The Performance of Johnson Distributions for Computing
Value at Risk and Expected Shortfall

Abstract

Option pricing plainly depends on the probability distribution of the underlying asset return, at least the portion of the distribution for which the option is in the money. Risk management also depends on the return distribution, but standard risk measures like value at risk (VaR) and expected shortfall (ES) concentrate only on its tails. While the lognormal may be (arguably) adequate for modeling option values, empirically, “risk” is inherently tied up with fat-tailed return processes. This result has led to interest in methods to approximate an unknown distribution by matching its first four moments.

The Cornish–Fisher and Gram–Charlier expansions are frequent choices. Simonato argues, however, that these techniques do not actually work very well because the range of densities for which the approximations are valid is quite limited. For example, the density approximation may have negative portions, even when skewness and kurtosis seem quite reasonable. He proposes adopting the Johnson family of densities instead, which also uses four parameters to match the first four moments of an empirical distribution. A simulation study shows that with Johnson distributions, tail fitting is accurate and available over the full range of parameter values.