Gauss Versus Cauchy: A Comparative Study on Risk

In risk management, the probability distribution must adequately capture the risks that matter most, namely the outliers. I estimate the parameters as if the sample were upper censored to put the focus to the most negative observations (non-censored outliers) and then only estimate the probability that I obtain an “ordinary” observation (censored). The left tail is assumed to follow a location-scale Student’s t-distribution, yet no distributional assumptions are made outside the tail. Cauchy’s and Gauss’ distributions are two special cases of Student’s t-distribution. For empirical stock index returns, the left tail is neither Gauss nor Cauchy, a Student’s t-distribution with around four degrees of freedom offers the best fit. As an empirical application, I compare the pricing of a short-term far out-of-the-money put option on the Dow Jones with a Gauss, a Cauchy, and a Student left tail. The Gaussian Black–Scholes model underprices such insurance contracts, Cauchy overprices them, and the Student model, however, is a good match.