A transform approach to compute prices and greeks of barrier options driven by a class of Levy processes

TL;DR

This paper introduces a transform method for efficiently computing prices and greeks of barrier options driven by Levy processes, including hyper-exponential jumps, with applications to popular models like VG and NIG.

Contribution

It develops an analytical transform-based approach for barrier option pricing under Levy models, enabling rapid and accurate computations and convergence proofs.

Findings

Transform method yields fast, accurate prices and greeks.

Method compares favorably with Monte Carlo simulations.

Applicable to various Levy models used in finance.

Abstract

In this paper we propose a transform method to compute the prices and greeks of barrier options driven by a class of Levy processes. We derive analytical expressions for the Laplace transforms in time of the prices and sensitivities of single barrier options in an exponential Levy model with hyper-exponential jumps. Inversion of these single Laplace transform yields rapid, accurate results. These results are employed to construct an approximation of the prices and sensitivities of barrier options in exponential generalised hyper-exponential (GHE) Levy models. The latter class includes many of the Levy models employed in quantitative finance such as the variance gamma (VG), KoBoL, generalised hyperbolic, and the normal inverse Gaussian (NIG) models. Convergence of the approximating prices and sensitivities is proved. To provide a numerical illustration, this transform approach is compared with Monte Carlo simulation in the cases that the driving process is a VG and a NIG Levy process. Parameters are calibrated to Stoxx50E call options.