Multi-Curve Interest Rate Framework & Tenor Basis Swap Lab

Simulate modern post-crisis multi-curve interest rate analytics. Model OIS and SOFR collateral discounting, floating forward projections, tenor basis swaps (3M vs 6M), and single-curve pricing discrepancies.

Multi-Curve Controls
Collateral Discount Curve (OIS / SOFR)
decimal
Risk-free collateral discounting rate (e.g. 0.0425 = 4.25%).
Tenor Basis & Credit Spreads
bps
Credit/liquidity premium for 3-month borrowing over collateral rate.
bps
Term liquidity premium of 6-month borrowing over 3-month borrowing.
Contract Specification
Dual-Curve Par Swap Rate

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OIS Discounted
Single-Curve Par Rate

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Traditional Pre-2008
Pricing Divergence

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Single vs Dual Error
3M/6M Basis Spread

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Fair Tenor Basis Swap
Payer Swap NPV

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Mark-to-Market Value
Swap DV01 (PV01)

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Dollar Value of 1 bp

Compares the OIS collateral discounting curve against the 3M and 6M forward rate projection curves, illustrating the persistent vertical spread required to account for term liquidity and credit risk.

Bars depict the pricing error (in basis points) resulting from pre-2008 single-curve discounting. The blue line depicts the breakeven tenor basis spread \(s_{3M/6M}\) across maturities.

Payment-by-payment breakdown of projected 3M floating rates (bars) alongside corresponding OIS discount factors \(P^D(0, T)\) (line).

Institutional Sensitivity Matrices
3M-OIS vs 6M-3M on 10Y Basis Spread (bps)
3M-OIS \ 6M-3M 4 bps 8 bps 12 bps 20 bps 30 bps
Swap Tenor vs OIS Rate on Par Swap Rate (%)
Tenor \ OIS 2.0% 3.0% 4.0% 5.0% 6.0%
Institutional Framework & Mathematical Derivations
1. The Demise of the Single-Curve Paradigm

Prior to the 2007–2008 Global Financial Crisis, interest rate markets operated under the single-curve hypothesis. Interbank lending rates (LIBOR in USD, Euribor in EUR) were treated as the risk-free benchmark, meaning that a single yield curve was used simultaneously to:

  1. Project future floating cash flows: \(F(t, T_{i-1}, T_i) = \frac{1}{\tau_i} \left( \frac{P(t, T_{i-1})}{P(t, T_i)} - 1 \right)\)
  2. Discount all cash flows to present value: \(\text{PV} = \sum c_i P(0, T_i)\)

Under this naive formulation, the par swap rate for a fixed-for-floating swap simplified via telescoping sums into:

\[S_{\text{single}}^* = \frac{1 - P(0, T_n)}{\sum_{i=1}^n \tau_i P(0, T_i)}\]

In August 2007, the subprime crisis shattered this assumption. Interbank lending was revealed to carry significant credit risk and term liquidity premia. The 3-month LIBOR to Overnight Index Swap (LIBOR-OIS) spread, historically steady at 10 basis points, blew out to over 360 basis points following the collapse of Lehman Brothers.

2. Collateralized Discounting & CSA Mechanics

Under standardized Credit Support Annex (CSA) agreements, bilateral derivative counterparties post daily cash collateral to eliminate counterparty credit risk. Because this collateral earns the risk-free overnight rate—Overnight Indexed Swap (OIS), Secured Overnight Financing Rate (SOFR), or Euro Short-Term Rate (€STR)—the no-arbitrage discount rate for any collateralized contract is strictly the collateral overnight rate, represented by the discount curve \(P^D(0, T)\).

3. Dual-Curve Valuation of Interest Rate Swaps

In the modern multi-curve architecture, the valuation of a plain vanilla interest rate swap divorces discounting from forward rate projection:

\[\begin{aligned} \text{PV}_{\text{Fixed}} &= S \sum_{i=1}^n \tau_i P^D(0, T_i) \\ \text{PV}_{\text{Float}} &= \sum_{i=1}^n \tau_i P^D(0, T_i) F_{\text{proj}}(0, T_{i-1}, T_i) \end{aligned}\]

Setting \(\text{PV}_{\text{Fixed}} = \text{PV}_{\text{Float}}\) yields the exact Dual-Curve Par Swap Rate:

\[S_{\text{dual}}^* = \frac{\sum_{i=1}^n \tau_i P^D(0, T_i) F_{\text{proj}}(0, T_{i-1}, T_i)}{\sum_{i=1}^n \tau_i P^D(0, T_i)}\]

Because \(P^D(0, T) \neq P^{\text{proj}}(0, T)\), the floating leg no longer telescopes to \(1 - P(0, T_n)\). A swap valued at par under single-curve formulas carries substantial hidden mark-to-market value and basis risk.

