ABSTRACT
Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 Starting kit for extreme value analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2.1 Sample maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.2.2 Generalized extreme value distribution (GEV) . . . . . . . . . . . . . . . . . . . 109 3.2.3 The POT-method (peaks over threshold) . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.4 Estimating tails and quantiles by the POT-method . . . . . . . . . . . . . . . . 112
3.3 Continuous-time diffusion models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.4 The AR(1) model with ARCH(1) errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.4.1 Stationarity and tail behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.4.2 Extreme value analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.5 Optimal portfolios with bounded Value-at-Risk (VaR) . . . . . . . . . . . . . . . . . . 141 3.5.1 The Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.5.2 The exponential Le´vy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.5.3 Portfolio optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
In this chapter we review certain aspects around the Value-at-Risk, which is nowadays the industry benchmark risk measure. As a small quantile (usually 1%) the Value-atRisk is closely related to extreme value theory, and we explain an estimation method based on this theory. Since the variance of the estimated Value-at-Risk may depend on the dependence structure of the data, we investigate the extreme behaviour of some of the most prominent time series models in finance, continuous as well as discrete time models. We also determine optimal portfolios, when risk is measured by the Valueat-Risk. Again we use realistic financial models, moving away from the traditional Black-Scholes model to the class of Le´vy processes.
