Regularization and analytic option pricing under $\alpha$-stable distribution of arbitrary asymmetry

TL;DR

This paper develops an analytic, regularized option pricing formula for models driven by $oldsymbol{ ext{alpha}}$-stable distributions with arbitrary asymmetry, extending classical models and enabling efficient market calibration.

Contribution

It introduces a Mellin regularization method for $oldsymbol{ ext{alpha}}$-stable driven models, deriving a closed-form option pricing formula valid for all $oldsymbol{ ext{alpha}}$ in (1,2] and any asymmetry, unifying and extending previous models.

Findings

The formula is computationally efficient and versatile.

It successfully calibrates to market data.

It recovers classical models like Black-Scholes and Carr-Wu.

Abstract

We consider a non-Gaussian option pricing model, into which the underlying log-price is assumed to be driven by an $\alpha$-stable distribution. We remove the a priori divergence of the model by introducing a Mellin regularization for the L\'evy propagator. Using distributional and $\mathbb{C}^n$ tools, we derive an analytic closed formula for the option price, valid for any stability $\alpha\in]1,2]$ and any asymmetry. This formula is very efficient and recovers previous cases (Black-Scholes, Carr-Wu); we calibrate the formula on market datas, make numerical tests, and discuss its many interesting properties.