Generalized Extreme Value Distribution and Extreme Economic Value at Risk (EE-VaR)

Abstract

In 2000, Ait-Sahalia and Lo have argued that Economic VaR (E-VaR) calculated under option market implied risk neutral density (RND) is a more relevant measure of risk than historically based VaR. As industry practice requires VaR at high confidence level of 99%, Extreme Economic Value at Risk (EE-VaR) based on the Generalized Extreme Value (GEV) distribution has been proposed as a new risk measure. This follows from a GEV option pricing model developed by Markose and Alentorn in 2005 which shows that the GEV implied RND can accurately capture negative skewness and fat tails, with the latter explicitly determined by the market implied tail index. Here, the term structure of the GEV based RNDs is estimated which permits the calibration of an empirical scaling law for EE-VaR, and thus, obtain daily EE-VaR for any time horizon. Backtesting results for the FTSE 100 index from 1997 to 2003, show that EE-VaR has fewer violations than historical VaR. Further, there are substantial savings in risk capital with EE-VaR at 99% as compared to historical VaR corrected by a factor of 3 to satisfy the violation bound. The efficiency of EE-VaR arises because an implied VaR estimate responds quickly to market events and in some cases even anticipates them. In contrast, historical VaR reflects extreme losses in the past for longer.