I. Outside the Benchmark Lens
In the previous essay, we defined the Numerical Stability Index (NSI) over a standardized temporal window: 15 to 45 Days to Expiration (DTE).
That boundary was deliberate. It isolates the active institutional surface while excluding structural distortions introduced by extreme maturities.
We now examine the region intentionally excluded from the benchmark: contracts with two days or fewer to expiration (DTE ≤ 2).
The 0DTE market is often discussed in terms of behavioral speculation or gamma-driven price impact. Here, we examine it exclusively through the lens of computational topology.
What emerges is not a reaction to macroeconomic stress. It is a structural boundary condition.
II. The Geometry of Expiration
As time to expiration \( T \to 0 \), the geometry of the pricing function transforms.
In classical option pricing frameworks:
- Vega scales approximately with \( \sqrt{T} \). As expiration approaches, Vega collapses toward zero.
- Gamma increases sharply near-the-money, scaling inversely with \( \sqrt{T} \).
- Vomma, the curvature of Vega, similarly concentrates near expiration.
The stabilizing and destabilizing forces of the Stability Signal therefore move in opposite directions.
Recall:
\[ s = \frac{\text{Pricing Error} \cdot \text{Vomma}}{\text{Vega}^2}. \]
As \( T \to 0 \):
- The denominator (\( \text{Vega}^2 \)) shrinks rapidly.
- The numerator (error × curvature) remains finite or increases near-the-money.
The stabilizing cushion disappears.
Even modest pricing errors interact with concentrated curvature to produce large values of \( s \). Contracts migrate into Divergent and Singular regimes not because of sentiment, but because of asymptotic geometry.
The fragility of the ultra-short surface is therefore not episodic. It is mathematically inevitable.
III. Finite-Precision Consequences
When Vega collapses toward machine-scale magnitudes, computational realities become dominant.
In floating-point arithmetic:
- Division by small slopes amplifies rounding error.
- Linear approximations become highly sensitive to perturbation.
- Step magnitudes become unstable relative to scale.
Convergence may still occur.
But it occurs with reduced numerical cushion.
Intermediate iterates experience elevated Path Stress. Greeks computed mid-trajectory exhibit heightened volatility. Stability is no longer governed primarily by curvature; it is constrained by hardware limits.
The solver operates at the boundary between mathematical geometry and machine precision.
IV. Empirical Confirmation
We now apply the NSI aggregation methodology exclusively to the 0DTE lens under calm market conditions.
Figure 12. Structural separation between institutional benchmark and ultra-short boundary condition (Q1 2020 average). - The 15–45 DTE lens resides within a high-integrity band, while the 0DTE boundary operates in a persistently lower structural range due to expiration asymptotics.
Figure 13. Institutional benchmark vs. 0DTE boundary (Q1–Q2 2020). - The 0DTE lens remains structurally lower across regimes, reflecting expiration geometry rather than macro sentiment. The institutional benchmark compresses during systemic shock but recovers as curvature concentration dissipates.
Even in calm markets:
- The 15–45 DTE benchmark frequently resides in the high 80s or 90s.
- The 0DTE lens typically operates in the 50s.
This divergence is not temporary.
The ultra-short surface is not unstable because markets are stressed. It is unstable because time is nearly exhausted.
V. Structural Property vs. Sentiment
Interpretation requires discipline.
If the 15–45 DTE benchmark were to decline to the low 50s, it would indicate severe macro-structural compression, comparable to systemic shocks.
An NSI in the 50s within the 0DTE lens, by contrast, is structurally typical.
This distinction reinforces a central principle of Numerical Governance:
Computational stability is distinct from market sentiment.
The fragility of the 0DTE surface reflects a geometric boundary condition. It is a property of expiration asymptotics, not an indicator of fear.
VI. The Measurement Problem
The 0DTE Fragility Report exposes a methodological hazard.
We now observe two structurally distinct regimes:
- A stable institutional benchmark (15–45 DTE).
- A persistently hostile ultra-short regime (≤2 DTE).
If a risk system attempts to compute a single “overall” stability metric by averaging the entire listed surface, the result becomes uninterpretable.
The stability of long-dated contracts and the fragility of ultra-short contracts mathematically dilute one another.
The blended metric masks localized structural risk.
In the next essay, we formalize this measurement problem. We examine expiry dilution and demonstrate why full-surface aggregation inherently fails to capture computational topology.