Interest Rate & FX Derivative Pricing — Multi-Curve / Hull-White / Black-Bachelier / SABR Family

Not all members are production-ready. The optionality and cross-currency members rest on uncalibrated placeholder inputs (A-054 FX correlation is default-zero; DS-058 swaption-vol / FX-basis source is planned, with the strike-grid swaption IV history a structural paid-only gap — CME DataMine barred by DR-007). Per Decision 053 §4 these inputs are not fabricated; instead the affected members are scoped non-production until calibration lands.

The non-production members re-promote via 2L re-review once A-054 is calibrated and/or DS-058 is sourced (tracked under PL-07 / WS3). The mirror of this table lives in model_registry.yaml::M-132.production_scope. The swaption snapshot below is retained as an illustrative (non-production) example, clearly labelled.

M-132 is the interest-rate and FX derivative pricing family (Tier 2). It prices linear rate products, cross-currency products, FX forwards, and the optionality layered on top of them. The family is FinView-resident: every engine lives under finview.L0/finview.L1 in [redacted], and there is no InsModel-side implementation — the InsModel virtual-env merely re-runs the same FinView source (confirmed reproducible below).

Multi-curve discounting (the foundation)

All members share one discounting spine. Curves are bootstrapped from par instruments by finview.L0.curve_builder (finlib/curve_builder.py::bootstrap_sofr_curve and build_sofr_curve):

• Maturities ≤ 1Y are stripped as simple money-market instruments (df = 1 / (1 + cT)); maturities > 1Y are stripped by sequential semi-annual coupon bootstrapping, treating each input as a par rate.
• Zero rates are continuously compounded (z = -ln(df)/T); inter-tenor forward rates are computed from adjacent discount-factor ratios.
• Interpolation is log-linear on discount factors by default, with an optional PCHIP (monotone cubic Hermite) mode that produces smooth, non-oscillating forwards (_interpolate_df, interpolate_curve).

The multi-curve framework proper is the CurveSet dataclass (finview.L0.curve_set): it holds separate Treasury, OIS/SOFR discount, and SOFR projection curves plus rating-keyed credit-spread curves, with graceful fallback to the Treasury curve when a discount or projection curve is absent (has_multi_curve, get_discount_dfs, get_projection_fwds). This is the OIS-discounting / forward-projection separation that post-LIBOR rate pricing requires: the float leg is projected off the projection curve while cashflows are discounted off the OIS curve.

Family members (engines)

Gold-copy challengers exist for the derivative engines (finlib/gold_test_derivatives.py, finlib/gold_test_derivatives frozen reference values, e.g. CME_ERIS_10Y) and provide an independent cross-check lane for the IRS/futures math.

Representative pricer in detail — vanilla IRS (IRSPricer)

The cleanest-running, most load-bearing member is the vanilla interest-rate swap, so it is the snapshot exemplar. IRSPricer.compute_fair_value:

1. Builds the semi-annual payment grid to maturity (_payment_times).
2. Discounts each payment time off the input curve, re-deriving zero rates from the discount factors and re-interpolating (_discount_at).
3. Fixed leg PV = N · c · Δt · Σ df(tᵢ).
4. Float leg PV = N · Δt · Σ fwd(tᵢ) · df(tᵢ) when a projection forward curve is supplied (the multi-curve path); otherwise it collapses to the single-curve par identity N · (1 − df_last).
5. DV01 is computed by full reprice under a parallel +1bp bump of zero and forward rates (_compute_dv01) — a genuine bump-and-reprice, not a closed-form approximation.
6. Key-rate DV01 bumps each of the {1,2,5,10,20,30}Y buckets with a triangular weighting kernel (_compute_key_rate_dv01, _triangle_weight) and reprices, giving the bucketed rate exposure.

The par swap rate and the annuity are emitted directly by compute_fair_value. The engine builds the annuity as Δt · Σ df(tᵢ) and the par rate as float_leg_pv / (N · annuity) — the rate that zeroes the swap — and returns both, exposing par_rate, its consumer-facing alias par_swap_rate, and annuity in the result dict (irs_pricer.py lines 18, 86–95, sha 22d4662, FinView #92, 2026-06-04). The snapshot reads these straight from the engine output rather than re-deriving them; the contract's declared outputs and the returned dict now agree.

