Computational Modeling of Non-Gaussian Option Price Using Non-extensive Tsallis' Entropy Framework

Financial markets have always been subject to various risk constraints which are necessary for better market prediction and accurate pricing. In this context, we derive stock price distribution subject to first and second moment constraints along with the normalization constraint in terms of theq-lognormal distribution. The derived distribution is validated against six high-frequency empirical datasets. To characterize the extreme fluctuation of empirical stock returns, we derive an analytical expression for complementary cumulative distribution function of theq-Gaussian distribution in terms of Hypergeometric2F1 function. However, for the computation of the non-extensive parameter 'q', we provide a precise algorithm. The estimated value of 'q' clearly describes the well-known stylized facts such as tail fluctuation, non-Gaussian intra-day returns, and cubic power-law behavior. As the option price depends on the underlying dynamics of the stock price, we derive a accurate and closed expression for option price usingq-lognormal distribution.