>VOLATILITY AND ITS MEASUREMENTS

The volatility index, sometimes called by financial professionals and academics as “the investor gauge of fear” has developed overtime to become one of the highlights of modern day financial markets. Due to the many financial mishaps during the last two decades such as LTCM (Long Term Capital Management), the Asian Crisis just to name a few and also the discovery of the volatility skew, many financial experts are seeing volatility risk as one of the prime and hidden risk factors on capital markets. This paper will mainly emphasize on the developments in measuring and estimating volatility with a concluding analysis of the historical time series of the new volatility indices at the Deutsche Boerse.

As a result of the volatility’s increasing importance as a risk indicator and hedging instrument, many financial market operators and their institutional clients have pioneered and ventured out into developing methods of estimating and measuring volatility based on various well established academic models and eventually have even based their estimations on self-made models. Some established models have proven not to withstand the test of time and empirical data. The Black-Scholes Options Pricing model for instance, does not allocate for stochastic volatility (i.e. skewness). On the other hand, two models have gained importance over the years, namely the Stochastic Volatility Model and the GARCH (1,1). An insight into these three models will be carried out in this paper.

Two measurements which are widely used by financial and risk management practitioners to determine levels of volatility risk are the historical (realized) volatility, and the implied volatility. These two perspectives of volatility will be viewed with the emphasis being placed on the latter.
Two volatility trading strategies would be introduced, namely the straddle and trading in volatility and variance swaps. Then the old and new methodologies of calculating the volatility index at the Deutsche Börse AG will be discussed and the business case behind the concept of a volatility index will then be presented. Finally an analysis and interpretation of the calculated historical time series between years 1999 and 2004 of the new volatility indices will be done.

Volatility and its Measurements
An option is a financial contract which gives the right but not the obligation to buy (call) or to sell (put) a specific quantity of a specific underlying, at a specific price, on (European) or up to (American), a specified date. Such an option is called a plain vanilla option. An underlying of an option could be stocks, interest rate instruments, foreign currencies, futures or indices. Option buyers (long positions) usually pay an option premium (option price) to the option seller (short positions) when entering into the option contract. In return, the seller of the option agrees to meet any obligations that may occur as a result of entering the contract.

The options called exotics include Path-dependent options whereby its payoffs are dependent on the historical development of the underlying asset, such as the average price (Asian Option) or the maximum price (Lookup option) over some period of time. Then there are other options in which their payoffs are anchored on whether or not the underlying asset reaches specified levels during the contractual period. They are called Barrier options. Option traders are constantly faced with a dynamically altering volatility risk. While many speculate on the course volatility will take in the near future, some may tend to seek to hedge this risk. For instance Carr and Madan1 suggested a strategy that combines the holding of static options, all the out-of-the money ones, and dynamically trading the underlying asset. Such a strategy is very costly and most of the time not convenient for most traders. That’s why advances have been made to develop new products and strategies which allow investors and traders to hedge their portfolios of derivative assets as well as portfolio of basic assets against pure volatility exposure. Brenner and Galai2 were one of the first researchers to suggest developing a volatility index back in 1989.

Then in 1993, Robert Whaley developed the first volatility index on S&P 100 options for the Chicago Board of Options Exchange (CBOE) which was then subsequently introduced in the same year. Called the VIX, it used the model described by Harvey and Whaley [1992]3 in their research article. One year afterwards in December 1994, the Deutsche Boerse started publishing its own volatility index on DAX options called the VDAX on a daily basis. The Deutsche Boerse even went on further to introduce the first futures on volatility based on the VDAX called VOLAX in 1998.

To understand the concept behind a volatility index one must first understand the differences between the methods of volatility measurements and their forecasting abilities. Using the formula derived by Black and Scholes4 to price options, one needs among other things, the parameter volatility. They derived a formula for plain vanilla options using the parameters listed below as input.

1. The current price of the underlying at time t= S
2. The strike price of the option = K
3. The time to expiration of the option = T -t
4. The risk free interest rate = r
5. The annualized volatility of the underlying (based on lognormal returns) = s

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