Curve Construction — No-Arbitrage Bootstrap / Multi-Curve Family

M-137 is the curve-construction foundation of the FinView pricing stack. It takes a set of market instruments quoted as par rates — money-market deposits / Treasury bills at the short end (≤ 1Y) and par swaps / par bonds at the long end (> 1Y) — and produces a self-consistent term structure of discount factors, continuously-compounded zero rates, and forward rates that every downstream pricer consumes. The engine family is FinView-resident (finlib/curve_builder.py); the InsightALM model registry binds it as M-137 via three engine entries: finview.L0.curve_builder (the bootstrap), finview.L0.curve_set (the multi-curve container), and finview.L0.interpolation (the log-linear / PCHIP interpolators).

Sequential bootstrap. bootstrap_sofr_curve strips discount factors one tenor at a time via an initial forward pass (each node sees only earlier nodes), exploiting the par condition that a par instrument prices to 1.0:

• Short end (T ≤ 1Y) — treated as a single-payment money-market instrument: DF(T) = 1 / (1 + c·T), where c is the par (simple) rate. No coupon stripping is required because there is one cash flow.
• Long end (T > 1Y) — treated as a semi-annual par coupon bond. The par condition 1 = Σ (c/2)·DF(t_j) + DF(T) is solved for the unknown terminal DF(T). The PV of the prior coupons (t_j < T) is evaluated against the discount factors already bootstrapped at earlier tenors, with intermediate coupon-date factors filled in by log-linear interpolation of discount factors (_interpolate_df). Rearranging gives DF(T) = (1 − PV_prior_coupons) / (1 + c/2).

Self-consistency refinement and self-check (post-INV-028, #92). The engine now contains two consistency facilities, both implemented but defaulted OFF:

• refine_passes (default 0) — after the forward pass, runs N additional full-curve sweeps that re-solve every coupon node against the complete current curve (rather than only earlier nodes), driving the bootstrap to reprice its own calibrating instruments to par. With refinement on the long-end residual collapses toward ~0 (see Output Snapshot).
• self_check_tol_bps (default None) — when set, the engine calls reprice_residuals() after building and emits a logger.warning if the max reprice residual exceeds the tolerance. The self-check is therefore implemented but not yet enabled-by-default; it warns, it does not raise.

Both knobs default OFF deliberately, to preserve the frozen gold curves downstream pricers calibrate against — so default/gold output (refine_passes=0, self_check_tol_bps=None) still carries the long-end residual documented below. Turning either on by default changes every downstream gold curve and is held for ratification (INV-028 follow-up; see Limitations note).

Zero and forward rates. Once the discount-factor curve is built, zero rates are recovered by inverting the continuous-compounding relation z(T) = −ln DF(T) / T, and instantaneous-ish forward rates between adjacent grid tenors are recovered from the discount-factor ratio f(T_{i-1}→T_i) = (DF_{i-1}/DF_i − 1) / Δt. The first forward is pinned to the first zero.

Interpolation scheme. Discount factors are interpolated log-linearly (linear in ln DF, equivalent to piecewise-constant forward rates between knots) — the workhorse used inside the bootstrap itself. For off-grid zero-rate queries, interpolate_curve offers either plain linear interpolation (default, backward-compatible) or PCHIP (monotone cubic Hermite, the Hagan–West "monotone-convex"-family choice that suppresses the spurious forward oscillation linear schemes can produce). PCHIP is also available for the discount-factor interpolation path.

Multi-curve framework. Post-2008 / post-LIBOR pricing requires separating the discounting curve (OIS / SOFR) from the projection curve (the forward-rate curve for the floating leg) and from government (Treasury) benchmarks. build_sofr_curve bootstraps an OIS/SOFR discount curve from a SOFR overnight rate plus a dictionary of par swap rates (2Y…30Y), with optional short-end calibration from 90-day / 180-day SOFR averages. The CurveSet dataclass is the container that holds Treasury, discount (OIS/SOFR), and projection curves side by side, plus a rating-keyed dictionary of credit spreads. Its get_discount_dfs / get_projection_fwds accessors fall back to the Treasury curve when an OIS/projection curve is absent, so single-curve and multi-curve callers share one interface.

