[excerpt] Dynamic Risk Modeling Handbook, Chapter 5:

자료: http://www.casact.org/research/drm/handbook/chapter5_071905.doc

Text source: Dynamic Risk Modeling Handbook, Chapter 5:

5. ASSET RISK MEASUREMENT

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5.2.5 Extreme Value Theory

Traditionally, returns are modeled as Gaussian or normally distributed, which gives the probability density function (PDF) of the returns a familiar bell shape. However, financial market data strongly contradicts this assumption of normality.

• Most markets experience one or more daily moves of at least four standard deviations each year. Under normality, one would expect to see only a single four standard deviation move every 125 years.
• Clearly, using a normal distribution for risk management would grossly underestimate the true risk exposure (for this reason, VaR measures are often multiplied by a correction factor).

The most common alternative to normality is to model the distribution of the return shocks as a Student's t distribution. The PDF of a t-distribution has a similar shape to the normal distribution, but with fatter tails controlled by an integer parameter called the "degrees of freedom" (df) of the distribution; t-distributions with four to six degrees of freedom have tails fat enough to match the empirical frequency of extreme moves.

A more advanced approach is to model directly the distribution of extreme moves using Extreme Value Theory (EVT). In EVT, the distribution of extreme moves beyond a certain threshold (e.g., four standard deviations) is given by a generalized Pareto distribution. This distribution is controlled by two parameters, which allow the risk manager to control both the fatness and shape of the tail, and thereby match even more closely the empirical distribution of extreme moves.

The theory of univariate EVT was developed in the early 1920s through 1940s for the statistical modeling of radioactive emissions, astronomical phenomena, flood and rainfall analysis, and seismic events. It has gradually made its way from uses in the natural sciences to actuarial sciences and finance. EVT is by now a standard statistical tool for risk models in the insurance industry. Current developments of EVT in finance deal primarily with multivariate extensions to measure and model correlations of extreme. The hope ultimately is to use these new statistical techniques to improve risk management during financial crises, such as the ERM crisis in 1992 and the Russian crisis in 1998, when financial market correlations tend to increase.

Modeling the distribution of extremes, however, is sometimes as much an art as it is a science. The problem is that, by the definition of extremes being rare events, we tend to have too few extreme observations to draw precise inferences about their distribution. Successfully modeling extremes therefore requires extensive experience with the various modeling techniques. (생략)