Laplace transform approach to option pricing for time-changed Brownian models

ABSTRACT

In this article, we consider European option pricing for time-changed Brownian models using Laplace transform. We obtain a general formula for the option price as the integral of a real-valued function involving the Laplace transform of the random time change. Unlike the usual Fourier transform technique, our method does not suffer from difficulties specific to complex integration, such as the evaluation of multiple-valued functions, and allows for a model-independent analysis of the truncation error. In the numerical analysis part, we compare option prices in variance gamma (VG), normal inverse Gaussian (NIG), and generalized hyperbolic (GH) models obtained by Laplace transform with those obtained by the Fourier transform method introduced by Carr and Madan in 1999. The results show that our method converges faster than the Fourier approach when the Laplace transforms of the subordinators decay exponentially, for examples like NIG and GH models.

KEYWORDS:

• Fourier transform
• Laplace transform
• Option pricing
• Time-changed Brownian models.

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60G20
• Secondary 65C50, 65G20

Acknowledgments

This work was done while the first author was visiting the Laboratoire de Probabilités et Modéles Aléatoires of Université Paris-Diderot (Paris 7). She would like to thank Professor Peter Tankov for his useful suggestions on this article.