2007issue C021-5
Constructing a random-walk baseline from ordered returns
A random-walk baseline for asset prices can be constructed so that only positive prices are admitted, by treating logarithmic returns as normally distributed. Once that Gaussian construction is explicit, a Hurst exponent and any fractal-dimension index can be judged as a separate overlay.
- A positive-price random-walk baseline models logarithmic returns as normally distributed, using a location mean and a variance that sets the width of the bell curve.
- An arithmetic construction of the price level itself can produce negative prices because the driving motion is unbounded below.
- The Hurst exponent is the slope of the logarithm of rescaled range against the logarithm of sample length, and financial series use log returns rather than raw prices as the input.
- A fractal-dimension index built from that exponent is presented as a filter for trend-sensitive systems.
Why the baseline is written first
The random-walk hypothesis test compares an explicit quantitative baseline with an out-of-sample result. The comparison uses ordered price, volume, or breadth observations over a defined sampling interval and lookback. The first construction task is to state that baseline in terms of ordered returns.
A positive-price Gaussian walk
A random-walk baseline for asset prices can be constructed so that only positive prices are admitted by modeling logarithmic returns as normally distributed. That model is geometric Brownian motion, and it is the classical random-walk baseline reconstructed here. The construction fully specifies a Gaussian return distribution with two parameters: a location mean and a variance that sets the width of the bell curve.
Price level versus logarithmic returns
An arithmetic Brownian construction of the price level itself can produce negative prices because the driving motion is unbounded below. The archive workflow therefore places the random walk on logarithmic returns rather than on the raw price series.
Recovering a Hurst exponent
A Hurst exponent is recovered by plotting the logarithm of rescaled range against the logarithm of sample length and reading the slope of that line. Rescaled range is the range of a cumulative mean-deviation series divided by the series standard deviation. For financial series the rescaled-range input is the log return of successive prices, not the raw price level. The resulting scaling exponent locates a series between mean-reverting, random-walk, and persistent regimes.
A filter built from that slope
A fractal-dimension index built from the Hurst exponent is presented as a construction that can be applied as a filter to trend-sensitive systems. It is a trend-state overlay derived from the Hurst exponent, not a restatement of the random-walk baseline.
All readings on this track · 10 readings
- 1984When random walks fail to explain market averages
- 1988Building probability-density adaptive weights and a chi-square trend filter
- 1990Withhold a random-walk label until grouping and lookback are checked
- 1991Constructing a variable-lookback random walk trend filter
- 1992Ranking futures trend tendency against a random walk
- 1995A normalized dispersion screen for trend-following
- 1997Why failed random-walk tests still need careful Monte Carlo work
- 2007Constructing a random-walk baseline from ordered returns
- 2011Constructing a cycle-free trend filter against an ATR-scaled random walk
- 2013How to evaluate a hybrid growth and money-flow scan