Closed-form option pricing for exponential Lévy models: a residue approach

Abstract

Exponential Lévy processes provide a natural and tractable generalization of the classic Black–Scholes–Merton model which account for several stylized features of financial markets, including jumps and kurtosis. 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 path independent 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 options, including vanillas (European), 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:

• Lévy process
• Stable process
• Variance Gamma process
• Normal inverse Gaussian process
• Stochastic volatility
• Option pricing
• Mellin Transform

AMS subject classifications (MSC 2020):

• 60E07
• 60E10
• 60H35
• 65C30
• 65T50
• 91G20
• 91G30

JEL Classifications:

• C00
• C02
• G10
• G12
• G13

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Source code is made publicly available at: https://github.com/jkirkby3/PROJ_Option_Pricing_Matlab.

2 To be fully rigorous, one should group the gamma functions in the numerator of (A13(A13) $\begin{array}{rl}\omega & =\frac{\mathrm{\Gamma }({\underset{\_}{a}}_1.\underset{\_}{s}+b_1)\dots \mathrm{\Gamma }({\underset{\_}{a}}_m.\underset{\_}{s}+b_m)}{\mathrm{\Gamma }({\underset{\_}{\tilde{a}}}_1.\underset{\_}{s}+{\tilde{b}}_1)\dots \mathrm{\Gamma }({\underset{\_}{\tilde{a}}}_l.\underset{\_}{s}+{\tilde{b}}_l)}{x}_1^{-s_1}\dots x_n^{-s_n}\frac{\mathrm{d}{s}_1\dots \mathrm{d}{s}_n}{(2i\pi {)}^n},\\ & x_1,\dots x_n\in \mathbb{R},\end{array}$(A13) ) into n subfamilies called divisors and ensure that each face of Π is intersected by at most one of each divisors. We deliberately omit these technical details which are automatically satisfied by construction in all our proofs. Interested readers are invited to refer to earlier papers such as Aguilar (2020b) or to the theoretical papers Passare et al. (1994) and Zhdanov and Tsikh (1998).