Tail Risk Killers: How Math, Indeterminacy, and Hubris Distort Markets

CHAPTER 37A Characterization of Dark Risk

The difficulty with modeling markets in a dynamical way is that their essence is free human choice, whereas the central core of dynamical systems is determinism. Determinism means that the past determines the future and, as will be demonstrated, vice versa. This definition implies that zero topologic entropy is a requirement for predictability. This will take some unpacking.

THEOREM (KAMINSKI)1

A system (X, T) is topologically predictable if for every continuous function f in C(X) we have f in <1, T, f, T², f, …> where <F> ⊂ C (X) denotes the closed algebra generated by a family F ⊂ C (X). (X, T) is topologically predictable if and only if every factor of (X, T) is invertible, where a factor is a system ...