Noisy Plateaus: Do Prices Remember?

A Working Paper on Temporary Price Reference States in Electronic Markets

Scott LeBrun
Working Paper — October 2026

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Abstract

Financial prices are usually represented as continuously changing processes. But at very short time scales, markets often appear to behave differently: price fluctuates around a relatively stable local region for a period, moves away from it, and then either establishes a new region or returns to the previous one.

This paper calls these temporary regions noisy plateaus.

The central question is not whether such patterns can be found. They clearly can. The more difficult question is whether a previously established price region retains information after the market leaves it.

Put simply:

Do prices remember where they were?

A useful answer requires separating several very different claims. A market may display plateau-like behaviour without investors consciously anchoring to previous prices. A previous region may help forecast subsequent behaviour without creating a profitable trading strategy. And apparent recaptures may emerge automatically from ordinary diffusion, mean reversion, market microstructure, or the way the regions themselves are selected.

Development tests using Nasdaq limit-order-book data found thousands of plateau-transition episodes, but did not find evidence that the historical characteristics of those plateaus improved aggregate forecasts beyond conventional price-history, geometry, and microstructure variables. Simulated processes without behavioural memory also produced plateaus and recaptures.

The Noisy Plateaus hypothesis therefore remains unconfirmed.

What survives is a narrower and potentially more useful proposition: markets may be usefully represented as transitions among temporary local price states, and the predictive relevance of previous states can be tested prospectively without assuming that those states represent fundamental value, investor psychology, or exploitable inefficiency.

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1. The Basic Idea

Imagine watching a highly liquid stock trade for several minutes.

The price does not necessarily wander smoothly across every possible value. Instead, it may spend some time moving back and forth within a narrow area:

$101.18
$101.20
$101.19
$101.22
$101.17
$101.21

Then something changes.

The stock begins trading around:

$101.41
$101.43
$101.40
$101.44

The market appears to have moved from one temporary region to another.

The important word is temporary.

A noisy plateau is not assumed to be fair value. It is simply an observable period during which price remains unusually concentrated around a local centre despite continuous trading noise.

The resulting picture of price behaviour resembles a staircase whose steps are covered in vibration:

plateau → displacement → resolution → plateau

But the transition between steps is uncertain.

After leaving an established region, the market might:

1. return and stabilize in the previous region;
2. stabilize around a new region;
3. continue moving without forming either state; or
4. remain ambiguous for the observation period.

This transition is the interesting part.

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2. Candidate States

The framework distinguishes an established plateau from what can be called a candidate state.

Suppose a stock has traded around $100 for some time and then moves toward $100.30.

The market has not necessarily established $100.30 as a new state.

It has merely left the old one.

The move can therefore be interpreted as a candidate transition:

The market is testing whether a different price region will persist.

Two broad outcomes are possible.

Recapture

The displacement fails and price returns to the previous plateau strongly enough to re-establish local stability.

The market has not merely touched the old price. It has recaptured the old region.

Replacement

Price instead stabilizes sufficiently far from the original region to establish an independently identifiable successor plateau.

The previous state has been replaced.

This distinction avoids an important problem with conventional reversal analysis. A brief return toward a previous price is not necessarily economically meaningful. Neither is a temporary move away from it.

The hypothesis concerns persistent states rather than individual ticks.

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3. What Does “Memory” Mean?

The word memory can easily imply more than the evidence supports.

Noisy Plateaus does not initially require psychological memory.

The hypothesis can be stated operationally:

After a price region has been established and the market leaves it, does information about that specific historical region improve our ability to predict what happens next?

That is a statistical question.

It is different from saying:

Traders remember the old price and consciously attempt to return to it.

Limit-order-book data cannot establish that.

Market participants may respond to previous price levels for many reasons:

• inventory management;
• latent liquidity;
• algorithmic execution;
• clustered orders;
• previous trading activity;
• technical strategies;
• market-making behaviour;
• risk limits;
• conventional benchmarks;
• or genuine psychological reference dependence.

Observable price memory and human memory are therefore not the same thing.

The first can potentially be measured without establishing the second.

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4. Why a Plateau Might Matter

A previously occupied price region could remain relevant because it summarizes more than its numerical price.

It may encode a history of successful trading around that level.

Market participants have previously demonstrated willingness to buy and sell there. Orders may have accumulated. Algorithms may incorporate previous trading ranges. Portfolio managers may have executed around the level. Technical traders may identify it. Participants may regard it as a reference price simply because it has recently been repeatedly observed.

In that sense, a plateau may become a temporary coordination point.

A displacement then tests that coordination.

If a new region becomes sufficiently stable, the coordination point may shift.

This suggests a possible sequence:

temporary consensus → disruption → competition between states → resolution

The term consensus must be used cautiously. Nothing in anonymous order-book data demonstrates that traders literally agree on value.

The observable phenomenon is simply concentrated trading around a local region.

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5. The Behavioural Interpretation

Behavioural finance provides one possible interpretation of the model.

