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
K
L
(
Q
∥
P
)
=
∑
i
=
1
N
q
i
ln
(
q
i
p
i
)
subject to the requirement that the expected return is
r
, where
r
is the risk-free rate.
