Calibration and dependent curves

Curve calibration chooses discount factors so that the prices or rates of selected instruments reproduce market observations. A quote-based CurveConfiguration specifies the instruments, interpolation rule and market index of the resulting curve.

Some calibration instruments need other curves. For example, a term-rate projection curve may use an overnight curve for discounting. MultiCurveBootstrapper resolves these dependencies, then solves a nonlinear system for all pillars of each curve.

Residuals and the nonlinear solve

Let \(x=(P_1,\ldots,P_n)\) be unknown factors at the calibration pillars, \(q\) the associated market quotes, and \(z\) already solved parent-curve inputs. Calibration imposes

\[ F(x,q,z)=0. \]

Each residual follows the quoted contract. Deposits compare implied and quoted rates; par swaps and basis swaps contribute NPV conditions; futures compare implied and quoted rates with supported convexity adjustment. FX forwards use their outright/points strategy. Bond anchors compare model price with either a quoted price or a yield-converted target, normally in units per 100. These bond strategies do not constitute OAS calibration.

The bootstrapper uses a Newton solve with finite-difference residual Jacobians. Trial bumps and accepted steps preserve positive discount factors. The current setup starts each factor at 0.99, with tolerance 1e-12 and 200 iterations. Inspect convergence status and repricing residuals: iteration exhaustion can return a result without SolverErr.

Dependencies and collateral

Calibration instruments may project one index and discount on another. SOFR discounting can therefore be a parent of a Term SOFR projection curve calibrated from basis swaps. Foreign-currency payments under USD collateral may require Collateral(CLP, USD) and an FX spot store. A topological sort orders the configured dependencies; absent parents and cycles are errors.

The discount policy makes these roles explicit. For USD collateral, USD payments use the selected USD collateral index and CLP payments use their USD-collateralized CLP curve. A local ICP curve can separately project CLP coupons. Only the instruments actually chosen as collateral-curve pillars determine whether ICP is a dependency of that calibration.

The bootstrapper determines solve order from dependencies, so configurations can be supplied in any order. The same list can include quoted zero-rate curves and composites. A composite depends on both component curves and can itself become a parent of calibrated curves. Duplicate identities, self-references and cycles are rejected.

Overnight-index calibration instruments

An OIS exchanges fixed coupons against compounded overnight interest. OIS_CCY_Index_Tenor identifies this contract and builds an OvernightIndexSwap for calibration. The preset uses coupon-only legs with annual schedules, index calendars and day counts, Modified Following accrual adjustment and Following payment adjustment. SOFR uses two business days to settlement and two business days of payment lag; SONIA uses zero settlement days and CORRA one. Other index presets use two settlement days and zero payment lag.

Optional positional fields in an OIS identifier override those contract conventions. Empty intermediate fields retain defaults: for example, OIS_USD_SOFR_1Y___0_3 retains frequencies and sets settlement/payment lag to zero/three. Calendars use canonical names such as UnitedStates(Sofr), not an assumed bare-country alias.

Standard future compounded coupons use discount-ratio telescoping. Seasoned coupons compound historical business-day observations with their accrual weights, then project the remainder. Same-day observations are used when available and forecast otherwise. Payment lag affects discounting, not accrual. Lookback, lockout, observation shift and daily spread compounding alter observation semantics and must match the actual contract. Deterministic valuation supports nonstandard overnight settings that the legacy XVA claim converter rejects.

Swap_CCY_Index_Tenor[_FixedFreq[_FloatFreq[_SettlementDays[_FixedDC[_FloatDC]]]]] uses single-fixing floating coupons, including when the index is overnight. Its defaults come from the index’s swap_conventions(tenor) metadata, which is separate from the rate’s deposit/forward quoting definition. The tenor runs from settlement on the index calendar. Swaption underlyings and direct swap builders use the same defaults.

Euribor defaults to annual Thirty360 (bond basis) fixed coupons, floating coupons at the index tenor with ACT/360 accrual, TARGET and two settlement days. TIBOR defaults to semiannual ACT/365 fixed coupons, while NIBOR uses annual 30/360. SOFR uses annual ACT/360 on both legs; SONIA uses annual ACT/365 and zero settlement days. There is no universal fixed-leg default.

Optional fields override each convention independently. For example, Swap_EUR_EURIBOR3m_1Y_Annual_Quarterly_1_Thirty360_Actual360 supplies both frequencies, one settlement day and both accrual conventions. Swap_EUR_EURIBOR3m_1Y___1_Actual365 retains the index frequencies and floating day count while changing settlement and fixed accrual. Empty fields retain defaults. Rust builders and serialized quote details can supply the same terms through their individual setters and fields.

Settlement lag delays the start of accrual; regular swap payments occur at accrual end. Use OIS quotes for compounded daily observations and their separate payment-lag conventions. To reproduce curves made with the former semiannual/quarterly defaults, supply those frequencies and the former day counts explicitly.

To reproduce a supplied market curve, match the calibration contracts, interpolation and valuation assumptions as well as the quoted rates.

In particular, a quote’s extraction date need not be its effective date, and discount_factor(T) is anchored at the curve reference date. Compare a factor measured from a later start S using discount_factor(T) / discount_factor(S). The curve’s day counter measures interpolation time; it does not override the swap quote’s fixed or floating coupon day counts. See the executable Euribor 3M reconciliation for an annual-versus-semiannual comparison and an independent cashflow check.

Differentiating calibration

At a converged solution, implicit differentiation gives

\[ \frac{\partial x}{\partial q} =-J^{-1}\frac{\partial F}{\partial q},\qquad \frac{\partial x}{\partial z} =-J^{-1}\frac{\partial F}{\partial z}, \quad J=\frac{\partial F}{\partial x}. \]

The implemented own-quote derivative is diagonal by calibration-instrument construction. Parent dependencies are finite-differenced. These matrices reconnect solved factors to tape inputs through a local first-order relation:

\[ x_i^{\rm AD}=x_i^\ast +\sum_j S_{ij}(q_j^{\rm AD}-q_j^\ast) +\sum_k C_{ik}(z_k^{\rm AD}-z_k^\ast). \]

The relation preserves the calibrated value and its local market derivatives, including chained parent-curve risk. It does not replay the Newton iterations or establish exact second-order differentiation of the calibration map. Finite-difference Jacobians and solver tolerances remain numerical approximations.

Credit calibration

CreditCurveBootstrapper strips piecewise-constant hazards from CDS par spreads. A CreditCurveElement uses a term-structure representation whose curve value is survival probability \(Q(t)\), not a risk-free discount factor. Configuration supplies credit identity, payment currency, recovery, premium frequency and the discount index for the premium/protection legs.

For each maturity the new hazard interval is solved with earlier intervals fixed. The implemented bisection search uses bounds \(10^{-12}\) to 20 and 200 iterations. A finite-difference IFT step connects survival to CDS quotes. Empty quote sets and duplicate maturities are rejected. Discount and credit curves therefore share construction and risk concepts, but their numerical values have different economic meanings.