Directional entropy and tail uncertainty, with applications to financial hazard

Abstract

“Mine is a long and sad tale”, said the Mouse, turning to Alice and sighing. “It is a long tail certainly,” said Alice, looking down with wonder at the Mouse's tail; “but why do you call it sad?” And she kept on puzzling about it while the mouse was speaking … Financial risk management metrics such as value at risk (VaR) can be illuminated by means of a regime-specific concept of directional entropy. This enables a change of measure via a rescaling function to an equivalent logistic distribution, one that has the same total and directional entropies at the chosen critical point. VaR rescaling adjusts the critical probability to capture the long tail entropy. The scaling function can be used as a comparative metric for tail length, or equivalent conditional value at risk, even where moments do not exist. Directional entropy can also be used to identify regions of maximal exposure to new information, which can actually increase entropy rather than collapse it.

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