Calibration of the risk-neutral density function by maximization of a two-parameter entropy

In the present work, a two-parameter entropy is maximized to calibrate the risk-neutral probability density function of the future asset price using options data subject to the expectation and the variance constraint. In the variance constraint, the volatility is assumed to be mean-reverting and following a quadratic path. The desired power law distribution is verified for the density function obtained, and it contains both the entropy parameters giving an additional degree of freedom. The calibrated density function is used to price the European call options for different strikes. The results thus obtained are discussed for the one-parameter Renyi and Tsallis entropies.