Modeling of rare events using non-normal distributions : an application in finance with the sinh-asinh distribution

Abstract

In 2008, the financial crisis put forward the relative inaccuracy of the market risk forecasting models in the financial industry. In particular, extreme events were shown to be regularly underestimated. This problematic, initially developed in the seminal work of Mandelbrot (1963), is mainly due to financial models using the normal law while empirical evidence show strong leptokurticity in financial time series. This stylized effect is particularly damaging the forecasting of indicators like Value-at-Risk (VAR). In this study, we try to tackle problem by testing a newly-developed probability distribution, never used in finance: sinh-arcsinh function. By creating different datasets from non-parametric and GARCH models, we adjust common functions (normal, t location-scale, GED, gen. hyperbolic) and sinh-arcsinh function on the data. We show that, regarding the leptokurtic datasets extracted from the DJA and the NIKKEI 225, the sinh-arcsinh function performs a better adjustment than any other function tested. We also tested simple VAR models using normal laws, Student’s t or sinh-arcsinh functions, to assess the operational efficiency of the sinh-arcsinh function. We show that models using sinh-arcsinh functions provide more accurate and better in-sample and out-of-sample VAR forecasts than any other model using the normal laws