Two-dimensional FX scaling for trend and reversal systems

Read size and duration from scale

Self-similar FX series can be viewed at different bar intervals. Self-similarity is a structure that keeps the same form when the observation altitude or the chart interval changes. That scale-independent structure is used to forecast expected trend size and expected trend duration for model construction.

An early FX scaling-law links mean absolute logarithmic return to the sampling interval through a drift-exponent. The drift-exponent is the power in that link. The early relationship is treated as historically useful rather than as a practical trading rule.

Map count and payoff to one zigzag-threshold

A later scaling-law maps average directional-change count, or ndc, to a zigzag-threshold that starts a new zigzag segment. A directional-change is a declared up or down turn in a zigzag path once price has moved a set threshold from the extreme of the current segment. Historical observations were described as fitting the power-law form of that later link.

Average return per zigzag segment is also mapped to the same zigzag-threshold. Expected segment payoff and the threshold that defines the segment therefore stay in the same price-change units, usually stated in pips.

Add volatility and collapse to joint surfaces

Volatility, defined as the average absolute close-to-close bar change and stated in pips, is introduced as a second measurement scale. The directional-change count and, more weakly, average segment return each follow a scaling-law in that measure.

Separate one-scale equations are collapsed into two joint surfaces under the assumption that the power-law slopes in threshold and in volatility stay common across regions. That joint form is two-dimensional-scaling: expected counts or expected returns depend on threshold and volatility at the same time, replacing the separate one-scale equations.

After those surfaces are fit, average directional-change count and therefore average segment-length can be read from measured volatility and a chosen zigzag-threshold. Segment-length is the average number of bars per zigzag segment, obtained by dividing the sample size by the directional-change count. In the historical workflow, predicted counts were presented as agreeing with the measured counts.

Invert the return surface, then read duration

Both mechanical procedures invert the joint return surface to set the zigzag-threshold from a chosen expected return and current volatility. They then read expected segment-length from the joint directional-change surface. The zigzag-threshold is allowed to move as volatility changes.

The trend-strategy is a mechanical rule that enters with a newly declared zigzag segment once the close-to-close move clears the threshold. It uses the predicted averages together with the unfinished segment's length in the decision.

The reversal-strategy is a mechanical rule that fades the unfinished segment once its length and price change both exceed the scaling-law averages. The same pair of surfaces supplies those averages for both procedures.