Hull-White / SABR for optionality

Optionality is priced two ways in-repo:

• SwaptionPricer computes the forward par swap rate and the annuity off the curve, then applies either Black-76 (lognormal) or Bachelier (normal) to the forward, returning premium, delta and vega. This is the standard market-model swaption, reusing the swap annuity rather than re-deriving swap math.
• SABR (Hagan's approximation) lives in vol_surface.py (sabr_implied_vol(F, K, T, alpha, beta, rho, nu), calibrate_sabr, SABRSurface). It produces the smile-consistent implied vol at a given strike that would feed the Black/Bachelier swaption — i.e. SABR supplies the vol surface, the swaption pricer consumes it. The snapshot shows the ATM SABR lognormal vol alongside the Bachelier-priced swaption to make the surface→pricer handoff explicit.
• Hull-White 1-factor short-rate engine (finview.L1.hull_white_1f, derivatives/hull_white_1f.py, INV-006 1L build #2434, 2026-06-08) models the short rate as dr = [θ(t) − a·r] dt + σ dW. The time-dependent drift θ(t) is implied by the exact A(t,T)/B(t,T) affine reconstruction so that P_hw(0,T) ≡ P_market(0,T) by construction — i.e. the model is calibrated to the canonical curve by identity, making it a genuine no-arbitrage term-structure model rather than a comment reference. It derives the instantaneous forward f(0,t) = −d/dt ln P(0,t) analytically from the pchip log-DF interpolant (NOT curve_builder's discrete simple forward_rates[] — the forward-curve fix mandated by the 1L design review). It prices a European ZCB option analytically (HW bond-option formula), hence caplets/floorlets, and a European swaption via the Jamshidian (1989) decomposition (solve for r* that makes the coupon bond hit the strike; decompose into a portfolio of ZCB options). The analytic prices are independently verified against an exact-transition Monte-Carlo simulator in the test suite. The HW analytic swaption is cross-checked against the existing Bachelier swaption — the family's first genuine cross-model challenger lane for optionality (documented soft, non-gated tolerance because a,σ are uncalibrated). a (A-284) and σ (A-285) are exogenous, caller-supplied, uncalibrated inputs (literature-typical defaults a=0.03, σ=0.01); the bond/curve analytics are exact (production) and the option outputs are non-production until calibration lands (DS-058 strike-grid swaption IV is the paid-only DR-007 gap). See Limitations 1 and the Production Scope table.

Prose. The discounting assumptions are the post-LIBOR consensus: OIS/SOFR discounting with a separable projection curve. The two material modelling choices a validator should scrutinise are (a) the uniform day-count shortcut (Δt = 1/frequency rather than true accrual fractions), which the curve_builder docstring itself flags, and (b) the absence of a convexity adjustment for any futures-derived calibration point. Neither is hidden; both are documented in code and carried into Limitations. The vol assumptions (Bachelier default + SABR β=0.5) are deliberately the low-rate-regime defaults, which is correct for a USD swaption book.

Run via python scripts/model_snapshots.py M-132. The snap_M_132() provider resides in InsModel (ecosystem/InsModel/Models/scripts/model_snapshots.py) and path-inserts FinView ([redacted]) before importing the finlib engines — the model is FinView-resident but the snapshot harness is InsModel-side (BV-032 cross-repo pattern). It re-runs IRSPricer and SwaptionPricer against a frozen par curve. The IRS/swaption/curve engines are regression-asserted by tests/test_irs_pricer.py, tests/test_swaption_pricer.py, and tests/test_curve_builder.py (34 passed at capture).

Instrument: 5-year pay-fixed vanilla IRS, $10,000,000 notional, semi-annual, fixed at 4.50% (deliberately off-market) on a frozen upward-sloping SOFR/Treasury par curve (4.30% → 4.45% across 3M→30Y).

