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Boletín de la Sociedad Matemática Mexicana

, Volume 22, Issue 2, pp 711–731 | Cite as

Advantages of the Laplace transform approach in pricing first touch digital options in Lévy-driven models

  • Oleg Kudryavtsev
Original Article

Abstract

Motivated by the pricing of first touch digital options in exponential Lévy models and corresponding credit risk applications, we study numerical methods for solving related partial integro-differential equations. The goal of the paper is to consider advantages of the Laplace transform-based approach in this context. In particular, we show that the computational efficiency of the numerical methods which start with the time discretization can be significantly enhanced (often, in several dozen of times) by means of the Laplace transform technique. As an additional result, we provide a new Wiener–Hopf factorization formula which admits an efficient numerical realization by means of the Fast Fourier Transform. We propose two new efficient methods for pricing first touch digital options in wide classes of Lévy processes. Both methods are based on the numerical Laplace transform inversion formulae and a numerical Wiener–Hopf factorization. The first method uses the Gaver–Stehfest algorithm, the second one deals with the Post–Widder formula. We prove the advantages of the new methods in terms of accuracy and convergence by using numerical experiments.

Keywords

Jump processes Factorization theory Laplace transform Computational methods Mathematical finance First passage probabilities 

Mathematics Subject Classification

60G51 62P05 60-08 60J75 47A68 42A85 

Notes

Acknowledgments

The author is grateful to participants of The International Workshop “Wiener–Hopf method, Toeplitz operators and their application”, Veracruz, Mexico, November 3–7, 2015, and especially to Professor Sergei Grudsky for discussions and useful comments.

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Copyright information

© Sociedad Matemática Mexicana 2016

Authors and Affiliations

  1. 1.Department of InformaticsRussian Customs Academy Rostov BranchRostov-on-DonRussia

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