Where it sits in the stack. M-137 is the Tier-2 gate that the FinView derivative pricers stand on: M-130 (bond pricer), M-132 (interest-rate swap pricer — its swap_terms input is paired with a curve_set), M-133 (swaption pricer), and M-135 (SOFR-futures pricer) all consume the curves M-137 produces. If the curve is wrong, every price above it is wrong — which is why this card is the foundation the others reference. The full dependent set (6 models: M-130–M-135) is enumerated in the Dependencies section above.

Prose. The model is deliberately a par-rate bootstrapper, not a global least-squares / spline-fit calibrator. The advantage is that every grid knot reprices its calibrating instrument exactly (residual ≈ 0 at the knot — see Output Snapshot), the method is deterministic and order-stable, and there are no optimizer convergence concerns. The cost is that accuracy between widely-spaced knots depends entirely on the interpolation scheme: coupon cash flows that fall between sparse long-end tenors are priced off log-linearly interpolated discount factors, which is where the only material reprice error appears (Limitation 1).

Deterministic curve construction from a synthetic par-instrument set — reproducible, requires no live market data (BV-032 immune: the input is a hand-specified upward-sloping par curve, not a market pull). Reproduce with python scripts/model_snapshots.py M-137; the underlying engine behaviour is asserted by FinView tests/test_curve_builder.py (12 tests).

Input: Treasury par curve, tenors [0.25, 0.5, 1, 2, 3, 5, 7, 10, 20, 30]y, par yields 4.30% → 5.30% (upward-sloping; mirrors the upward_curve test fixture). OIS/SOFR overlay: SOFR overnight 4.30% + par swaps 2Y 4.10% / 5Y 4.20% / 10Y 4.30% / 30Y 4.50%.

Reprice residual is the validation that matters. A correct bootstrap must reprice its own calibrating instruments to par. Per-tenor residuals are effectively zero at the short end and small through 10y (0.00 bps at ≤1y, 0.33 bps at 2y, 1.55 bps at 5y, 4.62 bps at 10y) and then balloon at the sparse long end (66.10 bps at 20y, 12.69 bps at 30y). This is not a bootstrap bug — the terminal DF(T) knot itself is solved exactly; the residual comes from re-pricing the coupon cash flows that fall on the 10y→20y→30y gaps, where log-linear discount-factor interpolation over a 10-year window is a coarse approximation. Tightening the long-end tenor grid (adding 15y/25y knots), switching the coupon-date fill to PCHIP, or running the engine's refine_passes self-consistency loop collapses this residual — with refine_passes=6 on the same fixture the max residual drops to ~0.14 bps. Refinement is off by default to preserve frozen gold curves, so the held/default figure remains 66.10 bps; default-on enablement + gold re-freeze is the INV-028 follow-up held for ratification (see Limitation 1).

The off-grid zero-rate interpolation is well-behaved: at 4y/8y/15y the log-linear and PCHIP zeros agree to within ~3 bps (e.g. 15y: 5.1342% log-linear vs 5.1636% PCHIP), confirming the curve is smooth enough that interpolation choice is second-order for rate queries.

