Do your volatility smiles take care of extreme events?

Abstract

In the Black-Scholes context we consider the probability distribution function (PDF) of financial returns implied by volatility smile and we study the relation between the decay of its tails and the fitting parameters of the smile. We show that, considering a scaling law derived from data, it is possible to get a new fitting procedure of the volatility smile that considers also the exponential decay of the real PDF of returns observed in the financial markets. Our study finds application in the Risk Management activities where the tails characterization of financial returns PDF has a central role for the risk estimation.