Relative Entropy Criterion and CAPM-Like Pricing

Abstract

The minimal relative entropy criterion for the selection of an equivalent martingale measure in an incomplete market seems to still hold some mystique in its financial interpretation. In this paper we work toward this interpretation by suggesting and exploring the idea of relating a martingale measure selection criterion to a CAPM-like pricing scheme. We examine this idea in the case of the minimal relative entropy criterion and we present some preliminary results. We work within a one-period financial market and show that the minimal relative entropy pricing criterion is equivalent to some CAPM-like pricing scheme where the classical beta coefficient formula has been replaced by some “entropic beta” and the market portfolio by some “appropriate” reference portfolio. Furthermore, we show that if the assets involved have returns that are jointly normal, then this “entropic beta” formula coincides with the classical beta coefficient. Additionally and for comparison reasons, we briefly illustrate that if our criterion for the choice of the martingale measure was the minimization of the variance of the Radon–Nikodym derivative, then the resulting martingale pricing and the pricing implied by the classical CAPM scheme would be the same.