Stable Models in Risk Management

It is a well known fact that the Gaussian assumption on market data is not supported by empirical evidence. Particularly, the presence of skewness and a large kurtosis can dramatically affect the risk management analysis, specially, the Value at Risk (VaR) calculation through quantile estimators. The stable distribution has nevertheless two major drawbacks: the density probability function has no explicit form except in the cases of the Cauchy and the Normal laws. Numerical methods are needed to compute it. Also, second and higher moments do not exist; which constitutes a challenge to most statistical methods. In the next section the family of stable laws and its properties are introduced. The next section reviews some calibration and simulation methods for stable distributions. Next, a maximum likelihood approach (m.l.e.) is considered under the framework of ARMA processes driven by stable noises. Asymptotic properties are studied and numerical methods are discussed. Finally, we present some simulation results for stable GARCH processes. The Value at Risk (VaR) for these stable models is calculated and compared with its Gaussian counterpart, revealing important differences between them. The procedure is also illustrated in real financial data.