Random Walk Index construction with an adaptive lookback

What the Random Walk Index measures

The Random Walk Index is constructed as an observed price excursion divided by the move a random walk would be expected to cover over the same span. Average daily range is the mean high-to-low range over the n days immediately preceding the current bar. The expected random walk is that average daily range multiplied by the square root of a candidate lookback length.

High-side and low-side readings

The high-side reading uses the gap from the current high to the low of n days earlier, scaled by average daily range times the square root of n. The low-side reading uses the gap from the high of n days earlier to the current low, scaled by the same average-range and square-root-of-n denominator.

A worked comparison of two windows

With an average daily range of 100, the expected random-walk distance is 200 over 4 days and 300 over 9 days. In the worked example, a current low 250 points under the high from 4 days earlier is treated as more overextended than a 300-point gap under the high from 9 days earlier, because only the shorter window exceeds its expected walk.

An adaptive lookback from the data

The day’s index is the lookback n that produces the largest ratio, so the window is selected from the data rather than fixed in advance. That integer n is the adaptive lookback used as that day’s published reading. The construction is presented as containing no arbitrary constants or parameters.

Beside a raw percent-K stochastic

On a Treasury bond futures example, a raw percent-K stochastic is plotted with the Random Walk Index. Raw percent-K is the unsmoothed stochastic oscillator series shown on the same price history. Selected intermediate highs and lows, including May 9, 18, and 25 in an uptrend and August 15 in a downtrend, are marked where the index exceeds 1.0 while the stochastic does not flag them. An index value of 1.0 is the unit threshold, at which the observed move equals the expected random walk.