Multi-Curve Bootstrapping

Why multi-curve?

Before 2008 one LIBOR curve did both jobs: it discounted cash flows and it projected floating-leg rates. That is the curve we built in Chapter 01.

The 2008 crisis broke it. The spread between overnight indexed swap (OIS) rates and 3-month LIBOR, a handful of basis points before 2007, passed 300 bp at the October 2008 peak and never returned to zero. So which rate discounts a collateralised trade? Cash collateral earns the overnight rate set in the credit support annex (CSA), so the discount rate is a property of the collateral agreement, not of LIBOR and not of your own credit (Henrard, 2009). If the cash I am posted pays ESTR, I should not discount at LIBOR.

That basis is two things at once: the default risk of lending to a bank for three months rather than overnight, and the cost of tying up funding for a longer horizon. Which dominates is regime-dependent (Filipović and Trolle, 2013). It is not an arbitrage; it compensates for real risk. Ametrano and Bianchetti (2013) give the arbitrage-free construction, Crépey (2015) the CSA-consistent XVA version.

Multi-curve swap pricing

Subscript $D$ names the discount curve throughout and $F$ the projection curve, on bond prices and zero rates alike.

In the single-curve world, the floating leg telescopes to $P(0,0) - P(0,T_n)$ and no explicit forward rates are needed. In multi-curve, that simplification breaks down because discounting and projection use different curves.

Fixed leg paying on $T_{\alpha+1}, \ldots, T_\beta$, floating leg resetting on $t_0, t_1, \ldots, t_m$, with $\tau_j$ and $\delta_i$ the two accruals of Chapter 01:

where the forward rate from the projection curve is:

$S$ is the quoted par rate and $F_i$ abbreviates $F(0;\,t_{i-1},t_i)$, the Brigo-Mercurio simply-compounded forward. $P_D$ is already calibrated and frozen; $P_F$ is still unknown. At par, $\text{PV} = 0$:

where each $F_i$ depends on the unknowns through $P_F(0, T_k) = e^{-R_k \, T_k}$.

The system of equations and Jacobian

The unknowns are the projection curve's zero rates $R_k := R_F(0,T_k)$, entering through $P_F(0,T_k) = e^{-R_k T_k}$. On the common annual pillar grid the forward rate is:

The residual of swap $k$ is floating PV minus fixed PV at par, the matched annual schedule collapsing $\delta_i = \tau_j = T_i - T_{i-1}$ to a single accrual $\tau_i$:

The accruals cancel before you differentiate: $\tau_i F_i = e^{T_i R_i - T_{i-1} R_{i-1}} - 1$, so the day-count convention never reaches the Jacobian $J_{ki} = \partial g_k / \partial R_i$. Its diagonal is:

Below the diagonal, $R_i$ reaches $g_k$ twice, once raising period $i$ and once lowering period $i+1$:

No $k$ survives on the right. Pillar $i$ enters every swap longer than it in the same way, so under the diagonal a column of $J$ is one number repeated: column 1 reads 0.0239956 in all nine rows beneath it. Only $\mathbf{R}$ is differentiated; every $P_D$ is a frozen weight.

$J$ is lower-triangular because $F_i$ involves only $P_F(0, T_{i-1})$ and $P_F(0, T_i)$, so $R_{k+1}$ never appears in residual $k$. The same holds one level up: no ESTR residual contains a projection rate, so the joint system is block lower-triangular and the sequential solve is the joint solution. Every $P_D$ and every exponential is positive, so the diagonal is positive, $J$ is non-singular at every iterate, and the Newton step always exists. Started at the quoted swap rates, convergence is locally quadratic (Ortega and Rheinboldt, 1970, Thm. 12.6.1), which is why three iterations clear both curves.

Newton-Raphson with autograd

The Newton step $\mathbf{R}^{(k+1)} = \mathbf{R}^{(k)} - J(\mathbf{R}^{(k)})^{-1} \mathbf{g}(\mathbf{R}^{(k)})$ is the one derived in Chapter 01. Only $J$ has changed.

