FinancePy Merton jump pricing: a zero probability can erase an option value

Watch the 37-second English evidence video on YouTube. Original GERO graphics and synthetic narration; current-source experiment, synthetic worksheet, no measured customer losses.

Zenodo · DOI 10.5281/zenodo.22916584 · Hugging Face · LinkedIn summary

GitHub report · Evidence archive

Archive SHA-256: 6b1611caee952ec8f0ed05db46c682f49de7f7f6e35149479f84fea5c57f25df. Original English report and executable evidence. Verified distribution links appear above.

Independent GERO research by Xamit Kadirbekov, 23 September 2026. AI-assisted preparation and actual public-source execution.

Result

At Poisson exposure lambda*T=800, the current FinancePy MertonJumpDiffusion.value returns 0.0 for a call whose independent Black–Scholes value is approximately 23.50586700. All price inputs are finite. Setting the jump size to zero makes every jump multiplier one, so changing the event rate must not change the underlying process or its option value.

The investigation is pinned to current-source commit 7a68a8a8b3be507da6829793861d36865c98c547. The latest PyPI 1.1.2 wheel does not contain this module. Its version banner must not be mistaken for evidence that this defect is present in that release.

Same call, zero-size jumps	Current source	Local candidate	Independent reference
Exposure 0	23.50585896	23.50585896	23.50586700
Exposure 740	23.50770001	23.50585896	23.50586700
Exposure 745	23.95368337	23.50585896	23.50586700
Exposure 800	0.00000000	23.50585896	23.50586700

Inputs: stock 120, strike 100, expiry one year, interest 0.03, dividend yield 0.01, diffusion volatility 0.2, zero jump mean and volatility. max_jumps=4000 deliberately provides ample terms; default-budget truncation is a separate question. Small differences between the candidate and reference are the existing normal-CDF approximation, not the error counted here.

Nonzero jumps also reproduce the loss. With exposure 800, jump volatility 0.01 and log-jump mean −0.00005, the current call is 0.0, the 70-digit reference is approximately 28.14771759, and the candidate is 28.14773001.

Cause and candidate

The algorithm initializes the Poisson mass with exp(-lambda*T) and generates the next mass by multiplication. At exposure 800, the initial mass rounds to zero in double precision. Every subsequent mass consequently remains zero, including near the distribution's mode. Before complete underflow, a subnormal initial mass can already distort the entire distribution.

The local candidate evaluates subsequent masses from their log PMF, using gammaln(n+2) for the factorial term. This allows representable probabilities near the mode to recover even when the first mass is unrepresentable. candidate.patch changes this probability computation only. It is a proposed correction, not an accepted upstream fix.

Executed checks

• 112 observations: 84 zero-jump-size call/put scenarios and 28 scenarios with nonzero jump volatility.
• 52 → 0 → 52 discrepancies for current source, candidate and restoration of the original source. Restored outputs match baseline byte for byte.
• Both baseline and candidate pass all 12 functions invoked by the existing Merton jump-diffusion regression script. The full FinancePy suite was not run.
• Formal absolute tolerance: 2e-7 * max(S, K, 1). The exploratory probe's stricter 1e-6 flag also flagged the normal-CDF approximation and is not used for the final defect count.
• Python 3.12.14, NumPy 2.3.5, SciPy 1.16.3, Numba 0.67.0, llvmlite 0.49.0. One configured CPU worker; no GPU.

The independent oracle imports neither FinancePy nor SciPy. It uses mpmath at 70 decimal digits: direct Black–Scholes for zero-size jumps and a conditional expectation with independently accumulated high-precision Poisson weights for nonzero jumps. Chernoff bounds put omitted price tails below 1e-25 in those checks. This does not certify all parameters or replace a full model validation.

Measured chain: implementation → document → decision

A GERO synthetic worksheet multiplies the exposure-800 call price by a synthetic quantity of 100 and compares it with an explicitly chosen review threshold of 2000:

Variant	Worksheet value, rounded	Worksheet decision
Current	0.00	Below threshold
Independent reference	2350.59	Review
Candidate	2350.59	Review
Restored source	0.00	Below threshold

The model price comes from actual FinancePy code. The CSV, quantity, threshold and review rule are demonstration logic, not FinancePy features or a real institution's policy. No customer positions, losses, deployment or regulatory violations were established.

History, disclosure and limits

The source history examined lists the module's introduction in commit bb10c3936e078a5212694d017746453175151c09 on 16 September 2026. A bounded review of current all-state issue searches, the recent 100 pull requests, source history and the 120-entry GERO catalog found no exact existing report or correction. FinancePy issue 263 concerns Merton firm calibration and monetary scaling, a different model and root cause. This search is not an exhaustive originality guarantee.

The developer report was submitted and its exact body verified at FinancePy issue 273. No acknowledgement or acceptance is claimed. The latest released wheel's absence of this module was checked against the official PyPI hash.

The tested inputs include deliberately high Poisson exposures. The candidate does not fix or certify payoff-weighted stopping criteria, an insufficient max_jumps budget, arbitrary extreme conditional prices, Greeks, calibration across all inputs, or deployed systems. These results establish a reproducible current-source numerical defect and a measured synthetic consequence.

Sources and reproduction

• Pinned model source
• Existing upstream regression script
• Official package releases

Run run_checks.py for current/candidate/restoration and the independent oracle, upstream_checks.py for the 12 upstream tests, and measure_downstream.py for the synthetic worksheet. Raw evidence and exact dependency versions accompany the archive. Platform publication receipts are separate; a prepared file is not a completed upload.