THE RANGE PROCESS AND VOLATILITY

9.9 THE RANGE PROCESS AND VOLATILITY

The range process of a time series is measured by the difference between the largest and the lowest values the time series assumes within a given time interval. It provides another indication for a process volatility with some noteworthy differ- ences between the range and the process standard deviation (or variance). Explic- itly, when a series becomes more volatile, the series standard deviation estimate

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varies more slowly than that of the range. Thus, a growth surge in volatility might

be detected more quickly using the range. By the same token, when the volatility declines, the range process will be stabilized. These properties have been used, for example, in the R/S (range to standard deviation statistic) applied in financial analysis to detect volatility shifts. Both the variance and the range processes are therefore two sources of information which are important. The Bloomberg, for example, provides such a statistic for financial time series, also named the Hurst exponent (Hurst, 1951) or the R/S index. This index is essentially a parameter that seeks to quantify the statistical bias arising from self-similarity power laws in time series. In other words, it expresses the degree of power nonlinearity in the variance growth of the series. It is defined through rescaling the range into a dimensionless factor.

Calculations for the range and the R/S statistic are made as follows. Samples are of fixed length N are constructed, and thus the sample range is given by:

R t,N = Max{y t,N } − Min{y t,N }

while the sample standard deviation is calculated by:

where ⌢ y

t,N is the sample average. A regression, (R/S) = (Const N ) H provides an estimate of H, the Hurst exponent, or using a logarithmic transformation:

In N

= a + bH ; b = log(α N );

With the notation: H = Hurst exponent, R = sample’s range, S = sample’s stan- dard deviation and finally α = a constant. For random (Normal) processes, the Hurst index turns out to equal 0.5. While for any values larger than 0.5 obtained in a regression, it may indicate ‘long-term dependence’. Use of the Hurst index should be made carefully and critically, however. The origins of the Hurst ex- ponent are due to Hurst who began working on the Nile River Dam project and studied the random behaviour of the dam and the influx of water from rainfall over the thousand years data have been recorded. The observation was made that if the series were random, the range would increase with the square root of time –

A result confirmed by many time series as well as theoretically for normal pro- cesses. Hurst noted explicitly that most natural phenomena follow a biased ran- dom walk and thus characterized it by the parameter H expressing as well a series’ dependence called by Mandelbrot the ‘Joseph effect’ (Joseph interpreted Pharaolc’s dream as seven years of plenty followed by seven years of famine). Explicitly, a correlation C between disjoint increments of the series is given by C=2 2H −1 − 1. Thus, if H = 0.5, the disjoint intervals are uncorrelated. For

0.5, the series are correlated, exhibiting a memory effect as stated above (which tends to amplify patterns in time series). For H < 0.5, these are called

H>

301 ‘anti-persistent’ time series. Such analyses require large samples N , however,

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which might not be always available. For this reason, such analyses are used when series are long, such as sunspots, water levels of rivers, intra-day trading stock market ticker data etc.

An attempt to represent these series, expressing a persistent behaviour (or alternatively a nonlinear variance growth) was reached by Mandelbrot who in- troduced a fractional Brownian motion, denoted by B H (t) (see also Greene and Fielitz (1977, 1980) for an application in finance). A particular relationship for fractional Brownian motion which is pointed out by Mandelbrot and Van Ness (1968) is based on the self-similarity of the power law for such processes which means that the increment for a time interval s are in distribution proportional to

s H , or:

H (t + 1) − B H (t)] where i.d. means in distribution. Furthermore, the increments variance is:

B H H (t + s) − B H (t) → s [B

i.d.

E [B (t)] 2 H 2H (t + s) − B H =s E [B H (t + 1) − B H (t)] 2 which means that the variance for any time interval s is equal to s 2H times the

variance for the unit interval. Of course, it is now obvious that for H = 0.5, the variance is linear (as is the case for random walks and for Brownian motion) and it is nonlinear otherwise. In this sense, assuming a relationship between

the Hurst exponent (which is also a power law for the series) and the notion of long-run dependence of series (modelled by fractional Brownian motion), an estimate of the one is indicative of the other. From the finance point of view, such observations are extremely important. First and foremost, long-run dependence violates the basic assumptions made regarding price processes that are valued under the assumption of complete markets. As such, they can be conceived as statistical tests for ‘fundamental’ assumptions regarding the underlying process. Second, the Hurst index can be used as a ‘herd effect’ index applied to stocks or other time series, meaning that series volatility that have a tendency to grow, will grow faster over time if the index is greater than 0.5 and vice versa if the index is smaller than 0.5. For these reasons, the R/S index has also been associated to ‘chaos’, revealing series that are increasingly unpredictable.