probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the minimal-entropy martingale measure (MEMM) is the risk-neutral probability measure that minimises the

entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

difference between the objective probability measure,

P

, and the risk-neutral measure,

Q

. In

incomplete market In economics, incomplete markets are markets in which there does not exist an Arrow–Debreu security for every possible state of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transfer ...

s, 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_(Q\, P) = \sum_^N q_i \ln\left(\frac{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). Martingale theory Game theory