Option price calibration from Rényi entropy

The calibration of the risk-neutral density function for the future asset price, based on the maximisation of the entropy measure of Rényi, is proposed. Whilst the conventional approach based on the use of logarithmic entropy measure fails to produce the observed power-law distribution when calibrated against option prices, the approach outlined here is shown to produce the desired form of the distribution. Procedures for the maximisation of the Rényi entropy under constraints are outlined in detail, and a number of interesting properties of the resulting power-law distributions are also derived. The result is applied to efficiently evaluate prices of path-independent derivatives.