Entropic risk measure

Mathematical definition

The entropic risk measure with the risk aversion parameter \theta 0 is defined as : \rho^{\mathrm{ent}}(X) = \frac{1}{\theta}\log\left(\mathbb{E}[e^{-\theta X}]\right) = \sup_{Q \in \mathcal{M}_1} \left{E^Q[-X] -\frac{1}{\theta}H(Q|P)\right} , where H(Q|P) = E\left[\frac{dQ}{dP}\log\frac{dQ}{dP}\right] is the relative entropy of Q

Acceptance set

The [acceptance set for the entropic risk measure is the set of payoffs with positive expected utility. That is : A = {X \in L^p(\mathcal{F}): E[u(X)] \geq 0} = {X \in L^p(\mathcal{F}): E\left[e^{-\theta X}\right] \leq 1} where u(X) is the exponential utility function.

Dynamic entropic risk measure

The conditional risk measure associated with dynamic entropic risk with risk aversion parameter \theta is given by :\rho^{\mathrm{ent}}_t(X) = \frac{1}{\theta}\log\left(\mathbb{E}[e^{-\theta X} | \mathcal{F}_t]\right). This is a time consistent risk measure if \theta is constant through time, and can be computed efficiently using forward-backwards differential equations{{cite journal .

References

References

1. (July 21, 2008). "Festschrift in celebration of Prof. Dr. Wilfried Greckschs 60th birthday". Shaker Verlag.
2. (2004). "Stochastic finance: an introduction in discrete time". Walter de Gruyter.
3. (October 8, 2008). "Convex and Coherent Risk Measures".
4. Penner, Irina. (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability". Humboldt University of Berlin.
5. (2019). "An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior". Finance and Stochastics.