The European Physical Journal B

, Volume 79, Issue 1, pp 47–53 | Cite as

Adiabaticity conditions for volatility smile in Black-Scholes pricing model

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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.

Keywords

Probability Density Function Stock Price Option Price Call Option Implied Volatility 

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Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità CattolicaBresciaItaly
  2. 2.FinecoBank S.p.A, Unicredit Group, via Marco D’Aviano 5MilanoItaly
  3. 3.Theoretical Division, MS-B213Los Alamos National LaboratoryLos AlamosUSA
  4. 4.I.N.F.N. Sezione di PaviaPaviaItaly

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