Probability-Mass Centroids in Options Markets

Research Status

Retained concept. Centroid-derived measurements are part of the current production feature framework. Individual centroids are not published as deterministic price targets.

The options market is a matrix.

Each contract occupies a location defined by:

• Strike.
• Expiration.
• Call or put side.
• Open interest.
• Volume.
• Implied volatility.
• Gamma.
• Vega.
• Other contract sensitivities.

One way to summarize this matrix is to calculate weighted centers.

Vyreon calls these probability-mass or structural centroids.

The word "probability" describes the normalized weighting used to summarize the matrix. It does not mean every weight is a literal risk-neutral probability.

Why Centroids Were Investigated

The early research asked whether market structure could be understood through the location and movement of mass across the options matrix.

Traditional analysis often inspects:

• One strike.
• One expiry.
• Put-call ratios.
• Largest open-interest walls.
• Implied-volatility skew.

Those measurements can be useful, but they do not provide one coherent description of the full matrix.

A centroid compresses a weighted distribution into a location measurement.

It answers:

Where is this selected form of structural mass concentrated?

Basic Form

For strike values K_i and nonnegative weights w_i, a weighted centroid is:

centroid = sum(K_i * w_i) / sum(w_i)

The meaning depends entirely on the weight.

Examples include:

• Open-interest-weighted strike center.
• Implied-volatility-weighted center.
• Gamma-weighted center.
• Vega-weighted center.
• Combined sensitivity field center.

There is no universal "true" centroid.

Each weighting scheme measures a different structural object.

KOI

KOI is an open-interest-weighted strike center.

It summarizes where contract inventory is concentrated.

KOI can be calculated:

• Per expiry.
• Per maturity bucket.
• Across a selected expiry range.
• For calls and puts separately.
• Across both sides.

KOI does not reveal:

• Who initiated the contracts.
• Whether the position is long or short.
• Dealer exposure sign.
• Trader motive.
• Future hedge behavior.

It is inventory geometry.

K-Star

K-star is a volatility-sensitive structural center.

Its exact production construction is proprietary, but conceptually it describes the location of a selected implied-volatility structure.

K-star is not:

• A guaranteed equilibrium.
• A fair-value estimate.
• A price target.
• A support or resistance line.

It is a summary of where the volatility-sensitive structure is located.

Field Mean And Field Width

The field mean combines sensitivity-weighted information into a central strike measurement.

Field width describes dispersion around the center.

Together, they answer:

• Where is the selected structure centered?
• How concentrated or broad is it?
• Is the field becoming tighter or wider?
• Is the center moving relative to spot?

A narrow field can indicate concentration.

A wide field can indicate dispersed structure.

Neither automatically identifies direction.

Why Parametric Distribution Fits Were Not Retained

Early research attempted to fit smooth distributions to the options matrix.

Candidates included skew-normal and mixture-like forms.

These methods created practical problems:

• Slow optimization.
• Sensitivity to starting values.
• Unstable behavior on multimodal chains.
• Difficulty fitting single-stock structures.
• Additional assumptions about distribution shape.
• Poor real-time reliability.

The research returned to the actual requirement.

The system needed a stable summary of weighted location and dispersion.

A discrete centroid was:

• Faster.
• Easier to audit.
• More robust to irregular chains.
• Better suited to live use.
• Less dependent on one assumed distribution family.

The simpler measurement survived.

Centroids As Features, Not Commands

The current model uses centroid-derived variables as features inside a broader state estimator.

A centroid can be informative because:

• Positioning structure changes through time.
• Different maturities move at different rates.
• Price can become displaced relative to structural centers.
• Width and curvature can change.
• The relationship among centers can contain information.

The model learns from these relationships alongside:

• Implied-volatility surface measurements.
• Price context.
• Trend and curvature.
• Field width.
• Multi-horizon target history.
• Adaptive state uncertainty.

The public report should not convert one center into a deterministic narrative.

