Strata: a zero-volatility Hessian prevents custom-model option pricing

English video · GitHub report · Hugging Face evidence · Zenodo DOI 10.5281/zenodo.22840914 · LinkedIn discussion · Vendor report · Evidence ZIP

Archive SHA-256: 3a6a81f884c71c72d76a93540140ec12ce2ad9f65e67267a50bce4fb3e93eff5. Original English report and executable evidence; video is not yet produced.

Xamit Kadirbekov · Independent GERO research · 19 September 2026 (Asia/Tashkent)

A finite option price does not guarantee that its derivatives are usable. In the official OpenGamma Strata 2.12.74 implementation, priceAdjoint2(110, 100, 1, 0, true) returns price 10 and correct first derivatives, but its entire 3-by-3 Hessian is NaN. Away from the payoff kink, the zero-volatility boundary has zero second derivatives.

The discrepancy also interrupts another actual Strata calculation. Through the documented custom-volatility-provider interface, a constant-zero test model causes SabrExtrapolationRightFunction to throw while constructing its price extrapolator. A narrow Black-formula correction restores the native constructor and subsequent price outputs; restoring the original implementation reproduces the interruption.

Executed chain: derivative → fitting decision → unavailable price

The following chain was executed on the official release, pinned current classes, local candidate and restored original:

1. BlackFormulaRepository.priceAdjoint2 produces a NaN strike curvature at exact zero volatility.
2. SabrExtrapolationRightFunction.computesFittingParameters incorporates it into priceK[2]. Its existing small-price fallback requires all three values to be small; the NaN comparison fails.
3. The native root bracketer then throws MathException: Failed to bracket root: function invalid at x = -1.0 f(x) = NaN. The pricing object is not constructed, so the requested prices cannot be returned through it.

For forward 0.03, cutoff 0.06, expiry 1, tail parameter 4 and a constant-zero volatility provider:

Measured output	Released / current / restored	Local candidate
Construct native extrapolator	Exception	Success
Call price at strike 0.015	Unavailable through this object	0.015
Put price at strike 0.09	Unavailable through this object	0.06
Call price at strike 0.09	Unavailable through this object	5.6699831977150385e-40

The six zero-provider scenarios use forwards 0.03, 0.05 and 1, cutoff twice the forward, and expiries 1 and 5. 6/6 constructions fail before, 0/6 fail with the candidate, and 6/6 fail after restoration. Fifteen other downstream scenarios retain identical outputs.

The constant provider is a GERO test fixture with exact zero first and second derivatives. The constructor, fitting logic, exception, fallback and pricing methods are actual Strata code. This is evidence about a supported extension path, not a deployed bank integration. The default Hagan provider floors its second-adjoint volatility at 1e-6 (or can become NaN earlier), so these results do not demonstrate this exact failure in the default Hagan/CMS configuration.

The native fallback uses parameters [-100, 0, 0]. Its positive call tail is an intentional small approximation, not exactly the deterministic zero price. The test does not claim that every downstream sensitivity is repaired; SABR-parameter sensitivities and other fitting regions were not measured.

Independent mathematical result

For positive finite forward F, strike K and expiry T, with F different from K, the zero-volatility payoff is locally linear in F and K. At F=110, K=100, T=1, a call therefore has price 10, gradient [1,-1,0,0] and zero Hessian in [F,K,volatility]. Volatility derivatives are interpreted from the right in the nonnegative domain.

For fixed non-ATM log-moneyness m=log(F/K), normal density in the Black formula contains exp(-m*m/(2*sigma*sigma*T)). As sigma approaches zero from above, this term vanishes faster than any inverse power of sigma. The mixed and volatility second derivatives tend to zero as well. This reasoning excludes ATM; assigning a zero Hessian at its payoff kink is not supported.

The existing second backward sweep divides by powers of volPeriod and combines zero densities with infinite intermediates. The candidate adds the ordinary deterministic limit before this sweep, restricted to finite positive inputs, exact zero volatility, F and K <= the class's LARGE threshold, and abs(F-K) >= its SMALL threshold. Near-ATM, large-level reference conventions and all positive-volatility paths are retained.

Versions and controls

• Official report-tool release: 2.12.74, JAR SHA-256 0fb4c6c778f25b7c6fe13c001f47a4e404868df55e4dd9841c752128bb864125.
• Current source pin: 987932ee95bf53e2baaff9a6b8e738a00f558b10.
• Black source SHA-256: 029f9e8708292cc015c5302d0119abfed1760eaa2d6ddc0fcb9274fba32dd22c.
• Execution: macOS arm64, OpenJDK 25.0.4.1, one configured JVM processor with SerialGC and interpreted execution. Current affected classes were compiled against released dependencies; this is not a complete build of main.

Direct test state	Non-ATM zero-volatility failures
Official release	180
Pinned current source	180
Local candidate	0
Original restored	180

The direct grid has 866 distinct input rows. All 686 non-target rows remain identical. Within those controls, 644 positive-volatility rows agree with an independently formulated 80-digit mpmath oracle within relative tolerance 2e-8 and absolute tolerance 1e-10. Fourteen rows reuse existing upstream test inputs; no full upstream JUnit run is claimed.

Remaining rows deliberately preserve 12 tiny-positive-volatility stress failures at 1e-200, six ATM conventions and 24 boundary controls. Downstream, six tiny-positive failures and the default-alpha-zero failure likewise remain unchanged. These are not counted as fixed. The 21-row downstream grid extends the evidence for the same implementation defect; it is not 21 new defects or a second discovery.

The archive contains the exact minimal Java snippet, candidate patch, pinned source, test fixtures, independent oracle, all raw output sets and replay receipts. No production frequency, bank/customer exposure, charges, losses, security issue or legal violation was measured. The effect demonstrated here is loss of availability of a pricing calculation under the stated synthetic custom-model conditions.

Sources, duplicate review and disclosure

• Pinned Black implementation.
• Native extrapolator fitting.
• Documented custom-provider interface.
• Hagan second-adjoint floor.
• Historical work PR843 introduced the method; PR939 changed the Hagan floor and separate finite differences. Both are credited, not presented as our work.

Bounded all-state searches included issue/PR titles, bodies and comments, actual relevant diffs, public documentation and the GERO catalog. No exact previous report or proposed correction for this specific Black zero-volatility Hessian was found. Private, unindexed or differently described reports may not have been captured. This is separate from the earlier Normal-IV initial-guess case.

Vendor-first report: OpenGamma Strata issue2797. Sent and publicly verified before distribution of this report. No acknowledgment or acceptance is claimed. Source review, test/code preparation and documentation were AI-assisted.