Finite differences need $2n$ function evaluations per iteration and carry truncation error from the choice of $\varepsilon$. Reverse-mode automatic differentiation (Adjoint Algorithmic Differentiation, AAD) returns the full $n \times n$ Jacobian in one torch.autograd.functional.jacobian call, exact to machine precision, with nothing to tune. It is the machinery production XVA engines use for market sensitivities.

Market data

Source: xvafoundations.data.estr, ESTR_INSTRUMENTS and EURIBOR_INSTRUMENTS, valuation date 15 Jan 2026, ACT/360 throughout. The ESTR curve also takes four short-end instruments: 1W deposit 2.37%, FRA 7x30 2.40%, FRA 30x90 2.42%, FRA 180x270 2.32%. A FRA fixes the curve at its end date, not its accrual length, so FRA 7x30 accrues 23 days and pins the 30-day pillar.

Step 1: ESTR OIS discount curve

Chapter 01's method, unchanged, on the ESTR quotes: $P_D(0,1) = 0.97749319$.

Step 2: EURIBOR 3M forward curve

The first pillar sits at $T_1 = 1.0138889$ (ACT/360 from 15 Jan 2026 to 15 Jan 2027) and it is the one point on either curve you can check on paper. The 1Y EURIBOR swap has a single payment, so $P_D(0,T_1)$ appears on both sides of the par condition and cancels, leaving $F_1 = S_1 = 2.75\%$ with no reference to the discount curve at all:

That is the calibrated value to eight decimals. The 2Y swap is where the difficulty starts: two payments, the discount factors no longer cancel, and the ESTR curve enters the answer for the first time. Snippet 3 below prints $P_F(0,1) = 0.97324094$, the curve read at the exact one-year point rather than at the pillar. Figure 5 uses that reading, not the pillar value.

Basis spread

The basis is the difference between the EURIBOR and ESTR forward rates for each annual period. The EURIBOR swaps quote above OIS at every tenor, and that gap is the premium on three-month unsecured lending.

The first bar is one subtraction. Both curves are read at $T = 1$: $F_{\text{ESTR}} = 1/0.97749319 - 1 = 2.3025\%$ and $F_{\text{EUR}} = 1/0.97324094 - 1 = 2.7495\%$, so the spread is 44.7 bp. Substitute the EURIBOR pillar value $0.97287437$ for the one-year reading and you get 48.6 bp, which is two curves evaluated at two different dates rather than a basis.

Only $P_D$ is a price. $P_F$ is a bookkeeping device whose ratios carry the forward rates: multiply the whole $P_F$ curve by any constant and every $F_i$ is unchanged. Nobody trades at $P_F$.

Source: xvafoundations.calibration.Stripper, both curves read on the integer-year grid; website/data/ir-02-multi-curve-bootstrapping.js, key annualGrid.

Zero rates from both curves. The gap is not the same number as Figure 5: that differences simply-compounded forwards, this differences continuously compounded zeros, so at 1Y it reads 43.6 bp against 44.7 bp.

Source: xvafoundations.calibration.Stripper, continuously compounded zero rates on the same grid; website/data/ir-02-multi-curve-bootstrapping.js, key annualGrid.

Two curves is the teaching case, not the desk case. Every tenor gets its own projection curve, 1M, 3M, 6M, 12M, and the gaps between them are quoted directly as tenor basis swaps (3s6s, 1s3s) rather than backed out. In practice the 3M curve is usually built as OIS plus those quotes, which prices the basis to the market instead of inferring it from two independent calibrations.

Python implementation

Nothing is redefined here. For the deposit, FRA and swap pricing formulas, see Chapter 01.