Prose. The signs and magnitudes are internally coherent: an off-market pay-fixed swap struck ~47bp above the 4.03% par rate carries a negative MV of roughly -$213k, which is close to DV01 × (4.50% − 4.0255%) × 10000 ≈ $4,545 × 47.45 ≈ -$215k, i.e. the NPV reconciles with the reported DV01 to within bootstrap/interpolation curvature. The key-rate DV01 correctly places exposure in the 2Y–5Y buckets and zero beyond 10Y (a 5Y swap has no 10Y+ sensitivity). The swaption's forward par rate (4.0071%) matches the spot par rate to a few bp, as expected on a gently-sloped curve, and the premium is consistent with a near-ATM 1Yx5Y payer at 75bp normal vol. The SABR ATM vol is shown as an illustrative surface input — it parametrises the smile that would supply the Black-76 vol, not a calibrated production surface.

Captured 2026-06-04 · deterministic, no live data.

1. Hull-White 1F is implemented but its vol parameters are uncalibrated (INV-006, 2026-06-08). A real Hull-White 1-factor short-rate engine (finview.L1.hull_white_1f) is now built and bound to M-132 (resolving the prior "not implemented" gap). It is calibrated to the canonical curve by construction (exact A/B refit; the curve/bond analytics are production), but its mean-reversion a (A-284) and short-rate vol σ (A-285) are exogenous, caller-supplied, and NOT calibrated to market swaption vols — the DS-058 strike-grid swaption IV history is the structural paid-only DR-007 gap (CME DataMine barred). Per Decision 053 §4 these are explicit documented inputs, not default-zero fabrication. Consequently the engine's option outputs are scoped non-production (illustrative until a,σ calibration lands; same honest pattern as the swaption member), while the bond/curve analytics are exact. The HW-vs-Bachelier swaption cross-check is therefore a soft, non-gated sanity/challenger lane, not a pass/fail on absolute price. Single-factor HW cannot fit the full vol surface/smile (that stays SABR's job) and is Gaussian (admits negative rates — acceptable for the post-LIBOR USD regime, consistent with the family's Bachelier default). Multi-curve basis inside HW is out of scope for v1 (uses the single discount curve). Genuinely multi-factor / Bermudan-callable term-structure needs remain future work.
2. Uniform day-count. Accrual fractions are taken as 1/frequency rather than true act/360, act/365, or 30/360 day-count fractions. The curve_builder governance docstring states this ("uniform day-count convention (actual/actual assumed)"). It introduces small accrual bias on stub and broken-dated trades.
3. No convexity adjustment. The curve builder applies no convexity adjustment for any futures-based calibration input (also stated in the curve_builder docstring). Futures-strip calibration would be biased at the long end without it.
4. Par rate / annuity engine emission — RESOLVED (2026-06-04). Previously IRSPricer.compute_fair_value did not return par_rate/annuity despite their appearing in the ENGINE_CONTRACT outputs list, so the snapshot re-derived them. This contract/implementation mismatch was resolved by FinView #92 (sha 22d4662): compute_fair_value now returns par_rate, the par_swap_rate alias, and annuity directly (irs_pricer.py lines 18, 86–95), and the snapshot reads them from the engine output. No open gap.
5. SABR surface is illustrative, not calibrated end-to-end in the pricer. sabr_implied_vol produces a smile-consistent vol, and calibrate_sabr can fit (α, ρ, ν), but SwaptionPricer takes a scalar vol input — it does not yet pull a strike-specific vol from a calibrated SABRSurface automatically. The surface→pricer wiring is manual.
6. Single-curve fallback is silent. When no projection/discount curve is supplied, IRSPricer collapses the float leg to the par identity N·(1−df_last) and CurveSet falls back to the Treasury curve without warning. This is convenient but can mask a missing-curve configuration.
7. Cross-repo / FinView-resident. This model is grounded entirely in FinView Python; there are no 10-K-derived inputs and no InsModel-side engine. The InsModel venv reproduces the numbers only because it imports the same FinView source. Governance treats this under BV-032 (cross-repo FinView-resident model) — the card is descriptive of in-repo code, not of a productionised market-data pipeline.

