Phase 3: implied variance — BSM, de-Americanization, MFIV, VIX replication

[NavyBlueCheese]

What was built

• black_scholes.py — BSM (Merton dividend-yield / Garman-Kohlhagen form), analytic greeks, bracketed-Brent IV inversion with static no-arbitrage checks.
• american.py — CRR binomial (European/American) and de-Americanization: FXE quotes are inverted against the American price, removing the early-exercise premium before any European machinery runs. The Brent bracket floor respects the CRR constraint sigma > |r-q| sqrt(dt).
• mfiv.py — the full VIX white-paper methodology: parity forward (with a carry-forward fallback for thin ETF chains), K0 selection robust to quoteless strikes, zero-bid wing truncation, minutes-based constant-maturity interpolation, AM/PM settlement conventions (third-Friday SPX 09:30 ET vs weeklies/ETF 16:00 ET).
• rates.py — DFF/ESTR to continuously-compounded ACT/365.
• scripts/build_implied.py — replicated VIX from scraped _SPX snapshots + the FXE MFIV (EVZ continuation), with reconciliation report; scripts/early_exercise_study.py — the section-1.5 study on library code (matches the Phase 0 scratchpad: worst case 6.57%).

Test results

104 passed, coverage 92.6%, mypy --strict clean. Ground-truth highlights: hypothesis IV round-trip over sigma in [0.001, 5.0]; greeks vs finite differences; CRR converges to BSM and the American call with q=0 equals the European (Merton); de-Americanization round-trips and naive European inversion demonstrably overstates vol on the predicted side; MFIV on a synthetic BS chain recovers sigma-squared within the measured 0.5% strike-discreteness and is insensitive to wing extension; the zero-bid truncation rule is pinned strike-by-strike.

Real-data findings (the interesting part)

1. First _SPX replication: 17.11 from 204+108 strikes (29.5/32.5-day bracket). The snapshot predates today's close, so the first matched published-VIX comparison lands with tomorrow's scrape; MAD accrues daily from here.
2. The 2026-07-08 FXE snapshot has zero-bid OTM wings on every expiry — ITM sides quoted, strip unusable. MFIV correctly refuses. Response (ADR 0002 addendum): the scrape task now fires twice daily with the mid-US-morning run as primary; term selection runs over usable expiries only (min_strip_strikes). This quote-quality regime is plausibly part of why CBOE decommissioned EVZ, and it goes in the paper.

Decisions taken autonomously (per gate delegation)

Carry-forward fallback with q = EUR overnight for FXE; K0 stepped to the nearest quoted strike when the literal white-paper K0 is quoteless; DFF/ESTR converted 365 ln(1+R/360); multi-root expiries resolved to the root with more two-sided quotes.