20. Volatility Risk of Option Portfolios

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: $ \tau$ (time to maturity in years), $ K$ (strike price), $ r$ (risk free, long-run interest rate) and $ S$ (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 $ \tau$. Here the term moneyness $ \frac{S}{K}$ refers to the ratio of the actual price $ S$ of the financial underlying and the strike price $ K$ 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.

Fig.: Implied volatility surface of the DAX option on July $ 18$, $ 1998$ 34673 SFEVolSurfPlot.xpl
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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.