Option pricing for log-symmetric distributions of returns

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.