Minimal Entropy Martingale Measure
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In
probability theory Probability theory or probability calculus 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 expre ...
, the minimal-entropy martingale measure (MEMM) is the risk-neutral probability measure that minimises the
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
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 transferr ...
s, this is one way of choosing a
risk-neutral measure In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price un ...
(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 In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price und ...
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 In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how much a model probability distribution is diff ...
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