4. Tenor Basis Swaps & Exact Closed-Form Pricing

A Tenor Basis Swap exchanges two floating rate legs of different tenors on the same currency—typically 3-month floating plus a basis spread \(s_{3M/6M}\) against 6-month floating flat.

Why is \(s_{3M/6M} > 0\)? Lending for 6 months locks up credit exposure and bank liquidity for twice as long as a 3-month loan. A lender executing two consecutive 3-month loans retains the real option not to roll over the loan at the 3-month mark if the borrower's credit quality deteriorates. This credit option and term liquidity premium create a persistent positive basis spread.

Equating the present values of both legs discounted along the collateral curve \(P^D\):

\[\sum_{j=1}^m \tau_{6M, j} P^D(0, T_j) F_{6M}(0, T_{j-1}, T_j) = \sum_{i=1}^n \tau_{3M, i} P^D(0, t_i) \left[ F_{3M}(0, t_{i-1}, t_i) + s_{3M/6M} \right]\]

Solving for the fair market quote yields the exact Breakeven Tenor Basis Spread:

\[s_{3M/6M} = \frac{\sum_{j=1}^m \tau_{6M, j} P^D(0, T_j) F_{6M}(0, T_{j-1}, T_j) - \sum_{i=1}^n \tau_{3M, i} P^D(0, t_i) F_{3M}(0, t_{i-1}, t_i)}{\sum_{i=1}^n \tau_{3M, i} P^D(0, t_i)}\]
Multi-Curve Framework Mastery Quiz

1. Why did derivatives pricing transition from the single-curve to the multi-curve framework after 2008?

2. Why is the 3M vs 6M tenor basis spread \(s_{3M/6M}\) strictly positive in practical interbank markets?

3. How does dual-curve discounting compute the par swap rate \(S_{\text{dual}}^*\)?

4. What is the danger of relying on legacy single-curve pricing models in modern trading environments?

Frequently Asked Questions

Prior to August 2007, financial institutions assumed interbank lending rates (LIBOR/Euribor) were risk-free, using a single curve to both project forward floating rates and discount cash flows. The 2008 financial crisis exposed substantial credit and liquidity risks in term interbank lending. Under standard Credit Support Annex (CSA) agreements, cash collateral earns the overnight rate (OIS, SOFR, €STR). Consequently, derivative pricing adopted the multi-curve framework: cash flows are discounted at the collateral OIS curve \(P^D(0, T)\), while floating rates are projected from separate tenor-specific forward curves.

A tenor basis swap is an interest rate agreement exchanging two floating rate streams of different frequencies on the same currency, such as 3-month floating rate plus a spread versus 6-month floating rate flat. Classical no-arbitrage theory dictated this spread should be zero. In practice, lending for 6 months exposes a bank to term credit and liquidity risk for half a year, whereas lending for 3 months with a rollover gives the bank the option not to roll if borrower credit deteriorates. This credit and term liquidity difference creates a persistent, positive basis spread \(s_{3M/6M}\).

Under the classical single-curve model, the par swap rate was \(S^* = \frac{1 - P(0, T_n)}{\text{Annuity}}\). Under the dual-curve framework, the discount factors \(P^D(0, T_i)\) are derived from the collateral curve (e.g., SOFR/OIS), while the forward floating rates \(F_{\text{proj}}(0, T_{i-1}, T_i)\) are projected from the floating index curve. The dual-curve par swap rate is \(S^* = \frac{\sum \tau_i P^D(0, T_i) F_{\text{proj}}(0, T_{i-1}, T_i)}{\sum \tau_i P^D(0, T_i)}\), breaking the identity \(S^* = \frac{1 - P(0, T_n)}{\text{Annuity}}\).

Using single-curve discounting in modern markets leads to severe mispricing—often 5 to 30 basis points in par swap rates and hundreds of thousands to millions of dollars on standard institutional portfolios. It also leaves the trading desk completely unhedged against tenor basis risk and collateral rate divergence, exposing the institution to dangerous basis arbitrage.

The breakeven tenor basis spread \(s_{3M/6M}\) is found by setting the present value of the 6-month floating leg equal to the present value of the 3-month floating leg plus spread, both discounted along the OIS collateral curve \(P^D\): \(s_{3M/6M} = \frac{\text{PV}_{6M} - \text{PV}_{3M}}{\sum \tau_{3M} P^D(0, t_i)}\). This yields an exact closed-form solution for the fair market quote.
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