Advantages of the Laplace transform approach in pricing first touch digital options in Lévy-driven models
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- 作者：Oleg Kudryavtsev
- 关键词：Jump processes ; Factorization theory ; Laplace transform ; Computational methods ; Mathematical finance ; First passage probabilities
- 刊名：Bolet¨ªn de la Sociedad Matem¨¢tica Mexicana
- 出版年：2016
- 出版时间：October 2016
- 年：2016
- 卷：22
- 期：2
- 页码：711-731
- 全文大小：686 KB
- 刊物类别：Mathematics, general;
- 刊物主题：Mathematics, general;
- 出版者：Springer International Publishing
- ISSN：2296-4495
- 卷排序：22
文摘
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.