Minimal-entropy martingale measure

{{Short description|Method used to minimise uncertainty between probabilities}}

{{Multiple issues|

{{context|date=May 2025}}

{{technical|date=May 2025}}

}}

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 $D\_\{KL\}(Q\backslash |P)\; =\; \backslash sum\_\{i=1\}^N\; q\_i\; \backslash ln\backslash left(\backslash frac\{q\_i\}\{p\_i\}\backslash right)$ subject to the requirement that the expected return is $r$, where $r$ is the risk-free rate.