Geometric Step Options with Jumps: Parity Relations, PIDEs, and Semi-Analytical Pricing

49 Pages Posted: 11 Mar 2020 Last revised: 18 Mar 2020

See all articles by Walter Farkas

Walter Farkas

University of Zurich - Department of Banking and Finance; Swiss Finance Institute; ETH Zürich

Ludovic Mathys

University of Zurich - Department of Banking and Finance

Date Written: February 23, 2020

Abstract

The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and parity relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

Keywords: Geometric Step Options, American-Type Options, Lévy Markets, Jump-Diffusion Disentanglement, Maturity-Randomization

JEL Classification: C32, C61, C63, G13

Suggested Citation

Farkas, Walter and Mathys, Ludovic, Geometric Step Options with Jumps: Parity Relations, PIDEs, and Semi-Analytical Pricing (February 23, 2020). Swiss Finance Institute Research Paper No. 20-11, Available at SSRN: https://ssrn.com/abstract=3543080 or http://dx.doi.org/10.2139/ssrn.3543080

Walter Farkas

University of Zurich - Department of Banking and Finance ( email )

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Zürich, 8001
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HOME PAGE: http://https://people.math.ethz.ch/~farkas/

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

ETH Zürich ( email )

Rämistrasse 101
ZUE F7
Zürich, 8092
Switzerland

Ludovic Mathys (Contact Author)

University of Zurich - Department of Banking and Finance ( email )

Plattenstr 32
Zurich, 8032
Switzerland

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