Extreme Value Theory using Parametric Approaches

Management of Extreme Values

To meet the challenge, researchers must assume a certain distribution. The assumed distribution will probably not be identical to the true distribution; therefore, some degree of error will be present. Researchers usually choose distributions based on measures of central tendency, which misses the issue of trying to incorporate extreme values. Researchers need approaches that specifically deal with extreme value estimation. Incidentally, researchers in many fields other than finance face similar problems. In flood control, for example, analysts have to model the highest possible flood line when building a dam, and this estimation would most likely require a height above observed levels of flooding to date.

Extreme Value Theory

One approach for estimating parameters is the Fisher-Tippett theorem. According to this theorem, as the sample size n gets large, the distribution of extremes, denoted Mn, converges to the following distribution known as the generalized extreme value (GEV) distribution:

For these formulas, the following restriction holds for random variable X:

The parameters μ and σ are the location parameter and scale parameter, respectively, of the limiting distribution. Although related to the mean and variance, they are not the same. The symbol ξ is the tail index and indicates the shape (or heaviness) of the tail of the limiting distribution. There are three general cases of the GEV distribution:

ξ > 0, the GEV becomes a Frechet distribution, and the tails are “heavy” as is the case for the t-distribution and Pareto distributions.

ξ = 0, the GEV becomes the Gumbel distribution, and the tails are “light” as is the case for the normal and lognormal distributions.

ξ < 0, the GEV becomes the Weibull distribution, and the tails are “lighter” than a normal distribution.

Distributions where ξ < 0 do not often appear in financial models; therefore, financial risk management analysis can essentially focus on the first two cases: ξ > 0 and ξ = 0. Therefore, one practical consideration the researcher faces is whether to assume either ξ > 0 or ξ = 0 and apply the respective Frechet or Gumbel distributions and their corresponding estimation procedures. There are three basic ways of making this choice.

The researcher is confident of the parent distribution. If the researcher is confident it is a t-distribution, for example, then the researcher should assume ξ > 0.

The researcher applies a statistical test and cannot reject the hypothesis ξ = 0. In this case, the researcher uses the assumption ξ = 0.

The researcher may wish to be conservative and assume ξ > 0 to avoid model risk.

DRAFT — RiskServ framework overview chapter v1.2a ( Bruce Haydon ACM Chapters (Buffalo/New York(NYC)/Toronto)/ HSB C CA 22 /

Copyright © 2020–2022 Bruce Haydon

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