Ratification status (RAT-132-v1.0.0 remediation, 2026-06-06). - INV-006 / COND-001 — CLOSED (doc-correction). "Hull-White" was dropped from the registry name + description and this card title + description; the declared methodology now matches the implemented SABR-fed Black-76/Bachelier code. The short-rate-engine alternative was not built (would be re-submitted for re-review if it is). See Limitation 1 and the Name-correction callout. - COND-002 — CLOSED (honest scoping). A-054 (FX correlation, default-zero) and DS-058 (swaption vols + FX basis, planned; strike-grid IV is the paid-only DR-007 gap) are not fabricated. The placeholder-dependent members (swaption, rate-option, xccy) are scoped non-production — see the Production Scope table at the top and model_registry.yaml::M-132.production_scope. Calibrating A-054 / sourcing DS-058 stays an honest pending build (PL-07 / WS3). - Validation evidence — Tier-2 pack assembled at modelling/validation_evidence/M-132/v1.0.0/README.md (real tests + reconciliation + gold-copy challenger lane; honest gaps preserved). The (T1) full pack is not required at Tier-2 (charter §4.2). - INV-032 — still tracked (non-gating, LG-132-03): FinView-resident model consumed by InsModel via sys.path insertion; no packaged interface / cross-repo drift gate.

Gaps relative to a full Tier-2 packet: no calibrated production vol surface, no HW short-rate validation, no day-count/convexity sensitivity tests. These trace directly to Limitations 1–3 and 5.

Multi-curve / OIS discounting: - Henrard, M. (2014). Interest Rate Modelling in the Multi-Curve Framework. - Bianchetti, M. (2010). "Two Curves, One Price." - Mercurio, F. (2009). "Interest Rates and the Credit Crunch: New Formulas and Market Models." (post-LIBOR projection vs discount separation)

SABR / smile: - Hagan, P., Kumar, D., Lesniewski, A., Woodward, D. (2002). "Managing Smile Risk." Wilmott — the SABR approximation implemented in vol_surface.py::sabr_implied_vol / calibrate_sabr. - Obłój, J. (2008). "Fine-tune your smile: correction to Hagan et al."

Hull-White / short-rate (implemented — finview.L1.hull_white_1f, INV-006 2026-06-08): - Hull, J. & White, A. (1990). "Pricing Interest-Rate Derivative Securities." - Brigo, D. & Mercurio, F. (2006). Interest Rate Models — Theory and Practice, ch. 3 (Hull-White affine bond/option formulas, §3.11 Jamshidian). - Jamshidian, F. (1989). "An Exact Bond Option Formula." J. Finance — the coupon-bond-option decomposition used for the analytic HW swaption. - Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering — exact Gaussian short-rate transition (test-only independent estimator).

Engine files (FinView-resident): - finlib/curve_builder.py — bootstrap_sofr_curve, build_sofr_curve, CurveSet (finview.L0.curve_builder / curve_set / interpolation). - finlib/derivatives/irs_pricer.py — IRSPricer (finview.L1.irs_pricer). - finlib/derivatives/swaption_pricer.py — SwaptionPricer (finview.L1.swaption_pricer). - finlib/derivatives/rate_option_pricer.py, inflation_swap_pricer.py, fx_forward_pricer.py, xccy_swap_pricer.py — remaining family members. - finlib/vol_surface.py — SABR (sabr_implied_vol, calibrate_sabr, SABRSurface) and SVI. - finlib/gold_test_derivatives.py — gold-copy challenger reference values.

Tests: - tests/test_irs_pricer.py, tests/test_swaption_pricer.py, tests/test_curve_builder.py, tests/test_fx_forward_pricer.py, tests/test_inflation_swap_pricer.py, tests/test_rate_option_pricer.py, tests/test_gold_derivatives.py, tests/test_hull_white_1f.py (the 12 HW acceptance tests — INV-006).

Internal: - BV-032 — cross-repo FinView-resident model boundary (no 10-K inputs; FinView Python is the source of truth for this card).

Card change history. Code-side change history lives in git log of the component files.