People frequently evaluate outcomes relative to reference points rather than entirely in absolute terms. Anchoring, reference dependence, the disposition effect, attention to round numbers, and similar phenomena could therefore make recently established prices relevant.

But this interpretation creates an important trap.

Suppose a stock falls below an earlier plateau and then rebounds.

It is tempting to say:

Investors regarded the lower price as a loss or bargain and bought the stock back toward the previous reference level.

That conclusion cannot be drawn from price data alone.

We do not know who owns the stock, their purchase prices, whether they are long or short, their hedges, their expectations, or what reference price they personally use.

Even the prediction that downward moves should recapture more frequently than upward moves is not automatic.

That asymmetry must itself be tested.

The behavioural interpretation therefore belongs near the end of the evidentiary chain, not at its beginning.

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6. Making the Idea Testable

An earlier version of the Noisy Plateaus project revealed a methodological problem common to this kind of research.

If a plateau is identified using future information, or its centre moves as subsequent observations arrive, the experiment can manufacture apparent predictability.

The revised framework therefore treats plateau creation as an online event.

At the moment a region is confirmed:

• its centre is fixed;
• its boundaries are fixed;
• its scale is fixed;
• its confirmation time is recorded.

Future observations cannot alter that historical region.

A candidate displacement must then occur after confirmation.

Likewise, a successor plateau must be established using observations occurring after the candidate transition begins.

This makes the sequence causal in time:

formation → confirmation → displacement → resolution

rather than identifying a pattern retrospectively and then testing the same observations used to construct it.

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7. What the Development Data Showed

The revised detector was applied to four previously examined Nasdaq MBP-10 datasets containing AAPL and INTC.

The primary procedure identified:

• 2,667 candidate transitions
• 774 durable recaptures
• 1,805 confirmed replacements
• 88 unresolved episodes

There were also 1,408 occasions on which price merely returned to the old region at least once.

Only 774 became durable recaptures.

That distinction is important.

A price touching an old level is much more common than the market actually re-establishing itself there.

The data therefore support the descriptive usefulness of distinguishing temporary states and transitions.

They do not, however, establish that the previous plateau has special predictive power.

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8. The First Important Negative Result

The strongest test was not whether recaptures existed.

It was whether information describing the previous plateau improved prediction after controlling for things we already know about the market.

A baseline model included variables describing:

• current price geometry;
• liquidity;
• recent price history;
• recent order flow;
• and time of day.

A second model added information specifically describing the historical plateau.

If the previous region contained useful additional information, the second model should forecast candidate outcomes better.

It did not.

Across the development sample, adding plateau-history variables slightly worsened aggregate forecast performance.

A second nonlinear model produced the same qualitative result.

Repeating the experiment with a slower sampling interval did not change the conclusion.

This is important because it prevents the theory from becoming unfalsifiable.

The appropriate conclusion is not:

The market remembers, but our model has not found the right way to prove it.

The appropriate conclusion is:

The particular reference-state information tested so far has not demonstrated incremental predictive value.

That is evidence against the stronger version of the hypothesis.

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9. Another Problem: Plateaus Can Appear Without Memory

There is an even more fundamental challenge.

What if plateau-like behaviour naturally emerges from processes that contain no reference state at all?

To examine this, the same detection procedure was applied to simulated price processes.

The simulations included:

• tick-rounded diffusion;
• a slowly moving centre with mean-reverting noise;
• and a simplified process in which persistent order-flow pressure influenced returns.

None was designed to remember a detected historical plateau.

Yet all produced candidate transitions.

They also produced recaptures and replacements.

Even simple diffusion occasionally generated convincing-looking returns to previous regions.

This means that a chart showing plateaus and recaptures is not strong evidence for the theory.

The selection process itself can manufacture visually persuasive patterns.

A successful Noisy Plateaus theory therefore has to beat relevant null models rather than merely produce attractive examples.

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10. The Downward-Recapture Idea Also Weakened

One early intuition was that downward departures might be particularly likely to fail.

That could be consistent with bargain seeking, anchoring, loss aversion, dip buying, or various market-structure mechanisms.

The larger corrected development sample did not support that pattern.

Downward candidates produced durable recaptures about 28% of the time.

Upward candidates recaptured about 30% of the time.

A constrained matched comparison did not reverse the result.

There was therefore no evidence in this development sample that downward departures were unusually likely to return.

That does not prove that directional asymmetry never exists.

It means it should no longer be treated as a central prediction of the framework.

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11. A More Interesting Hypothesis: Reference Replacement

The failure of the simple recapture hypothesis suggests a different question.

Perhaps the important property of a plateau is not that price is pulled back toward it.

Perhaps it is that reference states can be replaced.

Once a successor plateau becomes established, does the previous region gradually become less relevant?

This creates a different model of market behaviour.

Rather than:

price leaves equilibrium → price is pulled back

the process becomes:

state A → unstable transition → state A survives or state B becomes established

The market is not necessarily returning toward a permanent equilibrium.

It may be updating temporary coordination points.

Development data showed that, after a successor plateau was established, subsequent prices occupied the new region more often than the previous region.

