Directional entropy and tail uncertainty, with applications to financial hazard and investments

“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… Contexts such as value at risk or venture capital require local uncertainty measures, as distinct from properties of the entire distribution such as differential entropy. Applications such as value at risk and options pricing can be illuminated by means of a regime-specific concept of directional entropy. The latter enables a change of measure to an equivalent logistic distribution, one that has the same total and directional entropies at the given marker, e.g. value at risk critical point or option strike price. This is done via a scaling function that can be interpreted as a Radon-Nikodym derivative and used in its own right as a risk metric. Value at risk rescaling adjusts the critical probability to capture the long tail risk. Directional entropy can be used to identify regions of maximal exposure to new information, which can actually increase entropy rather than collapse it.