• 2026-06-08 (INV-006 — Hull-White 1F engine built, Decision 053 1L build #2434). Took the alternative the RAT-132-v1.0.0 COND-001 remediation explicitly left open: built a real Hull-White 1-factor short-rate engine (finview.L1.hull_white_1f, finlib/derivatives/hull_white_1f.py) and bound it to M-132. The engine is calibrated to the canonical curve_builder discount curve by construction (exact A/B affine refit; instantaneous forward f(0,t) derived analytically from the pchip log-DF interpolant — the forward-curve fix mandated by the 1L design review, NOT the discrete forward_rates[]), with analytic ZCB-option / caplet-floorlet pricing, a European swaption via Jamshidian decomposition, and a test-only exact-transition Monte-Carlo cross-checker. Added the HW-vs-Bachelier swaption challenger lane (soft, non-gated). 12 real deterministic tests (tests/test_hull_white_1f.py) all pass. Restored "Hull-White" to the family name (registry + this card title/description); rewrote the Methodology HW bullet and replaced Limitation 1 ("not implemented" → "implemented, vol params uncalibrated"). Registered the engine (engine_registry.yaml, total_engines 131→132) and A-284 (mean-reversion a) / A-285 (short-rate σ) as exogenous/uncalibrated assumptions (NOT default-zero fabrication; per Decision 053 §4). Production scope: HW bond/curve analytics are production, HW option outputs are non-production until a,σ are calibrated (DS-058 strike-grid IV is the DR-007 paid-only gap) — mirror in model_registry.yaml::M-132.production_scope. Tier-2 evidence pack at validation_evidence/M-132/v1.0.1/. Regenerated the FinView binding manifest. No engine outputs, back-test numbers, or validation results were fabricated. Prepared for 2L re-review (RAT-132-v1.0.1); 1L did not edit the RAT record or the 2L workspace.
• 2026-05-08 — stub created from registry data per Decision 023 Phase 5 / B-07.
• 2026-06-06 (1L remediation, RAT-132-v1.0.0 COND-001 + COND-002) — Remediated the two gating conditions under the Decision 053 1L charter §2.2 (no engine outputs, back-test numbers, or validation results fabricated). COND-001 (doc): dropped the unimplemented "Hull-White" from the model name
• description (registry + this card) — now "Multi-Curve / Black-Bachelier / SABR"; added the Name-correction callout. COND-002 (doc/scoping): split the family into a production linear-IR core and non-production placeholder-dependent members (swaption / rate-option / xccy) via a Production Scope table here and model_registry.yaml::M-132.production_scope; A-054 / DS-058 calibration is left as an honest pending build (PL-07 / WS3), not fabricated. LG-132-01 (doc): reconciled documentation_pack flags (model_card + change_log → present; validation_evidence → present, pack assembled). Assembled the Tier-2 validation-evidence pack at validation_evidence/M-132/v1.0.0/. Re-stamped doc-currency. Prepared for 2L re-review; 1L did not edit the RAT record or the 2L workspace.
• 2026-06-06 — 2L code-grounded documentation refresh against irs_pricer.py (sha 22d4662) + sofr_futures_pricer.py + the InsModel snapshot harness. Reconciled par_rate / par_swap_rate / annuity to engine emission (FinView #92, 2026-06-04): rewrote the Representative-pricer paragraph and the Output-Snapshot prose to read these from engine output rather than re-deriving, and converted Limitation 4 to a RESOLVED note. Added the Standards Coverage section (framework-level ASOP 56 / SR 26-2; no numbered standard prescribes the IR/FX pricing math). Tightened the Hull-White wording in Methodology and Limitation 1 to "no HW short-rate engine/class is implemented (grep finds only a comment reference in sofr_futures_pricer.py)". Corrected the snapshot-provider note: snap_M_132 resides in InsModel Models/scripts/model_snapshots.py and path-inserts FinView. Noted gated/ratification items (INV-006, Tier-2 validation evidence, A-054 / DS-058, INV-032) under Limitations. Non-output-changing: no model outputs, back-test numbers, or validation results were fabricated or changed.