But this is not sufficient evidence of reference replacement.

Price naturally begins near the successor because that is where the successor was just established.

A proper test must ask whether the successor retains unusual relevance after controlling for starting position, distance, volatility, liquidity, order flow, and recent history.

That experiment remains open.

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12. The Strong Version and the Weak Version

The project now contains two versions of the Noisy Plateaus hypothesis.

Strong version

Financial markets create temporary behavioural reference states that influence subsequent trading and are transferred when new reference states become established.

Current evidence does not establish this.

Weak version

High-frequency price behaviour can sometimes be usefully represented as persistent local states separated by uncertain transitions, and previous states can be tracked explicitly to determine whether their historical identity adds predictive information.

The evidence remains compatible with this version.

The distinction matters.

The weak theory is primarily a measurement framework.

The strong theory is an economic and possibly behavioural explanation.

The first can be valuable even if the second ultimately fails.

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13. What Would Change My Mind?

A useful theory should state in advance what evidence would count against it.

For Noisy Plateaus, several outcomes would weaken the proposition substantially.

The theory would lose support if, on genuinely new data:

1. historical plateau information consistently fails to improve prediction beyond strong recent-history and microstructure models;
2. simulated or conventional market models reproduce the same apparent effects;
3. successor states show no predictive relevance beyond current location and ordinary persistence;
4. independently salient references such as official opens, closes, VWAP levels, or predetermined numerical prices show no meaningful interaction with detected plateaus; and
5. any apparent effect is unstable across stocks, dates, venues, or market conditions.

Repeatedly changing the definition of a plateau until one of these tests becomes positive would not rescue the hypothesis.

It would destroy its falsifiability.

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14. What Evidence Would Strengthen It?

The most useful next experiment is surprisingly simple in principle.

Freeze the current detection rules.

Obtain market data from dates that have never been examined during development.

Identify plateaus prospectively.

Then compare two forecasting systems:

Model A: everything a reasonable microstructure model knows about current conditions and recent history.

Model B: exactly the same information, plus the identity and history of the previously established plateau.

If Model B consistently produces meaningfully better forecasts across independent market environments, then something about the way historical price states are represented appears to matter.

That still would not prove behavioural anchoring.

It would, however, demonstrate reference-state persistence in the operational sense used here.

Behavioural interpretation could then become a separate research question.

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15. Broader Implications

The most interesting possibility raised by Noisy Plateaus may have little to do with trading.

Many systems do not move smoothly from one stable condition to another.

They occupy temporary states, experience noisy transitions, and either restore the old state or settle into a replacement.

Financial markets may be one example.

The same abstract structure appears in:

• ecological regimes;
• political equilibria;
• social conventions;
• neural states;
• traffic systems;
• industrial processes;
• and other adaptive systems.

The deeper question is therefore not merely whether a stock price bounces from support.

It is whether an apparently continuous system can sometimes be better understood as a sequence of temporary attractor-like states connected by uncertain transitions.

Markets provide unusually rich data with which to investigate that possibility.

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16. Current Assessment

The Noisy Plateaus idea began with a visually intuitive observation:

prices seem to spend time somewhere before they go somewhere else.

That observation survived.

The stronger interpretations have not.

The initial continuation idea failed validation. The proposed downward-recognition effect weakened under corrected analysis. Historical plateau features have not yet improved forecasts. And simple processes with no behavioural reference mechanism can create convincing plateau-and-recapture patterns.

That is not a reason to discard the project.

It is a reason to narrow the claim.

The current hypothesis is therefore:

Previously established price regions may function as measurable temporary states. Whether the identity of those states contains information beyond conventional price history and market microstructure remains an open empirical question.

That question is specific enough to test and broad enough to matter.

For now, the correct status of Noisy Plateaus is neither discovery nor failure.

It is an idea worth examining.

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References

Barberis, N., & Xiong, W. (2009). What Drives the Disposition Effect? An Analysis of a Long-Standing Preference-Based Explanation. Journal of Finance, 64(2), 751–784.

Cont, R., Kukanov, A., & Stoikov, S. (2014). The Price Impact of Order Book Events. Journal of Financial Econometrics.

Huang, W., Lehalle, C.-A., & Rosenbaum, M. (2015). Simulating and Analyzing Order Book Data: The Queue-Reactive Model.

Osler, C. L. (2003). Currency Orders and Exchange-Rate Dynamics: Explaining the Success of Technical Analysis. Journal of Finance.

Wu, P., Rambaldi, M., Muzy, J.-F., & Bacry, E. (2019). Queue-Reactive Hawkes Models for the Order Flow.

LeBrun, S. (2026). Noisy Plateaus or Failed Continuation? A Frozen Candidate-Resolution Protocol for Limit-Order-Book Data. Working research note.

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Working-paper status

This document reports a developing hypothesis and retrospective development analysis. It is not a validated trading strategy, evidence of market inefficiency, or proof of behavioural reference dependence. Existing datasets used during development should not be treated as untouched confirmation samples. The next meaningful test requires genuinely new data and prespecified analysis rules.