Fractal derivative: Difference between revisions
en>Nsda →References: no self-promotion, please! two papers by W. Chen should be largely sufficient |
en>LokiClock →Physical background: Fractal geometry doesn't generalize Euclidean geometry to non-integer dimensions. |
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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, <math>P</math>, and the risk-neutral measure, <math>Q</math>. In [[incomplete market]]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 <math>Q</math> will always be equivalent to the measure <math>P</math> by construction. Another common choice of equivalent [[martingale measure]] is the minimal martingale measure, which minimises the variance of the equivalent [[martingale (probability theory)|martingale]]. For certain situations, the resultant measure <math>Q</math> will not be equivalent to <math>P</math>. | |||
In a finite probability model, for objective probabilities <math>p_i</math> and risk-neutral probabilities <math>q_i</math> then one must minimise the [[Kullback–Leibler divergence]] <math>D_{KL}(Q\|P) = \sum_{i=1}^N q_i \ln\left(\frac{q_i}{p_i}\right)</math> subject to the requirement that the expected return is <math>r</math>, where <math>r</math> 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). | |||
[[Category:Stochastic processes]] | |||
[[Category:Martingale theory]] | |||
[[Category:Game theory]] | |||
Latest revision as of 04:23, 2 October 2013
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, , and the risk-neutral measure, . 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 will always be equivalent to the measure 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 will not be equivalent to .
In a finite probability model, for objective probabilities and risk-neutral probabilities then one must minimise the Kullback–Leibler divergence subject to the requirement that the expected return is , where 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).