Constructing zero-net-lag price channels

Delay as a construction budget

Editorial: treat smoother delay as a construction budget that can be cancelled. Pair an exponential stage with a first-order lead so a residual-width price channel can sit on the latest bar, instead of waiting for a centered moving average.

Data lag and equivalent length

Data lag is the sample delay between a linear input path and a smoother output, used to match or cancel filters of different form. On a linear input path, an N-period moving average trails by (N-1)/2 samples, so a seven-period average trails by three samples.

An exponential smoother with gain alpha trails a linear path by (1/alpha)-1 samples, and matching that lag to an N-period average requires alpha equal to 2/(N+1). Equivalent length is the moving-average length chosen so its lag equals that of an exponential smoother with a given gain.

First-order lead and the cascade

A first-order lead is an exponential recursion plus a first-difference gain that can reduce lag or produce a lead. Adding a first-difference gain gamma to the exponential recursion changes lag to ((1+gamma)/alpha)-1, and a sufficiently negative gamma can produce lead rather than lag.

A lead-lag cascade is an exponential smoother followed or preceded by a compensating lead stage so net delay can be driven toward zero. Cascading an exponential smoother with a first-order lead whose gamma equals 2(alpha-1) is a construction that targets zero net lag at the same equivalent length.

Residual volatility as channel width

Residual volatility is the root-mean-square of close-minus-smoother residuals, taken over a longer window than the smoother itself. It is formed by averaging squared close-minus-smoother errors over twice the smoother length and taking the square root, because squared residuals are noisier than the level series.

For weekly residuals, a five-period variance prefilter followed by a thirteen-period average of those variances is the stated way to add usable smoothness to the volatility width.

Smoothness for the price-channel midline

The price-channel midline is the smoothed series that a residual-width band is drawn around, and extra smoothness is required before it is usable as a channel. A seven-week zero-lag smoother can track residual width similarly to a seven-week average advanced three weeks, but still lacks enough smoothness for a trend-channel midline. Raising N also inflates peak residual width after a trend break.

Preceding the seven-week zero-lag stage with a three-week exponential prefilter advanced one week, using gain 0.5, is the stated way to add smoothness without reintroducing net lag.