Minimal-entropy martingale measure explained

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 ^{N} _{i =1} q _{i} ln \left( (q _{i})/(p _{i}) \right) subject to the requirement that the expected return is r , where r is the risk-free rate.

References

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