Calibrating the weighted-moving-average exponent

Equal weights and linear weights in one family

A moving-average is a lookback filter that replaces an ordered run of price, volume, or breadth observations with one summary value at each sampling step. A simple moving average is the arithmetic mean of those observations over a chosen lookback and is equivalent to giving every point the same weight of one divided by that lookback. A linearly weighted moving average assigns integer weights that increase toward the most recent observation and then rescales those weights so they sum to one.

The general-weighted-moving-average contains both recipes as special cases. Each lookback weight is that observation’s index raised to a nonnegative exponent, the control-parameter-alpha, divided by the sum of those powered indexes. The result is still a weighted-moving-average: the lookback points receive explicitly assigned weights rather than a uniform share, and those weights are forced to sum to one.

How the recency exponent tilts the weights

Setting alpha to zero recovers equal weights. Setting it to one recovers linear weights. Setting it to two produces a square-weighted scheme. Setting it to one-half produces a square-root scheme between equal and linear weights. Letting alpha grow without bound concentrates nearly all weight on the newest observation.

For a twenty-period lookback, raising alpha from zero through one-half, one, and two progressively tilts the weight schedule toward more recent data. After the lookback length is chosen, simple and linearly weighted averages freeze that schedule. The extra alpha parameter lets one family be retuned for different targets, datasets, or markets.

A put-call ratio used to choose alpha

The historical workflow built a put-call-ratio from daily yen-dollar futures option put volume divided by call volume, then searched for the alpha that maximizes the correlation between a general weighted average of that ratio and the yen-dollar spot rate. That search is the construction of an optimal-weighted-moving-average: alpha is selected to maximize a stated objective.

On daily observations from 14 June 1993 through 9 August 1993, a trial-and-error search over alpha for lookbacks of eight, ten, and fifteen periods located peak correlations of 0.7075, 0.6953, and 0.69694 at alpha values of 0.60, 1.3, and 0.2. For those three lookbacks the correlation with the yen-dollar rate rose at small alpha, declined once alpha exceeded 3, and flattened once alpha exceeded 50.

When the same put-call-ratio was averaged at eight, ten, and fifteen periods with alpha held at 0.60, the shorter two traces tracked each other while the fifteen-period trace was smoother. Comparing several lengths was proposed as a way to read possible trend changes.