Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

TL;DR

This paper investigates how volatility smiles in the Black-Scholes model affect the distribution of future returns, revealing issues like negative probabilities and proposing adiabatic conditions to mitigate these problems.

Contribution

It introduces adiabaticity conditions for volatility smiles in the Black-Scholes model to ensure realistic probability distributions.

Findings

Volatility smiles can cause negative probabilities in return distributions.

Adiabatic conditions eliminate undesirable minima in the distribution.

The approach improves the realism of option pricing models.

Abstract

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile.