Advantages of the Laplace Transform Approach in Pricing First Touch Digital Options in Levy-Driven Models

20 Pages Posted: 11 Jan 2016 Last revised: 28 Jan 2016

See all articles by Oleg E. Kudryavtsev

Oleg E. Kudryavtsev

Southern Federal University - Faculty of Mathematics, Mechanics and Computer Science; InWise Systems, LLC; Russian Customs Academy Rostov Branch - Department of Informatics

Date Written: December 6, 2015

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

JEL Classification: 60G51, 62P05, 60-08, 60J75, 47A68, 42A85

Suggested Citation

Kudryavtsev, Oleg E., Advantages of the Laplace Transform Approach in Pricing First Touch Digital Options in Levy-Driven Models (December 6, 2015). Available at SSRN: https://ssrn.com/abstract=2713193 or http://dx.doi.org/10.2139/ssrn.2713193

Oleg E. Kudryavtsev (Contact Author)

Southern Federal University - Faculty of Mathematics, Mechanics and Computer Science ( email )

Milchakova str. 8a
Rostov-on-Don, 344090
Russia

InWise Systems, LLC ( email )

Eremenko 58/11
Rostov-on-Don, 344015
Russia

Russian Customs Academy Rostov Branch - Department of Informatics ( email )

Budennovskiy 20
Rostov-on-Don, 344011
Russia

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