Minimal-entropy martingale measure

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In probability theory, the minimal-entropy martingale measure (MEMM) is the risk-neutral probability measure that minimises the entropy difference between the objective probability measure, P, and the risk-neutral measure, Q. In incomplete markets, this is one way of choosing a risk-neutral measure (from the infinite number available) so as to still maintain the no-arbitrage conditions.

The MEMM has the advantage that the measure Q will always be equivalent to the measure P by construction. Another common choice of equivalent martingale measure is the minimal martingale measure, which minimises the variance of the equivalent martingale. For certain situations, the resultant measure Q will not be equivalent to P.

In a finite probability model, for objective probabilities p_i and risk-neutral probabilities q_i then one must minimise the Kullback–Leibler divergence D_{KL}(Q\|P) = \sum_{i=1}^N q_i \ln\left(\frac{q_i}{p_i}\right) subject to the requirement that the expected return is r, where r is the risk-free rate.


  • M. Frittelli, Minimal Entropy Criterion for Pricing in One Period Incomplete Markets, Working Paper. University of Brescia, Italy (1995).