Volatility in options formulae for general stochastic dynamics

TL;DR

This paper investigates the conditions under which the Black-Scholes formula can hold for general stochastic models, revealing that constant volatility is necessary under certain conditions and providing bounds on volatility variation otherwise.

Contribution

It establishes that the Black-Scholes formula implies constant volatility for a continuum of strikes and times, and derives bounds on volatility variation with finitely many strikes.

Findings

Constant volatility is necessary if the formula holds for a continuum of strikes and times.

A universal bound on volatility variation is derived for finitely many strikes.

Implied volatility is constant as strike sequences cover the entire half-line.

Abstract

It is well-known that the Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by financial markets. This paper addresses the question of whether an alternative model for stock price exists for which the Black-Scholes or similar formulae hold. The results obtained in this paper are very general as no assumptions are made on the dynamics of the model, whether it be the underlying price process, the volatility process or how they relate to each other. We show that if the formula holds for a continuum of strikes and three terminal times, then the volatility must be constant. However, when it only holds for finitely many strikes, and three or more maturity times, we obtain a universal bound on the variation of the volatility. This bound yields that the implied volatility is constant when the sequence of strikes increases to cover the entire half-line. This recovers the result for a continuum of strikes by a different approach.