An extremal fractional Gaussian with a possible application to option-pricing with skew and smile

TL;DR

This paper introduces an extremal fractional Gaussian model derived via Lévy processes, generalizing the Black-Scholes formula to better capture skew and smile effects in option pricing.

Contribution

It presents a novel extremal fractional Gaussian framework and a generalized, convergent option-pricing formula applicable to fractional markets.

Findings

Derivation of an extremal fractional Gaussian using Lévy-Khintchine theorem

Generalization of Black-Scholes-Merton formula for fractional markets

Analysis of implied volatility structure in the new model

Abstract

We derive an extremal fractional Gaussian by employing the L\'evy-Khintchine theorem and L\'evian noise. With the fractional Gaussian we then generalize the Black-Scholes-Merton option-pricing formula. We obtain an easily applicable and exponentially convergent option-pricing formula for fractional markets. We also carry out an analysis of the structure of the implied volatility in this system.