VOLATILITY DEFINED

9.1 VOLATILITY DEFINED

Volatility pricing, estimation and analysis are topics of considerable interest in finance. The value of an option, for example, depends on the volatility, which cannot be observed directly but must be estimated or guessed – the larger the volatility the larger the value of an option. Thus, trading in options requires that volatility be predicted and positions taken to profit from forthcoming high volatil- ity and vice versa from forthcoming low volatility. In many instances, attempts are also made to manage volatility, either by using derivative-based strategies or by some other creative means, such as ‘certification’. In a past issue of The Economist (18 August 2001, p. 56), an article on ‘Fishy Math’ pointed out that salmon certification may stabilize prices and thereby profit Alaska’s fishermen. To do so, options were used by the MSC (the Marine Stewardship Council, a not-for-profit agency that campaigns for sustaining fishing), to value the certi- fication of Alaska salmon, claimed to ensure a certain standard of fishery and environmental management which customers are said to value. For fishermen, a long-term benefit would be to reduce the volatility of salmon prices and thereby increase the value of their catch. The valuation of such profits was found by the MSC using Black–Scholes options. That is to say, the options prices implied by those two levels of volatility – what a reasonable person would expect to pay to hedge the price risk before and after certification – were calculated and compared, indicating a profit for fishermen, a profit sufficient to cover the cost of certification. Choosing a model of volatility is critical in the valuation of derivatives, however. In a stable economic environment it makes sense to use plain vanilla models. However, there is ample historical evidence that this may not be the case and therefore volatility, and in particular stochastic volatility, can be the cause of mar- ket incompleteness and create appreciable difficulties in pricing assets and their derivatives. The study of volatility is thus important, for both these and many other reasons. For example, the validation of fundamental financial theory presumes both the ‘predictability’ of future prices and interest rates, as well as other relevant

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INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

time series. Financial markets and processes where the underlying uncertainty is modelled by ‘random walks’ are such an instance, since they can provide future predictions, albeit characterized by a known probability distribution. The random walk hypothesis further implies, as we saw earlier; independent increments, inde- pendently and identically distributed Gaussian random variables with mean zero and a linear growth of variance. Statistically independent increments imply in fact, ‘a linear growth of uncertainty’. Technically, this is shown by noting that the functional relationship, implying independence, f (t + s) = f (t) + f (s) implies

a linear growth since it is uniquely given by the linear time function, f (t + s) = (t + s) f (0). This facet of ‘linear growth of uncertainty’ has been severely criticized as too simplistic, ignoring the long-term dependence of financial time series. Further, empirical evidence has shown that financial series are not always ‘well-behaved’ and thus, cannot be always predicted. For this reason, extensive research has been initiated seeking to explain, for example, the leptokurtic character of rates of returns distributions, the ‘chaotic behaviour’ of time series, underscoring the ‘unpredictability of future asset prices’. These approaches characterize ‘nonlin- ear science’ approaches to finance. Practically, ‘bursts’ of activity, ‘feedback volatility’ and broadly varying behaviours by stock market agents, ‘memory’ etc., are contributing to processes which do not exhibit predictable price pro- cesses and therefore violate the presumptions of fundamental finance. The study of these series has motivated a number of approaches falling under a number of themes spanning: fat tails (or Pareto–Levy stable) distribution analysis char- acterized by infinite variance; long-term memory and dependence characterized by explosive growth of volatility; chaotic analysis; Lyapunov stability analy- sis; complexity analysis; fractional Brownian motion; multifractal time series analysis; R/S (range to scale) analysis etc. Extensive study has been devoted to these methods (see, for example, the review papers of Mandelbrot (1997a) and Lo (1997)).

Volatility modelling and estimation is often specialized to the second mo- ment evolution of a price process, but it is much more. Generally, we say that

a random variable, say the returns x is more volatile than a random variable y if for all a > 0, the cumulative density functions of the returns distributions

F X (.), F Y (.) satisfies, F X (a) > F Y (a). The mathematics of ‘stochastic ordering’ consisting in comparing and ordering distributions, as above, has focused finan- cial managers’ attention on such measurements using terms such as ‘stochastic dominance’ (or first, second and third degree), ‘hazard rate dominance’, convex dominance, etc. These techniques have the advantage of being utility-free, but they are not easy to apply, nor is it always possible to do so. A practical measurement of volatility is thus problematic. When the underlying distribution of a process is Normal, consisting of two parameters, the mean and the variance, it makes sense to accept the standard deviation as a measure of volatility. However, when the un- derlying distribution is not Normal (as with leptokurtic distributions, expressing asymmetry in the distribution), the definition of what constitutes volatility has to be dealt with carefully. An appropriate measure of volatility is thus far from

273 being unique, albeit a process standard deviation is often used and will be used

MEMORY AND VOLATILITY

in this chapter. There are other indicators of volatility, such as the range R, the semi-variance, R/S statistics (see the last section in this chapter for a develop- ment and explanations of such statistics) etc. providing thereby more than one approach and more than one statistical measurement to express the volatility of a series.

Given the importance of volatility, a broad number of approaches and tech- niques have been applied to measure and model it. The simplest case is, of course, the constant (variance) volatility model implied in random walk models. When the variance changes over time (whether it is stochastic or not), models of volatil- ity are needed that are both economically acceptable and statistically measurable. We shall provide a brief overview of these techniques in this chapter since they are currently a ‘workhorse’ of financial statistics.