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

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.