Minimal-entropy martingale measure

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.