Closed-form Option Pricing for Exponential Lévy Models: A Residue Approach

53 Pages Posted: 12 Apr 2021 Last revised: 13 Dec 2021

See all articles by Jean-Philippe Aguilar

Jean-Philippe Aguilar

Société Générale

Justin Kirkby

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE)

Date Written: April 9, 2021

Abstract

Exponential Lévy processes provide a natural and tractable generalization of the classic Black-Scholes-Merton model which are capable of capturing observed market implied volatility skews. In the existing literature, closed-form option pricing formulas are sparse for exponential Lévy models, outside of special cases such as Merton's jump diffusion, and complex numerical techniques are required even to price European options.

To bridge the gap, this work provides a comprehensive and unified pricing framework for vanilla and exotic European options under the Variance Gamma (VG), Finite Moment Log Stable (FMLS), one-sided Tempered Stable (TS), and Normal Inverse Gaussian (NIG) models. We utilize the Mellin Transform and residue calculus to obtain closed-form series representations for the price of several European options, including vanillas, digitals, power, and log options. These formulas provide nice theoretical representations, but are also efficient to evaluate in practice, as numerous numerical experiments demonstrate. The closed-form nature of these option pricing formulas makes them ideal for adoption in practical settings, as they do not require complicated pricing methods to achieve high accuracy prices, and the resulting pricing error is reliably controllable.

Keywords: Levy process, Stable process, Variance Gamma process, Normal inverse Gaussian process, Stochastic volatility, Option pricing, Mellin Transform

JEL Classification: C00, C02, G10, G12, G13

Suggested Citation

Aguilar, Jean-Philippe and Kirkby, Justin, Closed-form Option Pricing for Exponential Lévy Models: A Residue Approach (April 9, 2021). Available at SSRN: https://ssrn.com/abstract=3823337 or http://dx.doi.org/10.2139/ssrn.3823337

Jean-Philippe Aguilar

Société Générale ( email )

Justin Kirkby (Contact Author)

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE) ( email )

765 Ferst Drive
Atlanta, GA 30332-0205
United States

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