Option pricing for log-symmetric distributions of returns

Fima C. Klebaner, Zinoviy Landsman

Research output: Contribution to journalArticlepeer-review


We derive an option pricing formula on assets with returns distributed according to a log-symmetric distribution. Our approach is consistent with the no-arbitrage option pricing theory: we propose the natural risk-neutral measure that keeps the distribution of returns in the same log-symmetric family reflecting thus the specificity of the stock's returns. Our approach also provides insights into the Black-Scholes formula and shows that the symmetry is the key property: if distribution of returns X is log-symmetric then 1/X is also log-symmetric from the same family. The proposed options pricing formula can be seen as a generalization of the Black-Scholes formula valid for lognormal returns. We treat an important case of log returns being a mixture of symmetric distributions with the particular case of mixtures of normals and show that options on such assets are underpriced by the Black-Scholes formula. For the log-mixture of normal distributions comparisons with the classical formula are given.

Original languageEnglish
Pages (from-to)339-357
Number of pages19
JournalMethodology and Computing in Applied Probability
Issue number3 SPEC. ISS.
StatePublished - Sep 2009

Bibliographical note

Funding Information:
Acknowledgements The authors wish to thank the Australian Research Council, EPSRC , Israel Caesarea Rothschild Institute and Zimmerman Foundation for the financial support.


  • Log-symmetric distribution
  • Martingale measure
  • Mixture of normal distributions
  • Option price
  • Returns

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics


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