In this chapter we analyze the principal factors in the dynamic structure of implied volatility at the money (ATM). The data used are daily Volatility-DAX (VDAX) values. By using principal component analysis we consider a method of modelling the risk of option portfolios on the basis of ``Maximum Loss''.
There is a close connection between the value of an option and the
volatility process of the financial underlying. Assuming that the
price process follows a geometric Brownian motion we have derived
the Black-Scholes formula (BS) for pricing European options in
Chapter 6. With this formula the option price is at
a given time point a function of the volatility parameters when
the following values are given: (time to maturity in
years),
(strike price),
(risk free, long-run interest
rate) and
(the spot price of the financial underlying).
Alternatively one can describe the observed market price of an
option at a specific time point with the help of the BS formula
using the so called ``implied'' volatility (see Chapter
6). In doing this one typically finds a U-shaped
form for the resulting surface of the volatility over different
times to maturity and strike prices. This phenomenon is also
referred to as the ``Volatility Smile''. Figure 19.1 illustrates the
typical form of a volatility surface
using DAX options. Shown is the implied volatility as a function of the moneyness and the remaining time
to maturity . Here the term moneyness
refers to the ratio of the actual price
of the
financial underlying and the strike price
of the respective
option. It should be noted that options are only traded on the
market on a discrete price basis and a discrete time to maturity.
In determining the volatility surface, as in Chapter
13, a smoothing technique needs to be applied.
By observing the volatility surface over time, distinct changes in the location and structure become obvious. Identifying the temporal dynamics is of central importance for a number of financially oriented applications. This is of particular importance for the risk management of option portfolios. To determine the volatility's dynamics, an application of principal component of analysis is quite suitable, see Skiadopoulos et al. (1998). The total temporal structure can be sufficiently represented by a small number of principal components so that the dimensions of the factor space for the purpose of risk analysis can be significantly reduced.