Option Pricing for Log-Symmetric Distributions of Returns

Fima C. Klebaner () and Zinoviy Landsman ()
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Fima C. Klebaner: Monash University
Zinoviy Landsman: University of Haifa

Methodology and Computing in Applied Probability, 2009, vol. 11, issue 3, 339-357

Abstract: Abstract 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.

Keywords: Martingale measure; Option price; Returns; Log-symmetric distribution; Mixture of normal distributions; 60G35; 60G42 (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-007-9038-2

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