Value-at-risk for Long and Short Trading Positions

In this paper we model Value-at-Risk (VaR) for daily stock index returns using a collection of parametric models of the ARCH family based on the skewed Student distribution. We show that models that rely on a symmetric density distribution for the error term underperform with respect to skewed density models when the left and right tails of the distribution of returns must be modelled. Thus, VaR for traders having both long and short positions is not adequately modelled using usual Normal or Student distributions. We suggest using an APARCH model based on the skewed Student distribution to fully take into account the fat left and right tails of the returns distribution. This allows for an adequate modelling of large returns defined on long and short trading positions. The performances of all models are assessed on daily data for the CAC40, DAX, NASDAQ, NIKKEI and SMI stock indexes. We also compute the expected short-fall and the average multiple of tail event to risk measure for the new model.