Ch5 Parametric Approach: Extreme Value (2)

2. POT (Peaks-Over-Threshold) Approach
- distribution of excess losses over a (high) threshold
- GPBdH(Gnedenko-Pickands-Balkema-deHaan) theorem: as u gets large, the distribution
Fu(x) converges to a GPD(generalized Pareto distribution)

where:
beta = positive scale parameter
ksi = shape or tail index parameter

- distribution function F(x): u = threshold value of X & x > 0 -> probability that a loss exceeds the threshold u by at most x
**tradeoff of choosing the threshold: it needs to be high enough so the GPBdH theory can apply, but it must be low enough so that there will be enough observations to apply estimation techniques to the parameters
- the distribution is defined for the following regions:
for x >= 0 for ksi >= 0 and 0 <= x <= beta/ksi for ksi <>ksi & beta can be estimated using maximum likelihood approaches or semi-parametric approaches

2-1. VaR & Expected Shortfall
- all distributions of excess losses converge to the GPD (natural model for excess losses)

where:
u = threshold (in %)
n = number of observations
Nu = number of observations that exceed threshold

*corresponding ES (in %):

- ES (expected shortfall, a.k.a. conditional VaR) viewed as an average or expected value of all losses greater than the VaR: E[LPLP > VaR]
http://www.youtube.com/watch?v=eHGJFOjyzr4

2-2. GEV vs. POT
- one might be more natural in a given context than the other
- GEV: additional parameters & block maxima approach can involve some loss of useful data relative to the POT
- POT: problem of choosing the threshold

3. Multivariate EVT
We can easily see how extreme values can be dependent on each other with MEVT.
- similar relationship between the occurrence of a natural disaster and a decline in financial markets as well as markets for real goods and services
Multivariate EVT has the same goal as univariate EVT in that the objective is to move from the familiar central-value distributions to methdos that estimate extreme events. The key issue is how to model the dependence structure of extreme events. Knowledge of variances and correlations suffices to specify the multivariate distribution. However, given non-elliptical distribution, correlation no longer suffices to describe the dependence structure.
MEVT tells us that the limiting distirbution of multivariate extreme values will be a member of the family of EV copulas, and we can model multivariate EV dependence by assuming one of these EV copulas. The copulas can also have as many dimensions as appropriate and congruous with the number of random variables under consideration. However, there is a curse of dimensionality.
The occurrence of extreme events is governed by the tail dependence of the multivariate distribution. The tail dependence is the central focus of MEVT.