Extreme Values and Their Applications in Finance

Extreme price movements in the financial markets are rare, but important. The stock market crash on Wall Street in October 1987 and other big financial crises such as the Long Term Capital Management and the bankruptcy of Lehman Brothers have attracted a great deal of attention among investors, practitioners and researchers. The recent worldwide financial crisis characterized by the substantial increase in market volatility, e.g., the VIX index, and the big drops in market indices has further generated discussions on market risk and margin setting for financial institutions. As a result, value at risk (VaR) has become the standard measure of market risk in risk management. Its usefulness and weaknesses are widely discussed. In this lecture, we consider the extreme value theory developed in the statistical literature for studying rare (or extraordinary) events and its application to VaR. Both unconditional and conditional concepts of extreme values are discussed. The unconditional approach to VaR calculation for a financial position uses the historical returns of the instruments involved to compute VaR. On the other hand, a conditional approach uses the historical data and explanatory variables to calculate VaR. The explanatory variables may include macroeconomic variables of an economy and accounting variables of companies involved. VaR is a point estimate of potential financial loss. It contains a certain degree of uncertainty. It also has a tendency to underestimate the actual loss if an extreme event actually occurs. To overcome the weaknesses of VaR, we discuss other risk measures such as expected shortfalls and the loss distribution of a financial position in the lecture. We use daily log returns of IBM stock to illustrate the applications of the methods discussed. Figure 1 shows the time plot of daily log returns of IBM stock from July 3, 1962 to December 31, 1998 for 9190 observations.