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, ${\displaystyle P}$, and the risk-neutral measure, ${\displaystyle Q}$. 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 ${\displaystyle Q}$ will always be equivalent to the measure ${\displaystyle P}$ 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 ${\displaystyle Q}$ will not be equivalent to ${\displaystyle P}$.

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