Entropic value at risk

In financial mathematics, the entropic value at risk (EVaR) is "the tightest possible upper bound obtained from the Chernoff inequality for the VaR". The EVaR is a coherent risk measure which depends on a parameter $\alpha \in (0,1]$. It is defined for every $X\in L^{\infty }$ and any $\alpha \in (0,1]$ by

$EVaR_{\alpha }(-X)=\inf _{z>0}\left[z^{-1}\log \left({\frac {M_{X}(z)}{\alpha }}\right)\right]$

where $M_{X}(z)=\mathbb {E} [e^{zX}]$ is the moment generating function for the random variable $X$.

The entropic value at risk is termed entropic since its dual representation (via convex conjugates) is given by

$EVaR_{\alpha }(X)=\sup _{\mathbb {Q} \in {\mathcal {Q}}}\mathbb {E} ^{\mathbb {Q} }[-X]$

where ${\mathcal {Q}}=\left\{\mathbb {Q} \ll \mathbb {P} :H(\mathbb {Q} |\mathbb {P} )\leq -\log(1-\alpha )\right\}$ with $H(\mathbb {Q} |\mathbb {P} )=\mathbb {E} ^{\mathbb {Q} }[\log({\frac {d\mathbb {Q} }{d\mathbb {P} }})]$ is the relative entropy of $\mathbb {Q}$ with respect to $\mathbb {P}$.

See also

• Risk measure
• Coherent risk measure
• Value at risk
• Average value at risk
• Entropic risk measure