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Rates & curves · front-office

Multi-curve valuation and basis

Separating collateral discounting from index-specific forward projection

BY THE END, YOU CAN

01Explain why discount and projection curves have different roles.

02Price floating coupons with an index curve and collateral discount curve.

03Derive a basis-spread par condition.

04Identify cross-curve sensitivities and bootstrap dependency order.

01
INTUITION

Identify the state variables and the behavior they add.

Post-crisis rates valuation separates the curve that discounts collateralised cash flows from curves that project index-specific fixings. Basis quotes reconcile cash-flow streams carrying different tenor, liquidity and credit characteristics.

01

One cash flow can depend on one curve for projection and another for discounting.

02

Each index tenor needs its own forward representation when basis is material.

03

A multi-curve build is a dependency graph; rebuilding order and joint sensitivities matter.

02
WHY MARKETS CARE

Ask which instruments can identify the dynamics.

Multi-curve valuation is required for swaps, basis swaps, caps/floors and swaptions whose index projection differs from collateral discounting.

INSTRUMENTS

OIS

term-index swaps

tenor basis swaps

cross-currency basis swaps

QUOTE CONVENTION

State discount collateral, projected index and tenor, basis-leg sign, spread placement, reset frequency and all schedules.

03
MATHEMATICS

Write the dynamics before interpreting parameters.

Formula · Short derivation

Projected floating leg

PVfltp,d=N∑iδiPd(0,Ti)Fp(0;Ti−1,Ti)PV_{flt}^{p,d}=N\sum_i\delta_iP_d(0,T_i)F_p(0;T_{i-1},T_i)

Projection supplies expected coupon rates; collateral discounting supplies present-value weights.

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Formula · Definition

Basis par condition

PVA+sAA=PVBPV_A+sA_A=PV_B

The quoted spread equalises two index-specific floating legs after common collateral discounting.

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Formula · Short derivation

Basis spread

s=PVB−PVAAAs=\frac{PV_B-PV_A}{A_A}

Spread is the residual value difference divided by the discounted accrual annuity of the spread-paying leg.

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Short derivation
Short derivation

Solving a tenor-basis spread

Price each projected floating stream separately, then solve for the spread that restores zero PV.

  1. 01

    Build collateral discounting

    Construct P_d from collateral-consistent OIS instruments.

  2. 02

    Project each index

    Use the A- and B-tenor forwarding curves to produce period-specific forwards on their exact schedules.

  3. 03

    Discount both streams

    Apply the same collateral discount curve to every payment while retaining distinct projections.

    PVA=N∑iδiAPd(TiA)FAiPV_A=N\sum_i\delta_i^AP_d(T_i^A)F_A^i
  4. 04

    Solve for the basis

    Add s to the designated leg and set net PV to zero.

    s=(PVB−PVA)/[N∑iδiAPd(TiA)]s=(PV_B-PV_A)/[N\sum_i\delta_i^AP_d(T_i^A)]
  5. 05

    Differentiate jointly

    Bump discount, A-projection and B-projection quotes independently and together to identify cross-curve dependencies.

Multi-curve valuation makes curve role explicit: discounting prices collateralised cash flows; index curves project contractual fixings.

Inputs
  • P_d(0,T): discount-curve factor
  • F_p(0;T_{i-1},T_i): projection-curve forward
  • s: quoted basis spread
  • A_d: discounted spread annuity
Assumptions and limits
  • The framework does not eliminate bank credit, funding or CSA optionality.
  • Legacy index fallback and cessation require contract-specific treatment.
  • Joint curve calibration can be ill-conditioned in illiquid tenors.
05
MODEL / PRICING

Calibrate, compute, and challenge the dynamics.

METHOD

Construct the OIS discount curve first, then projection curves in a documented dependency order using swaps and basis instruments; price with separate typed handles.

CALIBRATION

Reprice OIS, outright swaps and basis swaps after every dependent curve build. Report residuals and cross-curve Jacobians.

Implementation with current QuantLib

Use separate RelinkableHandle<YieldTermStructure> objects for discounting and each index. Bootstrap dependencies deterministically and avoid accidental handle aliasing between OvernightIndex and IborIndex projections.

API authority: upstream QuantLib reference pinned in the source registry.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
ARCHITECTURE
  • Keep market conventions and quote lineage at the boundary.
  • Solve curves and dynamics in framework-free deterministic kernels.
  • Return residuals, state and sensitivities with every value.
  • Test analytical limits, reconstruction identities and failure domains.
PYTHON 3 · NUMPY / SCIPY

Two-curve floating-leg valuation

Show how a projection spread changes PV under fixed OIS discounting.