Snippet 1: Imports

import torch
from xvafoundations.curves import YieldCurve, InterpolationMethod
from xvafoundations.instruments import build_instruments, pillar_maturities, initial_rates
from xvafoundations.calibration import Stripper
from xvafoundations.data.estr import ESTR_INSTRUMENTS, EURIBOR_INSTRUMENTS, VALUATION_DATE

torch.set_default_dtype(torch.float64)

Snippet 2: ESTR OIS discount curve

# ── Step 1: Bootstrap the ESTR OIS discount curve ──
pillars = torch.tensor(pillar_maturities(ESTR_INSTRUMENTS, VALUATION_DATE))
rates   = torch.tensor(initial_rates(ESTR_INSTRUMENTS))

def estr_factory(curve: YieldCurve) -> list:
    return build_instruments(ESTR_INSTRUMENTS, VALUATION_DATE, curve)

disc_stripper = Stripper(pillars, rates, estr_factory)
disc_stripper.calibrate()
disc_curve = disc_stripper.get_curve()

t0 = torch.tensor(0.0)
print(f"P_D(0, 1) = {disc_curve.zc_price(t0, torch.tensor(1.0)):.8f}")

Snippet 3: The projection curve, built on the frozen discount curve

The factory captures disc_curve from the enclosing scope; the Stripper passes fwd_curve as the argument being calibrated. Each OISSwap gets both.

# ── Step 2: Bootstrap the EURIBOR 3M forward curve ──
fwd_pillars = torch.tensor(pillar_maturities(EURIBOR_INSTRUMENTS, VALUATION_DATE))
fwd_rates   = torch.tensor(initial_rates(EURIBOR_INSTRUMENTS))

def euribor_factory(fwd_curve: YieldCurve) -> list:
    """disc_curve captured from enclosing scope; fwd_curve being calibrated."""
    return build_instruments(
        EURIBOR_INSTRUMENTS, VALUATION_DATE,
        curve=disc_curve,        # captured (frozen)
        fwd_curve=fwd_curve,     # being calibrated
    )

fwd_stripper = Stripper(fwd_pillars, fwd_rates, euribor_factory)
fwd_stripper.calibrate()
fwd_curve = fwd_stripper.get_curve()

print(f"P_F(0, 1) = {fwd_curve.zc_price(t0, torch.tensor(1.0)):.8f}")

Snippet 4: Basis spread computation

# ── Basis spread: F_EURIBOR(T) - F_ESTR(T) for each annual period ──
t0 = torch.tensor(0.0)
print("\nBasis spread (EURIBOR 3M vs ESTR):")
for T in range(1, 11):
    T_prev = torch.tensor(float(T - 1))
    T_i    = torch.tensor(float(T))
    F_eur  = fwd_curve.forward_rate(t0, T_prev, T_i)
    F_estr = disc_curve.forward_rate(t0, T_prev, T_i)
    spread_bp = (F_eur - F_estr).item() * 10_000
    print(f"  {T:2d}Y: F_EUR={F_eur.item()*100:.4f}%, "
          f"F_ESTR={F_estr.item()*100:.4f}%, "
          f"spread={spread_bp:.1f}bp")

What this enables

• Swaption pricing: the forward swap rate becomes $S_{\alpha,\beta}(0) = \sum_i \delta_i P_D(0,t_i) F_i \,/ \sum_j \tau_j P_D(0,T_j)$, with the $F_i$ from $P_F$ and every discount factor from $P_D$, so the at-the-money-forward (ATMF) strike moves with the basis. Chapter 03 stays on the single USD SOFR curve of Chapter 01 so the change-of-numéraire argument stands on its own; multi-curve changes the strike, not the measure.
• Hull-White calibration (Chapter 05, draft): the model must reproduce one initial term structure exactly, and in a full multi-curve workflow that target is the projection curve with OIS discounting. The examples here pass a single curve and treat multi-curve as a layer on top.
• XVA: collateralised exposure discounts on OIS while floating legs project at tenor rates, so the basis enters the valuation adjustment.

Summary

The size of a cash flow and its value come from different curves. That is the whole of the post-2008 reform, and it costs one extra calibration in the strict order Figure 1 draws. The basis it leaves, 44.7 bp at 1Y down to 17.9 bp at 10Y, shows up in every price downstream.