Positive-payoff quadrature for Black–Scholes: a frozen binary64 prototype

Abstract

We study an accuracy-oriented binary64 implementation of the positive-payoff integral for Black–Scholes–Merton option valuation. A locally frozen candidate and 288 new synthetic confirmation inputs are compared with an exact-binary multiprecision erfc reference, our previous prototype and an unchanged historical Jäckel implementation. All registered practical price/log/API gates pass. Among 202 nonzero reference prices, v2 is closer than our previous prototype in 192 cases, farther in 3, equal in 7; against the historical source: 146/30/26. A separate timing check finds a 15.4-fold slowdown relative to our previous Python detailed API; native C++ throughput is not measured. This is a bounded computational result, not a new financial model, novelty claim, universal accuracy guarantee or improvement in market prediction. Independent reproduction is invited; no human peer review is claimed. Authors: Khamit Kadyrbekov and Daniyal Kadirbekov. Khamit directs the research programme; Daniyal is designated to lead continuing software development. This prospective role does not claim that he independently implemented or validated the frozen v2. OpenAI Codex materially assisted analysis, implementation, tests, result checking and drafting. Code and complete evidence: https://github.com/kadyrbekovhamit-cyber/gero-numerical-observatory/tree/main/reports/black-scholes-positive-payoff-v2 Original text/data: CC BY 4.0. Original code: MIT. Included Jäckel/Cody source retains its original notices and terms; see vendor/jackel/PROVENANCE.md. Video summary: https://youtube.com/shorts/F6Wx5byxJtM Research announcement and discussion: https://www.linkedin.com/feed/update/urn:li:activity:7509495438547333120/