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

Abstract

Exponential L\'evy 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 L\'evy 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.