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

We derive an extremal fractional Gaussian by employing the L\évy-Khintchine theorem and L\évian 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.