Skip to main content
SpringerLink
Log in

Adiabaticity conditions for volatility smile in Black-Scholes pricing model

  • Published:
The European Physical Journal B Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.C. Hull, Options, Futures and Other Derivatives (Prentice Hall, Upper Saddle River, New Jersey, 2003)

  2. P. Willmott, Derivatives (John Wiley and Sons, Chichester, 1998)

  3. F. Black, M. Scholes, J. Polit. Econ. 81, 637 (1973)

    Article  Google Scholar 

  4. J.-P. Bouchaud, M. Potters, Theory of Financial Risk – From Statistical Physics to Risk Management (University Press, Cambridge, 2000)

  5. J.C. Jackwerth, Journal of Derivatives 7, 66 (1999)

    Article  Google Scholar 

  6. E. Derman, I. Kani, Risk Magazine 7, 32 (1994)

    Google Scholar 

  7. B. Dupire, Risk Magazine 7, 18 (1994)

    Google Scholar 

  8. E. Derman, I. Kani, The volatility Smile and its Implied Tree, Quantitative Strategies Research Notes (Goldman Sachs, New York, 1994)

  9. M. Rubinstein, J. Finance 49, 771 (1994)

    Article  Google Scholar 

  10. J. Jackwerth, Journal of Derivatives 5, 7 (1997)

    Article  Google Scholar 

  11. T. Herwing, Lecture Notes in Economics and Mathematical System 571, 11 (2006)

    Article  Google Scholar 

  12. M.R. Fengler, Quant. Financ. 9, 417 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Y. Ait-Sahalia, J. Duarte, J. Econom. 116, 9 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. N. Kahale, Risk Magazine 17, 102 (2004)

    Google Scholar 

  15. L.D. Landau, E.M. Lifshitz, Mechanics (Butterworth-Heinemann, Oxford, UK, 2000)

  16. A.M. Malz, Option-Implied Probability Distribution and Currency Excess Returns, FRB of New York Staff Report, 1997, Vol. 32

  17. G. Brown, C. Randall, Risk Magazine 12, 62 (1999)

    Google Scholar 

  18. R. Tompkins, Eur. J. Finance 7, 198 (2001)

    Article  Google Scholar 

  19. K. Toft, B. Prucyk, J. Finance 52, 1151 (1997)

    Article  Google Scholar 

  20. J. Campa, K. Chang, R. Reider, J. Int. Money Financ. 17, 117 (1998)

    Article  Google Scholar 

  21. V.F. Pisarenko, D. Sornette, Physica A 366, 387 (2006)

    Article  ADS  Google Scholar 

  22. P. Gopikrishnan, M. Meyer, L.A.N. Amaral, H.E. Stanley, Eur. Phys. J. B 3, 139 (1998)

    Article  ADS  Google Scholar 

  23. U. Kirchner, Market Implied Probability Distributions and Bayesian Skew Estimation, arXiv:0911.0805v1 [q-fin.PR] (2009)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università Cattolica, via Musei 41, 25121, Brescia, Italy

    L. Spadafora

  2. FinecoBank S.p.A, Unicredit Group, via Marco D’Aviano 5, 20131, Milano, Italy

    L. Spadafora

  3. Theoretical Division, MS-B213, Los Alamos National Laboratory, Los Alamos, 87545, NM, USA

    G. P. Berman

  4. I.N.F.N. Sezione di Pavia, Pavia, Italy

    F. Borgonovi

Authors
  1. L. Spadafora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. P. Berman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Borgonovi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Spadafora.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Spadafora, L., Berman, G. & Borgonovi, F. Adiabaticity conditions for volatility smile in Black-Scholes pricing model. Eur. Phys. J. B 79, 47–53 (2011). https://doi.org/10.1140/epjb/e2010-10305-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2010-10305-8

Keywords

Search

Navigation