calculation worst-case Value-at-Risk prediction using empirical data under model uncertainty

Quantification of risk positions under model uncertainty is of crucial importance from both viewpoints of external regulation and internal management. The concept of model uncertainty, sometimes also referred to as model ambiguity. Although we know the family of models, we cannot precisely decide which one to use. Given the set $\mathcal{P}$, the value of the risk measure $\rho$ varies in a range over the set of all possible models. The largest value in such a range is referred to as a worst-case value, and the corresponding model is called a worst scenario. Value-at-Risk(VaR) has become a very popular risk-measurement tool since it was first proposed. Naturally, WVaR(worst-case Value-at-Risk) attracts the attention of many researchers. Although many literatures investigated WVaR, the implications for empirical data analysis remain rare. In this paper, we proposed a special model uncertainty market model to simply the $\mathcal{P}$ to a set contain finite number of probability distributions. The model has the structure of the two-layer mixed distribution model. We used change point detection method to divide the returns series and then used EM algorithm to estimate the parameters. Finally, we calculated VaR, WVaR(worst-case Value-at-Risk) and BVaR(best-case Value-at-Risk) for four financial markets and then analyzed their different performance.