Hurst Exponents

The Hurst exponent, H, is a means o f classifying a time-series[Hurs51]. It was developed by Hurst, a hydrologist who was studying the changes in water levels at the Nile River Dam. It was natural to assume that the distribution o f reservoir levels would be normally distributed, as it was the product of a system with many degrees o f freedom. However, Hurst found that the distribution o f levels was not normally distributed, and so developed a classifying test statistic, the Hurst exponent, that did not depend on assumptions o f normahty. It is a test capable o f classifying time-series as random, mean-reverting, or persistent (trending).

Unlike many statistical tests, the time-series does not need to be normal distributed - this test can distinguish between random non-normal series and non-random non-

normal series. It is for this reason that this test is valuable. Market analyses that depend on assumptions of normality are to a degree tautological - as HuU[HuU93] notes on the empirical testing of Black-Scholes option pricing:

“The first problem is that any statistical hypothesis about how options are priced has to be a joint hypothesis to the effect that (1) the option pricing formula is correct; and (2) markets are efficient. If the hypothesis is rejected, it may be the case that (1) is untrue; (2) is untrue; or both (1) and (2) are untrue.”

It is not unfair to say that the option pricing formula will be derived from “conventional financial theory” and its assumptions/reliance on the notion o f market efficiency, and hence the tautology arises. While market efficiency does not necessarily imply that the market prices exhibit a random walk, (and therefore that assumptions of normality wiU hold), the price being a random walk (and hence being normal) does imply efficiency. Many statistical tests do depend on assumptions of normahty, and this is a stronger requirement than the market just being efficient. For example, the derivation o f the Black-Scholes pricing formula makes repeated use of generalised Wiener processes that foUow geometric Brownian motions. An imphed assumption here is that the markets must then be normal and not merely efficient.

These discussions are operationaUy irrelevant for the Hurst exponent, as few assumptions are made about the behaviour of the series under analysis, and no assumptions are made about market efficiency. The brief discussion given above is presented to lend credence to the appUcation o f a non-market specific technique to the market. The Hurst Exponent was originaUy pubhshed in [Hurs51], but an informal description is given here.

This technique works by examining how the rescaled range o f the series(the high-low range divided by the standard deviation o f the series over that interval), R, increases with time, T. For a random-walk series, the rescaled range increases with the square root of time: R = aT^^. The longer the period over which the range measurement is made, the larger the measured value - but the functional relationship between the two variables remains approximately constant. If the series is mean reverting, the range expands slower than if the series was random (R=aT", H<0.5) as the series has a

greater tendency to revert to its original value. Similarly, if the series is persistent, or trending, the range expands faster than if the series was random (R=aT", H>0.5), as the series has a greater bias to carry on moving in an estabhshed direction. This is summarised in Table 7.2[Pete91].

Table 7.2: Hurst Exponent Summary

Series behaviour Rate o f rescaled range expansion Hurst Exponent

Random Square root of time 0.5

Mean reverting Slower than random 0^H<0.5

Persistent(trending) Faster than random 0.5<H^1

7.4.1 Hurst Exponents and the Capital Markets

Peters[Pete91] has carried out a survey o f the Hurst exponents o f the monthly returns o f a range o f securities from various markets. This is shown in Table 7.3.

A Hurst exponent of 0.5 indicates that the underlying market is following a random walk, while a Hurst exponent greater than this indicates the presence o f trending behaviour. Clearly none of these securities except the Singapore dollar has a Hurst exponent o f 0.5. There are two points that are important here:

1. The Singapore doüar/US dollar exchange rate is a random walk, because the Singapore dollar is pegged to the US dollar. The randomness o f this series reflects the random timing of the trades required to bring the Singapore dollar back into line.

2. Clearly, the more information is available (the greater the period of data available), the lower the level of noise associated with the Hurst exponent sampling. Peters uses some rather ad-hoc reasoning to determine the amount o f data that is required, and (fortunately, perhaps?) determines that he has enough. However, Peter’s motive is to make the case that the Hurst exponents for the financial markets (except in cases o f pegging) are not 0.50. The most convincing demonstration of this is in the scrambling experiment he conducts on the S&P500.

Table 7.3: Hurst exponents and the Capital Markets

Security H u rst E xponent

Ind iv id u al Stocks IBM 0.72

Xerox 0.73 Apple Computer 0.75 Coca-Cola 0.70 Anheuser-Busch 0.64 McDonald’s 0.65 Niagra Mohawk 0.69

Texas State Utihties 0.54

Consohdated Edison 0.68

Stock M a rk e t Indices S&P 500 0.78

MSCI UK 0.68

MSCI Germany 0.72

MSCI Japan 0.68

D ollar exchange ra te Sterling 0.61

DeutscheMark 0.64

Yen 0.64

Singapore Dollar 0.50

G overnm ent Bonds Treasury Bill (yield) 0.65

US Bond (yield) 0.68

It can be seen from Table 7.3 that the Hurst exponent for monthly returns of the S&P500 is 0.78. When the series is scrambled, Peters reports that the exponent drops to 0.51 - i.e. random. This is a clear indicator that the scrambling process has destroyed the persistent, trending behaviour that was originally present in the S&P500, and consequently a demonstration that the S&P500 series is not a Markov process with no memory. That market movements are independent is one of the foundations of financial theory.

7.4.2 Approximate Entropy

The Hurst exponent is not the only means of assessing whether time-series are random or not. Working as what Stewart calls a “freelance mathematician”[Stew97], Pincus developed a test of randomness called Approximate Entropy (ApEn)[Pinc95] that, like the Hurst exponent, makes no assumptions about the process that generates the time- series. It works by examining blocks of (binary) data, for instance 101, and if this block is usually followed by a 1 then the process that generated the series must have a degree o f predictability. Similarly, if the block is followed equally often by a 0 as it is by a 1, then the series is unpredictable with respect to this block. The Approximate Entropy measure is the average unpredictabihty over all blocks. Pincus used this technique to examine whether stocks in the S&P500 are random or not, and concluded that they were far from random.

There was one intriguing exception to this, however. "Interestingly," says Pincus, "in the 1987 to 1988 time frame, there was a unique single two-week period in which ApEn was nearly maximally irregular— precisely the two weeks immediately preceding the stock market crash o f 1987."