7.7 New Approach Based on the Extreme Value Theory

The aforementioned approach to VaR calculation using the extreme value theory encounters some difficulties. First, the choice of subperiod length n is not clearly defined. Second, the approach is unconditional and, hence, does not take into consideration effects of other explanatory variables. To overcome these difficulties, a modern approach to extreme value theory has been proposed in the statistical literature; see Davison and Smith (1990) and Smith (1989). Instead of focusing on the extremes (maximum or minimum), the new approach focuses on exceedances of the measurement over some high threshold and the times at which the exceedances occur. Thus, this new approach is also referred to as peaks over thresholds (POT). For illustration, consider the daily returns of IBM stock used in this chapter and a long position on the stock. Denote the negative daily log return by rt. Let η be a prespecified high threshold. We may choose η = 2.5%. Suppose that the ith exceedance occurs at day ti (i.e., Inline). Then the new approach focuses on the data Inline. Here Inline is the exceedance over the threshold η and ti is the time at which the ith exceedance occurs. ...

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