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

1. Long-end reprice error under a sparse tenor grid. With knots at 10y/20y/30y, coupon cash flows on the 20y and 30y par bonds are discounted off log-linearly interpolated DFs across 10-year gaps, producing a worst-case ~66 bps reprice residual at 20y (vs < 5 bps through 10y) on the default forward pass. The terminal knot is exact; the residual is an interpolation-gap artifact. Mitigation: denser long-end knots, PCHIP coupon-date interpolation, or the engine's own refine_passes self-consistency loop (post-INV-028, #92) which collapses the residual toward ~0 when enabled. A calibration self-check (reprice_residuals + self_check_tol_bps warning path) is implemented but not yet enabled-by-default — it warns rather than fails, and refinement is off by default to preserve frozen gold curves (see Methodology and Validation Packet). Enablement-by-default + gold re-freeze is tracked as the INV-028 follow-up (held for ratification, see note below).
2. Uniform day-count / no calendar. Tenors are decimal years; there is no business-day adjustment, holiday calendar, or per-leg day-count basis (actual/360 for SOFR floating, 30/360 for fixed, etc.). The engine docstring states "uniform day-count convention (actual/actual assumed)." This biases accrual-sensitive prices for real-dated instruments and is acceptable only for benchmark / synthetic curves, not for trade-date production marking.
3. No futures convexity adjustment. The bootstrap calibrates to par deposit/swap rates, not futures prices. A futures-based short end would need a convexity correction (the engine docstring flags this explicitly); it is absent. M-135 (SOFR-futures pricer) consumes the curve but the curve is not itself futures-calibrated.
4. Additive parallel SOFR–Treasury basis is first-order. The sofr_treasury_spread_bps path applies a flat parallel spread to all par rates; a real SOFR–Treasury basis is tenor-dependent. The richer path (build_sofr_curve) bootstraps OIS/SOFR independently, which is preferred — the flat-spread option remains a convenience approximation.
5. No global no-arbitrage / monotone-convex guarantee. Log-linear DF interpolation gives piecewise-constant (non-smooth) forwards; PCHIP is monotone but not the full Hagan–West "monotone convex" forward interpolator. Neither path enforces a hard non-negative-forward constraint at calibration time — a steeply inverted input curve could in principle yield a negative interpolated forward without raising an error.
6. Cross-repo ownership / change-control (FinView-resident). The engine lives in [redacted], outside the InsightALM governance repos. The snapshot inserts the FinView source tree on sys.path to run. Change-control binding between the InsightALM model registry (M-137) and the FinView source is by registry path reference only; there is no automated drift gate that fails if the FinView signature changes — should be tracked as a governance issue, same class as the M-040 cross-repo finding.
7. Aspirational docstring line — no wired Bloomberg validation. The CurveBuilderGovernance engine docstring states "Validation: cross-check bootstrapped zeros against Bloomberg SOFR curve." This is an aspirational docstring line, not a performed validation: the engine and the snapshot are numpy-only and have no Bloomberg dependency. The actual implemented self-consistency check is reprice_residuals() against the model's own calibrating instruments (Validation Packet). Noted here so the docstring is not read as implying a market-data cross-check that does not exist.

Tracked for ratification (not applied in this documentation pass). The following are output-changing / modeling-code items left for ratification: (a) INV-028 follow-up — enabling refine_passes and/or self_check_tol_bps by default in bootstrap_sofr_curve / build_sofr_curve and re-freezing the gold curves (refinement-on changes every downstream gold curve and the M-137 snapshot); (b) INV-032 — a packaged/versioned interface + CI drift gate for the FinView-resident finlib library consumed cross-repo by InsModel via sys.path (Limitation 6); (c) a hard non-negative-forward / monotone-convex calibration-time constraint (Limitation 5); (d) day-count/calendar support (Limitation 2) and futures convexity (Limitation 3). None applied here.

Self-reprice invariant: at each grid knot the bootstrapped terminal DF(T) reprices its calibrating par instrument to 1.0 by construction; the only non-trivial residual is interpolation-driven at the sparse long end (Limitation 1), which is a known, bounded artifact rather than a correctness failure. This invariant is now machine-checkable via reprice_residuals() (implemented, defaulted off — Validation Packet row above).

Cross-venv reproducibility: the snapshot is numpy-only (no InsModel dependency) and produces byte-identical output under both the FinView venv ([redacted]) and the InsModel venv ([redacted], importing finlib off the FinView source on sys.path).

