Constructing a variable-lookback random walk trend filter

When a fixed window hides path shape

Fixed-length moving averages and last-close oscillators can assign identical readings to price paths that have different shapes, including cases where recent direction would differ. The missing piece is shape information: path geometry inside the window that those tools discard when they collapse many different trajectories onto one number.

A random-walk yardstick for highs and lows

In a random-walk model the largest expected temporary displacement over n steps scales with the square root of n. That expected random walk is the displacement a pure random path would be expected to reach over n bars, scaled as average daily range times the square root of n.

Average high-to-low channel heights, expressed as a ratio to the one-day range, tracked the square root of lookback length in tabulated futures series, with most markets sitting somewhat above that benchmark. The channel-height ratio is that average high-to-low span over a lookback, divided by the one-day span, and it is used to check whether market ranges grow like a random walk.

Channel-height ratios at or above the square-root schedule were read as a mixture of random walks and directional persistence. Trend was defined as any move beyond the expected random walk. The random walk hypothesis test is that comparison of observed high-low channel growth with the square-root-of-time excursion implied by a random walk. It is used both to measure overextension and to set the threshold that distinguishes trend from noise.

How the random walk index is scored

The Random Walk Index for a high or a low is the actual excursion from the opposite extreme n bars earlier divided by average daily range times the square root of n. The published reading uses the n that maximizes that ratio. That maximizing lookback is the data-chosen n that produces the largest Random Walk Index on a given bar, replacing any fixed-length parameter.

With a 100-point average daily range, a four-bar random walk is expected to span 200 points and a nine-bar walk 300 points, so a 250-point drop from a four-bar high is more overextended than a 300-point drop from a nine-bar high. The same actual move can therefore look stretched or ordinary once it is scored against the expected random walk for its own lookback.

Splitting one score at a data-marked corner

Distributions of the lookback that maximized the index were similar across several futures and stocks and showed a corner near eight days. That corner was used to split a two-to-seven-day short-term Random Walk Index from an eight-to-sixty-four-day long-term Random Walk Index.

The short-term Random Walk Index is the same construction restricted to short candidate lookbacks and read as a swing or overbought-oversold gauge on highs versus lows. The long-term Random Walk Index is the same construction restricted to longer candidate lookbacks and read as a trend gate when the reading exceeds one.

A swing gauge and a persistence filter

Short-term index peaks in highs aligned with price peaks and peaks in lows aligned with price valleys. A long-term index above 1.0 on highs was treated as an uptrend filter and the same threshold on lows as a downtrend filter. The trend filter is that constructed reading: directional persistence is treated as present only when a high or a low has traveled farther than the displacement expected from a random walk over the same variable lookback.

On a Treasury bond futures example, several intermediate lows inside an uptrend and an intermediate high inside a downtrend that a nine-day raw stochastic missed were marked when the Random Walk Index exceeded 1.0. Separate high and low indices can mark short-term oversold lows during a strong uptrend, whereas a single oscillator can remain one-sided through the same trend.