Regularization and analytic option pricing under α-stable distribution of arbitrary asymmetry

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We consider a non-Gaussian option pricing model, into which the underlying log-price is assumed to be driven by an α-stable distribution. We remove the a priori divergence of the model by introducing a Mellin regularization for the Lévy propagator. Using distributional and Cⁿ tools, we derive an analytic closed formula for the option price, valid for any stability α∈]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.