Constructing fade rules from an empirical deviation histogram

How the histogram is built

The construction begins by subtracting a moving average from price, binning those deviations, counting frequencies, and plotting the histogram. A cumulative percent-below curve is then formed by summing lower bins and dividing by the total count.

The moving average is the balance line, the center of the constructed model and later a flatten or reverse reference. The empirical deviation histogram counts how often price occupied each bin of distance from that average. The cumulative percent-below curve is the running share of observations that sat beneath a given deviation, and it is what labels ordinary, uncommon, and extreme prints.

Monte Carlo simulation, in the sense used here, takes ordered price prints over a stated interval and lookback, builds an explicit frequency baseline, and judges later prints against that baseline rather than against a textbook curve.

What the one-minute sample showed

On one-minute S&P futures deviations from a one-day average, the empirical histogram held more mass between minus one and plus one point than a matched normal curve, skewed left, and showed extra extremes near plus and minus 3.5 points.

The same sample placed 42 percent of prints below the one-day average and 47 percent within one point of it. A five-point rise above the average was treated as an extreme because 99 percent of observations were lower.

Most of those one-minute prints stayed within 4.5 points of the one-day average, yet seven- and eight-point departures still occurred.

Why a Gaussian baseline is set aside

A random uncorrelated series is assigned a Hurst exponent of 0.5. The S&P 500 measurement used in this construction is 0.78, which is presented as incompatible with a Gaussian random-walk baseline used in many option models.

A short path and the extreme band

On a six-day S&P 500 futures path, bands at the one-day average plus or minus three points were described so that 94 percent of prices sat below the upper band and 96 percent sat above the lower band.

Those thresholds function as an extreme band: a deviation taken from the histogram so new risk is not added when price already occupies a rare tail. The path study shows price can remain at the extreme, so the constructed fade still waits for a one-point reversal off that band or through the average.

One procedure for entry, exit, and abstention

Mean reversion, in the sense used here, treats large departures from a local average as states that historically resolve toward that average, without fixing when the return occurs or where the average will then sit. Example rules abstain from buying more than two points above the average or selling more than two points below it, flatten when price tags the mean, and only initiate after a one-point move up from a low extreme, down from a high extreme, or through the average.

The same example set caps any loss at 1.5 points for the life of the position. That cap is a stop-loss: a precommitted loss bound, set in price points before entry and held for the life of the position, so adverse excursion cannot expand without a forced exit. The deviation study must be rebuilt periodically because the frequency curves change.

The listed rules are offered only as a translation of the empirical model into a single back-testable procedure, not as a finished live system.