6.1 Introduction

Implied volatilities are the focus of interest both in volatility trading and in risk management. As common practice traders directly trade the so called "vega", i.e. the sensitivity of their portfolios with respect to volatility changes. In order to establish vega trades market professionals use delta-gamma neutral hedging strategies which are insensitive to changes in the underlying and to time decay, Taleb (1997). To accomplish this, traders depend on reliable estimates of implied volatilities and - most importantly - their dynamics.

One of the key issues in option risk management is the measurement of the inherent volatility risk, the so called "vega" exposure. Analytically, the "vega" is the first derivative of the BS formula with respect to the volatility parameter $ \sigma$, and can be interpreted as a sensitivity of the option value with respect to changes in (implied) volatility. When considering portfolios composed out of a large number of different options, a reduction of the risk factor space can be very useful for assessing the riskiness of the current position. Härdle and Schmidt (2002) outline a procedure for using principal components analysis (PCA) to determine the maximum loss of option portfolios bearing vega exposure. They decompose the term structure of DAX implied volatilities "at the money" (ATM) into orthogonal factors. The maximum loss, which is defined directly in the risk factor space, is then modeled by the first two factors.

Our study on DAX options is organized as follows: First, we show how to derive and to estimate implied volatilities and the implied volatility surface. A data decription follows. In section 6.3.2, we perfom a standard PCA on the covariance matrix of VDAX returns to identify the dominant factor components driving term structure movements of ATM DAX options. Section 6.3.3 introduces a common principal components approach that enables us to model not only ATM term structure movements of implied volatilities but the dynamics of the "smile" as well.