The Price-Attractor Hypothesis

Early notes often described KOI or K-star as attractors.

This language was useful for generating tests, but it became too strong.

A price attractor would imply that price has a reliable tendency to move toward the measured center.

The empirical record is more complicated.

Price can:

• Cross a centroid.
• Remain displaced.
• Move with the centroid.
• Move away during repricing.
• Respond to a different part of the matrix.
• Ignore the center during large external shocks.

The current interpretation is:

A centroid is a structural reference point whose relationship with price can be measured. Attraction is a hypothesis that must be tested under a defined regime.

Expiry And Centroid Reconfiguration

The expiry research extended the centroid concept.

When a large expiry disappears from the next option-chain snapshot, the aggregate open-interest centroid changes.

This is mechanical weighted-average behavior.

The research studies:

• The size of the field change.
• Whether the change can be estimated before expiry.
• How the observed post-expiry centroid differs from the mechanical estimate.
• The delayed market response.
• The volatility state that controls movement amplitude.

The expiry study does not establish that KOI alone causes price.

It establishes that centroid reconfiguration is a measurable event.

Multi-Horizon Centroids

The current public system divides the options matrix into four maturity buckets:

• 8 to 30 calendar days.
• 31 to 60 calendar days.
• 61 to 120 calendar days.
• 121 to 365 calendar days.

The buckets are not independent markets.

They are slices through one coupled maturity surface.

Each slice has different:

• Contract density.
• Gamma sensitivity.
• Vega sensitivity.
• Noise.
• Inertia.
• Repricing speed.
• Signal-to-noise ratio.

Centroid features are calculated within this horizon structure.

The system preserves cross-horizon disagreement rather than averaging everything into one location.

Why The Range Stops At 365 Days

Some options markets contain expiries beyond one year.

The current public model truncates the matrix near 365 days.

This was an empirical engineering choice.

The selected buckets provided useful signal-to-noise behavior while keeping the system computationally manageable and consistent across history.

The cutoff is not a claim that contracts beyond one year have no effect.

It is a defined boundary of the current model.

Invalidated Damping Interpretation

The early Market Physics program linked centroid relationships to damping and resonance analogies.

An apparent negative damping ratio became one of the early candidate signals.

The negative result was caused by a missing absolute value in the calculation.

Under the intended definition, the damping ratio did not become negative.

The signal was invalidated.

The broader lesson survived:

• Structural relationships can be measured.
• Physical analogies can generate hypotheses.
• The analogy does not validate the hypothesis.
• Every derived quantity requires dimensional, numerical, and causal checks.

What Centroid Measurements Are Good For

Centroids are useful for:

• Compressing the option matrix.
• Comparing structure across time.
• Measuring displacement from spot.
• Measuring cross-horizon differences.
• Studying expiry reconfiguration.
• Building causal model features.
• Explaining structural charts.
• Detecting when the location or width of market structure changes.

What Centroid Measurements Are Not Good For By Themselves

A centroid does not establish:

• Trader intent.
• Dealer sign.
• Guaranteed pinning.
• Support.
• Resistance.
• One exact future price.
• One exact trade.
• Causality.
• Fair value.

Those interpretations require additional evidence.

Current Production Role

The production feature set includes measurements related to:

• KOI.
• K-star.
• K-star relative to KOI.
• Field center.
• Field width.
• Implied-volatility level.
• Implied-volatility slope.
• Surface fit quality.
• Price trend and curvature.

The model standardizes and combines these features by horizon.

The exact learned coefficients and artifacts remain proprietary.

The public methodology explains how the measurements enter the broader state-estimation system.

Call And Put Structure

Calls and puts can be summarized separately.

This can reveal that their inventory centers, widths, or maturity distributions differ.

However, separate call and put centroids still do not reveal one directional vote.

A call contract can represent:

• A long speculative call.
• A covered call.
• A spread leg.
• A dealer inventory position.
• A hedge against another instrument.

The same ambiguity applies to puts.

