Derive inverse-contract variance replication for DVOL-aligned BVOL

[obchain]

Context

PR #38 / issue #37 diagnosed the +7.1 % BVOL vs DVOL bias surfaced in PR #36.

• Cheap implementation tweaks (quote source, near-expiry rule, intrinsic filter, DVOL lookahead) close none of it.
• Fitted-IV smoothing closes ~1.3 pp (→ +5.77 %), removing the Riemann-discretisation artefact on BTC's coarse strike grid.
• The residual ~5 % is structural: Deribit BTC + ETH options are inverse-quoted — payoff (K - S) / S in BTC, not (K - S) in USD. Their published mark_iv is calibrated under the inverse spec. Multiplying BTC quotes by spot and feeding them into the standard 1/K²-weighted Carr-Madan integral overstates variance by precisely the inverse-pricing offset (~3-7 % per option, ~5-7 % in vol terms after the integral).

See research/bvol-dvol-gap-diagnostics.ipynb §5 for the per-strike comparison showing quote-to-vanilla-BS ratio of 1.03-1.07 across the full 50 %-200 %-of-spot strike range.

Goal

Re-derive the Carr-Madan variance replication portfolio for the inverse-options contract spec Deribit uses, then implement, validate against DVOL on the existing 144-point dataset, and publish a "DVOL-aligned" BVOL variant.

Why this is a separate issue

The work is mathematically substantive (a fresh derivation, not a parameter tweak) and unrelated to M1 milestone close. M1 ships our CBOE-style BVOL as the public index with the documented +5-6 % DVOL gap; nothing in M1 depends on this issue resolving.

What needs doing

1. Read Deribit's DVOL whitepaper and any public-domain references on inverse-options variance replication (e.g. Bergomi, "Stochastic Volatility Modeling," or Carr-Madan extensions to inverse contracts).
2. Derive the replicating portfolio for a contract that pays Var(log S) settled in BTC. Expected outcome: kernel likely changes from 1/K² to 1/K and an extra forward-related term appears.
3. Implement as compute_dvol_inverse() in a new notebook (or extend bvol-dvol-gap-diagnostics.ipynb §4).
4. Validate against DVOL on the 144-point dataset (research/data/tardis/{ccy}/{date}/snapshot.parquet). Target: ≥95 % within ±5 %.
5. Document the resulting formula in docs/vix-notes.md alongside the CBOE-style spec (which stays as our public BVOL index).

Acceptance

• ≥95 % of 144 hourly comparisons within ±5 % vs DVOL using the inverse-contract derivation.
• Derivation written up in docs/vix-notes.md with the BTC-numeraire change-of-measure argument made explicit.
• Reference Python impl in research/.

Out of scope

• Replacing our public BVOL index with the inverse-aligned variant. BVOL = CBOE-style is the M1 deliverable; the inverse-aligned variant would be a separate "BVOL-DVOL-aligned" or "DVOL-clone" index if we choose to publish it.
• Rust port. Reference Python first, port later when public-facing.

Priority

priority:p1 — research depth without an M1 deadline. Pick up after M1 ships if there is no faster customer-facing demand.