Entropic risk measure

The entropic risk measure with the risk aversion parameter ${\displaystyle \theta >0}$ is defined as

${\displaystyle \rho ^{\mathrm {ent} }(X)={\frac {1}{\theta }}\log \left(\mathbb {E} [e^{-\theta X}]\right)=\sup _{Q\in {\mathcal {M}}_{1}}\left\{E^{Q}[-X]-{\frac {1}{\theta }}H(Q|P)\right\}\,}$[2]

where ${\displaystyle H(Q|P)=E\left[{\frac {dQ}{dP}}\log {\frac {dQ}{dP}}\right]}$ is the relative entropy of Q << P.[3]

The acceptance set for the entropic risk measure is the set of payoffs with positive expected utility. That is

${\displaystyle A=\{X\in L^{p}({\mathcal {F}}):E[u(X)]\geq 0\}=\{X\in L^{p}({\mathcal {F}}):E\left[e^{-\theta X}\right]\leq 1\}}$

where ${\displaystyle u(X)}$ is the exponential utility function.[3]

The conditional risk measure associated with dynamic entropic risk with risk aversion parameter ${\displaystyle \theta }$ is given by

${\displaystyle \rho _{t}^{\mathrm {ent} }(X)={\frac {1}{\theta }}\log \left(\mathbb {E} [e^{-\theta X}|{\mathcal {F}}_{t}]\right).}$

This is a time consistent risk measure if ${\displaystyle \theta }$ is constant through time.[4]

• Entropic value at risk