Problem bank

Problem 299 of 333MediumStochastic ProcessesP299

Two volatility regimes

  1. A desk models each day's return as normal with mean 0. On a calm day the volatility is 1%, and on a stressed day it is 3%. Each day is stressed with probability 0.2, independently of everything else.

    What is the unconditional daily volatility of the return? What is its kurtosis, E[R4]/Var⁡(R)2E[R^4]/\operatorname{Var}(R)^2? (A normal distribution has kurtosis 3.)