reflexive-options

Research code for Gamma-Shaped Dealer-Book Pressure and Endogenous Volatility Cycles: A Reduced-Form Fixed-Equilibrium Hopf Model and a Pre-Extraction Predictive Protocol (Patel, 2026).

Current claim

The paper gives a local physical-measure model for detrended log price $X$, instantaneous variance $v$, and filtered price memory $\chi$:

$$ \begin{aligned} dX_t&=[-\delta X_t-\tfrac12(v_t-\theta_v)+\kappa g(X_t,v_t,\chi_t)]dt +\sqrt{v_t},dW_t^S,\\ dv_t&=[\kappa_v(\theta_v-v_t)+\gamma v_t\chi_t]dt +\xi\sqrt{v_t},dW_t^v,\\ d\chi_t&=\alpha(\beta X_t-\chi_t)dt. \end{aligned} $$

The dealer-book functional is centered so $(0,\theta_v,0)$ is an equilibrium for every coupling $\kappa$. The $v\chi$ variance feedback leaves the boundary drift equal to $\kappa_v\theta_v>0$ at $v=0$.

For a Gaussian density of signed dealer positions in fixed log moneyness, the Routh--Hurwitz Hopf determinant is quadratic in $\kappa$. The transparent, non-calibrated example reproduces

• $\kappa^\star=31.4932976,\mathrm{yr}^{-1}$,
• $\omega^\star=47.1185670,\mathrm{rad},\mathrm{yr}^{-1}$,
• $\ell_1=-6.2888041$,
• a nonlinear attracting cycle at $1.02\kappa^\star$ with $v\in[0.04453,0.08187]$.

The same quadratic also has a remote second valid point at $\kappa^{\star\star}=16860.8961,\mathrm{yr}^{-1}$, where the pair crosses back into the stable half-plane. The API returns both roots; plots and phase maps labeled “threshold” use the first.

On the closest numerical grid, the actual nonlinear cycle amplitude scales as ${(\kappa-\kappa^\star)/\kappa^\star}^{0.509}$, close to the local Hopf exponent $1/2$. A gross-normalized mixture audit is deliberately less tidy: nearby same-sign mixtures remain supercritical, but dispersed and offsetting-sign books can be subcritical, and reversing the canonical sign orientation removes the valid positive root. The existence and type of the local bifurcation are therefore book-dependent.

This is a local deterministic possibility result for a postulated gamma-shaped pressure term inside an independently assumed price--memory--variance loop. It is neither an SPX calibration nor evidence that markets are near a Hopf threshold. The canonical crossing is driven mainly by the book kernel's variance sensitivity, not delta hedging alone.

Measurement boundary

Public option open interest counts outstanding contracts. It does not identify which side is held by a dealer. The theory's signed position density is therefore latent; an OI-weighted signed GEX series is a convention-dependent proxy, not observed dealer inventory.

Amendments A13--A16 replace the former event-selected directional GEX test with a pre-extraction registered-horizon protocol. On eligible OptionMetrics dates from 2017-01-03 through 2024-10-29 it will study four observable summaries:

1. nonnegative OI-gamma mass;
2. call--put composition;
3. gamma-weighted mean log moneyness;
4. gamma-weighted log-moneyness dispersion.

The contract universe, parity/carry forward rule, rate/dividend tuple, spot and return sources, OI availability rule, fractional settlement time, liquidity filters, and duplicate/attrition policy are fixed before access. Leads, lags, HAC offsets, and bootstrap blocks remain on the complete CRSP trading-session calendar, so a missing option date cannot compress the next-session outcome. These summaries predict next-session log squared CRSP returns with 22-session inference horizons, official Cboe VIX, log spot, a linear session trend, outcome-session weekday indicators, and one regressor-session monthly-expiration control. HAC and moving-block-bootstrap p-values are BH-adjusted as separate families. “Robustly associated” requires both adjusted p-values below 0.05 and a 95% bootstrap interval excluding zero; a one-family rejection is explicitly method-sensitive. Convention-signed GEX and stress-window interactions are secondary. Even a robust association is not causal or dealer-sign identification.

No registered WRDS, OptionMetrics, CRSP, or VIX dataset has been extracted or analyzed in the project. The 2017--2024 market path and named stress episodes were historically public when the plan was written; this is a retrospective pre-analysis plan, not a blinded prospective experiment. Access is expected in September 2026.

