subject areas

Computational Mathematics
Software
Applied Mathematics
Computer Networks and Communications

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10.1007/s10543-014-0490-4

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title = "Computable error estimates of a finite difference scheme for option pricing in exponential L{\'e}vy models",

abstract = "Option prices in exponential L{\'e}vy 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 L{\'e}vy process has infinite jump activity, then the jumps smaller than some (Formula presented.) 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 (Formula presented.). {\textcopyright} 2014 Springer Science+Business Media Dordrecht.",

author = "Jonas Kiessling and Raul Tempone",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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 for stokastiska differentialekvationer med tillampningar{"} 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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N2 - 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 (Formula presented.) 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 (Formula presented.). © 2014 Springer Science+Business Media Dordrecht.

AB - 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 (Formula presented.) 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 (Formula presented.). © 2014 Springer Science+Business Media Dordrecht.

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

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