For this reason, the current public interpretation avoids statements such as:

call centroid above spot -> bullish
put centroid below spot -> bearish

The measurements can enter a learned model, but their economic sign must be established through validation rather than contract labels.

Strike Space And Moneyness Space

Centroids can be calculated in raw strike space or in a normalized coordinate.

Raw strike space is intuitive for one asset on one date.

It becomes difficult to compare across:

• Different spot levels.
• Long histories.
• Different assets.
• Different strike grids.

A normalized moneyness coordinate can use a form such as:

m = log(strike / spot)

This expresses structural location relative to current price.

Benefits include:

• Better cross-date comparison.
• More stable distance interpretation.
• Easier cross-asset analysis.
• Reduced dependence on nominal price level.

The public structure chart converts selected centers back into price space so readers can interpret them directly.

Maturity Aggregation

A single aggregate centroid can hide important maturity structure.

For example:

• Near-dated inventory may cluster below spot.
• Medium-dated inventory may center near spot.
• Long-dated inventory may sit above spot.

One overall average can fall near spot even though the maturity surface is strongly fragmented.

The production system therefore keeps horizon-specific measurements.

An aggregate center can still be displayed as context, but it should not replace the per-expiry or per-bucket view.

Time-Series Transformations

A raw centroid level is only one feature.

The research also examines transformations such as:

• Daily change.
• Smoothed level.
• Smoothed change.
• Adaptive Z-score.
• Distance from spot.
• Distance between centers.
• Ratio between centers.
• Trend and curvature.
• Dispersion around the center.

These transformations ask different questions.

Examples:

center level
-> where is structure concentrated?

center change
-> how is the structure moving?

spot minus center
-> how displaced is price?

center A minus center B
-> how do two structural lenses disagree?

No one transformation is inherently the signal.

The production Ridge and Kalman layers determine how the feature relationships contribute to the estimated horizon state.

Data-Quality Problems

Centroid calculations depend on the option chain.

Potential problems include:

• Missing strikes.
• Stale open interest.
• Sparse expiries.
• Invalid implied volatility.
• Duplicate contracts.
• Changes in weekly listing density.
• Corporate actions.
• Different data-provider conventions.
• Extremely small total weight.

A centroid can be mathematically valid and economically meaningless when the underlying distribution is sparse or corrupted.

Production calculations therefore require:

• Finite weights.
• Minimum coverage.
• Consistent contract filtering.
• Stable moneyness limits.
• Bucket completeness.
• Validation against source snapshots.

How To Read Centroid Charts

A structure chart can show:

• Current price.
• Per-expiry positioning center.
• Per-expiry volatility center.
• Aggregate positioning center.
• Aggregate volatility center.
• Horizon ranges.
• Open-interest composition.

The correct reading order is:

1. Identify current price.
2. Observe how centers vary by expiry.
3. Check whether one expiry dominates open interest.
4. Compare near and long maturities.
5. Note whether the aggregate center hides fragmentation.
6. Read the forward expectation bands separately.
7. Avoid converting one crossing into a trade command.

The chart is a cross-sectional snapshot.

It does not show the time-series path unless multiple reports are compared.

Cross-Asset Limitations

Centroid behavior cannot be assumed to transfer unchanged from SPY to another asset.

Different assets have different:

• Option liquidity.
• Strike spacing.
• Weekly-expiry availability.
• Institutional use.
• Earnings or event risk.
• Skew.
• Open-interest concentration.
• Data quality.

Each asset requires separate preprocessing and validation.

A shared centroid formula can be portable.

Its statistical meaning is not automatically portable.

Current Conclusion

Probability-mass centroids are a durable part of Vyreon's market representation.

Their value comes from stable structural compression, not from treating them as magic price levels.

The strongest public claim is:

The options matrix contains measurable location and dispersion structure. Weighted centroids provide practical, causal summaries of that structure and become useful when interpreted inside a broader probabilistic model.