Reproduce the current result

uv sync --locked --all-extras --group dev

# Core analytic and empirical-protocol checks
uv run pytest -q tests/test_centered_model.py tests/test_oi_proxy_protocol.py

# Rebuild the actual-nonlinearity validation figure
uv run python -m reflexive_options.experiments.centered_hopf_validation

# Rebuild amplitude, sensitivity, mixture, and derivative robustness checks
uv run python -m reflexive_options.experiments.centered_model_robustness

# Rebuild the paper
cd paper && make pdf

# Full repository verification
cd .. && bash scripts/verify.sh

The principal implementation is in:

• src/reflexive_options/theory/centered_model.py
• src/reflexive_options/experiments/centered_hopf_validation.py
• src/reflexive_options/experiments/centered_model_robustness.py
• src/reflexive_options/empirical/oi_proxy_protocol.py
• paper/main.tex
• paper/pre_registration_amendments.md (preserved A13--A14 record)
• paper/pre_registration_amendment_a15.md (calendar and disclosure correction)
• paper/pre_registration_amendment_a16.md (timing, settlement, inference, and measurement correction)
• docs/wrds_day_one_validation_plan.md

What was withdrawn from v0.3

The repository retains older modules and artifacts for reproducibility, but the current paper does not claim:

• a non-zero stochastic threshold correction from the affine additive surrogate; its tangent cocycle is $e^{Jt}$ and the correction is zero. The full state-dependent stochastic variational equation remains unanalyzed;
• a Hawkes--SV equivalence theorem;
• global stability from absence of a Hopf root;
• a McKean--Vlasov threshold theorem for the centered model;
• an information-theoretic critical-edge theorem;
• stationary-tail or bimodality consequences of the local Hopf result;
• that synthetic RL, CSD, or event-window exercises validate the market mechanism.

Affected modules are marked archived/exploratory. The obsolete 4D noise scan is no longer an installed command.

Repository map

src/reflexive_options/
├── theory/          # centered current model plus archived legacy utilities
├── empirical/       # A13--A16 OI proxy utilities and legacy A9 reproducer
├── experiments/     # current validation plus archived experiments
├── simulator/       # legacy/full simulator infrastructure
├── baselines/       # comparison simulators
├── surface/         # IV-surface and arbitrage-filter utilities
├── rl/              # exploratory agent infrastructure
└── third_party/     # vendored ATLAS/RAT code

paper/
├── main.tex
├── main.pdf
├── references.bib
├── pre_registration.md
├── pre_registration_amendments.md
└── variants/

Pre-registration provenance

The original document and earlier amendments retain their historical OpenTimestamps proofs. A13--A16 are disclosed pre-extraction amendments made before anticipated WRDS access. The preserved A13--A14 amendment-file SHA-256 is 603e89366c0dbe49718e8c31f805d6f85d3c508e2e0ed4276a6310c80f5f9cd7, with receipt paper/pre_registration_amendments.md.ots pending Bitcoin consolidation. The A13-only snapshot paper/pre_registration_amendments.md.a13 remains independently verifiable at hash 83950ede7049cec9842246cb291307a144bd620923898e9793c2f1197f558e17 with receipt paper/pre_registration_amendments.md.a13.ots. Do not rewrite either historical state to imply that a later clarification existed earlier. The separate A15 correction has SHA-256 a5f694f99953d57563d4f17dc5646ef0b87452c45119ccbdc12fc90efd034a52 and receipt paper/pre_registration_amendment_a15.md.ots, pending Bitcoin consolidation at creation. The separate A16 correction has SHA-256 a5cbf9ef56c9a402ff05b61bb720d8487313b154f30393b4835b43fe5c33e61d and receipt paper/pre_registration_amendment_a16.md.ots, pending Bitcoin consolidation at creation.

Citation

@unpublished{patel2026dealergamma,
  author = {Patel, Mahimn},
  title  = {Gamma-Shaped Dealer-Book Pressure and Endogenous Volatility Cycles:
            A Reduced-Form Fixed-Equilibrium Hopf Model and a Pre-Extraction
            Predictive Protocol},
  year   = {2026},
  note   = {Working paper, v0.4.1},
  url    = {https://github.com/mahimn01/reflexive-options}
}

License

Code is MIT-licensed. The manuscript and its original figures are licensed under CC BY 4.0; see paper/LICENSE.md. Vendored ATLAS/RAT modules derive from mahimn01/trading-algo; see NOTICE.