Do your volatility smiles take care of extreme events?

TL;DR

This paper investigates how well volatility smiles in the Black-Scholes model capture the tail behavior of financial return distributions, proposing a new fitting method that incorporates empirical tail decay for improved risk management.

Contribution

It introduces a novel fitting procedure for volatility smiles that accounts for the exponential decay of return distribution tails, enhancing risk assessment accuracy.

Findings

The new method aligns the implied PDF tails with observed market data.

It improves the modeling of extreme events in financial returns.

Application to risk management shows better tail risk estimation.

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.