REUSABLE EXAMPLE
01from __future__ import annotations
02
03import math
04
05times = [1., 2., 3., 4., 5.]
06discounts = [math.exp(-0.032*t) for t in times]
07ois_forwards = [0.033, 0.034, 0.035, 0.036, 0.037]
08term_forwards = [f + 0.004 for f in ois_forwards]
09accruals = [1.0] * len(times)
10
11def leg(forwards: list[float]) -> float:
12 return sum(a*p*f for a, p, f in zip(accruals, discounts, forwards))
13
14annuity = sum(a*p for a, p in zip(accruals, discounts))
15basis = (leg(term_forwards) - leg(ois_forwards)) / annuity
16assert abs(basis - 0.004) < 1e-12
17print(f"par basis={basis:.2%}")
EXPECTED OUTPUTA 40bp par basis when every projected term forward is 40bp above OIS.
SANITY CHECKS

✓ Both legs share discounting.

✓ Each leg retains its projection curve.

✓ Basis sign matches the spread-paying leg.

07
INTERACTIVE LAB

Shock one parameter and trace the full response.

DISCOUNT CURVE ≠ PROJECTION CURVE

Multi-curve basis laboratory

Move OIS, term projection and basis independently; inspect swap PV and risk by curve role.

SYNTHETIC · CONTROLLED SCENARIOS
1Y basis31.1 bp
5Y basis26.3 bp
10Y projection4.19%
Annualised rate by Maturity (years)

Normal basis: Term projection above OIS.

  • OIS
  • term projection
  • basis
Maturity (years): 0.3Y. OIS: 2.857%. term projection: 3.194%. basis: 0.337%.

Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.

ACTIVE STATE

Normal basis — Term projection above OIS. Move the control and inspect every series with pointer or touch.

08
FRONT OFFICE

Where the model meets the book.

ON THE DESK
“‘The curve moved’ is incomplete until you say discount curve, projection curve and tenor basis.”
VISIBLE INPUTS

OIS curve

index-specific swaps

basis quotes

CSA currency

curve dependency graph

CALIBRATION

Build discounting before projection, then solve basis-linked curves and reprice the full dependency set after each update.

RISK

discount DV01

projection DV01

tenor basis

cross-curve Jacobian

DAILY WORKFLOW
  1. freeze all quote sets
  2. build OIS
  3. build projection curves
  4. reprice basis
  5. aggregate risk by role
Production failure modes
  • curve-handle aliasing
  • circular dependency
  • wrong basis sign
  • staggered quote timestamps
09MACRO CONNECTIONOpen the transmission channel.
MACRO CONNECTION

Liquidity and credit transmission through basis

Funding stress, index credit sensitivity and liquidity demand can widen the spread between overnight discounting and term-index projection.

01Funding/liquidity shocktransmits

changes term-index premium

02Basis quotestransmits

separate projection curves

03Multi-curve PVtransmits

revalues floating streams

04Cross-curve hedgeoutput

allocates OIS and basis risk

10COMMON PITFALLSOpen the failure checklist.
01

Projecting term coupons from the OIS curve.

02

Discounting collateralised cash flows on an index curve.

03

Reporting one net DV01 for dependent curves.

04

Building projection curves before their discount dependency is frozen.

11SOURCES / FURTHER READINGOpen sources and continue the track.
research

Curve construction, multi-curve, short-rate and HJM lectures

Research map for term-structure theory and numerical experiments; all platform explanations and code are original.

Source
Financial Engineering: Interest Rates & xVA
Author
L. A. Grzelak
Ref
main
OPEN ORIGINAL SOURCE ↗
research

Monte Carlo, stochastic calculus and calibration lectures

Mathematical and numerical cross-reference for model dynamics and diagnostics.

Source
Computational Finance Course
Author
L. A. Grzelak
Ref
main
OPEN ORIGINAL SOURCE ↗
implementation reference

Current bootstrapping, interpolation, curve, model, cap/floor and swaption tests

Implementation authority for production object boundaries and regression-test patterns.

Source
QuantLib upstream
Author
QuantLib contributors
Ref
v1.42.1
OPEN ORIGINAL SOURCE ↗
CONTINUE THE TRACKOIS and overnight compounding

Building collateral discounting from daily fixings, observation rules and policy expectations

→Curve construction and bootstrapping

Solving a dated system of market instruments with repricing residuals at every pillar

→Curve risk, carry and roll-down

Moving from one parallel DV01 to bucketed scenarios, hedge instruments and time passage

→