Extreme Value Theory

Analogous with the central limit theorem, where the normal distribution acts the limit for the distribution of the mean of a large number i.i.d. random variables, the extreme value theory (EVT) investigates the limit distribution of the sample maximum.

Empirical models of financial returns based on distributional assumptions such as Gaussian, Student’s t and GED are often chosen based on their ability to t data near the mode given that only a few observations fall in the distribution tails by definition. But effective risk management requires accurate estimation of the likelihood of rare events that could trigger catastrophic losses. Extreme value theory can be useful for this purpose because it is specifically aimed at modelling tail behaviour without requiring assumptions on the entire distribution, i.e. it provides a semi-parametric model for the tails of distribution functions.

Pros: much more accurate for applications focusing on the extremes
Cons: don’t have that many extreme observations

EVT can be useful to explicitly identify the type of asymmetry in the extreme tails.

Regardless of the overall shape of the distribution, the tails of all distributions fall into one of three categories as long as the distribution of an asset return series does not change over time:

  • Weibull: Thin tails where the distribution has a finite endpoint
  • Gumbel: Tails decline exponentially
  • Frechet: Tails decline by a power law

Block maxima and peaks-over-threshold are the two main EVT modeling methodologies.

Generalized extreme value distribution

Let \{x_{t}\},\: t=1,..,T, denote an iid process with distribution F\left(x\right). The maximum of a block of n<t observations,=”" called block maximum and denoted M_{n}=\max\left(x_{1},\ldots,x_{n}\right), follows asymptotically the probability distribution

    \begin{equation*} \textrm{P}\left[\frac{M_{n}-b_{n}}{a_{n}}\leqslant y\right]=F^{n}\left(a_{n}y+b_{n}\right)\rightarrow G\left(y\right),\qquad n\rightarrow+\infty \end{equation*}

as n\rightarrow+\infty for all y\in\mathbb{R}, where a_{n}>0 and b_{n} are appropriate constants, F^{n}\left(\cdot\right) is F\left(\cdot\right) raised to power of n, and G\left(\cdot\right) is a non-degenerate distribution function. According to the Extremal Types Theorem, the block maxima distribution G\left(\cdot\right) must be either Frechet, negative Weibull or Gumbel; these three distributions can be cast as members of the Generalized Extreme Value distribution (GEV) with cdf given by

    \begin{equation*} G\left(y\right)=\begin{cases} \exp\left\{ -\left(1+\xi\frac{y-\mu}{\beta}\right)^{-1/\xi}\right\} & \quad\xi\neq0\\ \exp\left\{ -e^{-\frac{y-\mu}{\beta}}\right\} & \quad\xi=0 \end{cases}, \end{equation*}

where \mu,\:\beta>0 and \xi are location, scale and shape parameters, respectively.

GED becomes the Frechet distribution for \xi>0, the negative Weibull distribution for \xi<0, and the Gumbel distribution for \xi=0.

Generalized Pareto distribution

Let \{x_{t}-u\}\: t=1,..,T, denote the exceedances or peaks-over-threshold process where x_{t}>u and u denotes a threshold loss. The exceedances distribution can be formalized as

    \[ \Pr\left[x_{t}-u\leqslant y\mid x_{t}>u\right]=\frac{F\left(y+u\right)-F\left(u\right)}{1-F\left(u\right)}\rightarrow H\left(y\right),\quad t=1,\ldots,T. \]

According to the Pickands-Balkema-de-Haan Theorem, for a sufficiently large threshold loss u, the exceedances distribution can be approximated by the Generalized Pareto Distribution (GPD) as

    \begin{equation*} H\left(y\right)=\begin{cases} 1-\left(1+\xi\frac{y}{\beta}\right)^{-1/\xi} & \quad\xi\neq0\\ 1-\exp\left\{ -\frac{y}{\beta}\right\} & \quad\xi=0 \end{cases}, \end{equation*}

where \beta>0 and \xi are scale and shape parameters, respectively. GPD nests the exponential distribution (\xi=0), the heavy-tailed Pareto Type I distribution (\xi>0) and the short-tailed Pareto Type II distribution (\xi<0).

The parameters of GPD are estimated by maximizing the corresponding log-likelihood function

    \begin{eqnarray*} \ln\mathfrak{L}(y_{1},\ldots,y_{N_{u}};\beta,\xi) & = & \sum_{j=1}^{N_{u}}\ln h\left(y_{j};\beta,\xi\right)\\  & = & -N_{u}\ln\beta-\left(1+\frac{1}{\xi}\right)\sum_{j=1}^{N_{u}}\ln\left(1+\xi\frac{y_{j}}{\beta}\right) \end{eqnarray*}

where N_{u} is the total number of observed exceedances y_{j}\equiv x_{j}-u for given threshold u.

Hill Method

Alternatively, one can use Hill method to estimate the tail distribution.

Finding the threshold

Several methods have been proposed to determine the optimal threshold.

  1. The most common approach is the eyeball method where we look for a region where the tail index seems to be stable.
  2. More formal methods are based on minimizing the mean squared error (MSE) of the Hill estimator

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