Open MRM items: (a) calibration self-check implemented (reprice_residuals, INV-028 fix #92) — enablement-by-default (self_check_tol_bps / refine_passes) + gold re-freeze pending (INV-028 follow-up, Limitation 1); (b) cross-repo FinView change-control / drift gate (Limitation 6 / INV-032). Both should be filed as tracked governance issues.

Curve construction / interpolation literature: - P. S. Hagan and G. West, "Interpolation Methods for Curve Construction," Applied Mathematical Finance 13(2), 2006 — the reference for log-linear vs monotone-convex discount-factor interpolation and the forward-oscillation pathologies the PCHIP path avoids. - P. S. Hagan and G. West, "Methods for Constructing a Yield Curve," Wilmott Magazine, 2008 — practitioner treatment of bootstrap + interpolation trade-offs. - F. M. Ametrano and M. Bianchetti, "Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping But Were Afraid to Ask" (2013) — the multi-curve (OIS-discounting / projection-curve) framework that build_sofr_curve + CurveSet implement. - L. B. G. Andersen and V. V. Piterbarg, Interest Rate Modeling, Vol. I, Ch. 6 — multi-curve discounting and forward-curve construction post-LIBOR.

Engine source: - FinView:finlib/curve_builder.py — bootstrap_sofr_curve, build_sofr_curve, interpolate_curve, _interpolate_df, and the CurveSet dataclass; governance sentinels CurveBuilderGovernance (finview.L0.curve_builder), CurveSetGovernance (finview.L0.curve_set), InterpolationGovernance (finview.L0.interpolation). - FinView:tests/test_curve_builder.py — 12-test validation suite (TestCurveBuilder, TestMultiCurve); fixtures flat_curve_5pct / upward_curve in tests/conftest.py.

Internal: - insightalm/modelling/engine_registry.yaml — engine entries finview.L0.curve_builder, finview.L0.curve_set, finview.L0.interpolation (all tier-2-gate / tier-3-infra, model_membership: [M-137] for the first two), owner engine-owner-foundation. - Downstream consumers: M-130 (bond pricer), M-132 (IRS pricer, input curve_set), M-133 (swaption pricer), M-135 (SOFR-futures pricer). - BV-032 (firm-data divergence) — this card's snapshot is BV-032-immune by construction (synthetic par inputs, no live market pull).

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

• 2026-05-08 — stub created from registry data per Decision 023 Phase 5 / B-07.
• 2026-06-04 — Tier-2 hand-authoring of Methodology, Key Assumptions and Their Justification, Output Snapshot, Limitations and Known Gaps, Validation Packet, and References from finlib/curve_builder.py + engine_registry + bound DS/A entries. Stub marker advanced to ``.
• 2026-06-06 — code-grounded documentation-accuracy pass against the post-INV-028 engine state (FinView #92). Added Standards Coverage (ASOP-56 reliance + SR 26-2 aggregate/concentration) and Dependencies (no upstream models; 6 downstream consumers M-130/M-132/M-133/M-135 + family) sections. Updated Methodology + Limitation 1 to reflect that reprice_residuals() self-check and the refine_passes self-consistency loop exist but default OFF (refine_passes=0, self_check_tol_bps=None) to preserve frozen gold curves — so default/gold output still carries the long-end residual; replaced "not yet wired into a calibration self-check" with "self-check implemented but not yet enabled-by-default". Reconciled the Output Snapshot residual by re-running scripts/model_snapshots.py M-137 (default forward pass reproduces 66.10 bps; noted the refinement-on improvement ~0.14 bps as available-but- off). Added Validation-Packet row for reprice_residuals and reworded open MRM item (a). Noted the engine docstring's "cross-check against Bloomberg SOFR curve" as an aspirational line, not a wired validation (numpy-only, no Bloomberg dependency — Limitation 7). Noted gated/ratification items (INV-028 default-on + gold re-freeze, INV-032 drift gate, non-negative-forward guard, day-count / futures convexity) under Limitations. No model outputs, back-test numbers, or validation results were fabricated or changed. Code-side history stays in FinView git log.