Eur. Phys. J. B 79, 47-53 (2011)
https://doi.org/10.1140/epjb/e2010-10305-8

Adiabaticity conditions for volatility smile in Black-Scholes pricing model

¹ Dipartimento di Matematica e Fisica, Università Cattolica, via Musei 41, 25121 Brescia, Italy
² FinecoBank S.p.A, Unicredit Group, via Marco D'Aviano 5, 20131 Milano, Italy
³ Theoretical Division, MS-B213, Los Alamos National Laboratory, Los Alamos, 87545 NM, USA
⁴ I.N.F.N. Sezione di Pavia, Pavia, Italy

Corresponding author: ^a l.spadafora@dmf.unicatt.it

Received: 13 April 2010
Revised: 17 October 2010
Published online: 22 November 2010

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 expression with volatility σ 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.