Bootstrapping a curve: zero-rate risk for free

Every calibration so far in this module has fit a model's own parameters. Bootstrapping a curve is a different problem wearing similar clothes: there is no model, just a stack of swap quotes that a discount curve and one or more forecast curves have to reprice exactly. A rates desk hedges each swap by its own PV01, but a risk system reports exposure by zero-rate bucket instead — someone has to compute the map between the two, d(zero rate) / d(quote), for every pillar against every quote. Re-bootstrapping the whole stack once per quote just to bump it would work, but it's the same one-Greek-at-a-time cost 2.1 already showed is avoidable.

The whole story

Two closed-form stages, no solver anywhere

MultiCurveBootstrap never iterates to find a discount factor. Because the annual grid gives each swap exactly one new pillar, every stage's par-swap equation solves in closed form for that one unknown:

// annual par swap: P(0,T) = (1 - q * prefix_{<T}) / (1 + q)
df = rec.constant(1.0).sub(q.mul(prefix)).div(rec.constant(1.0).add(q));

Stage 1 builds the OIS discount curve this way from OIS deposits and par swaps. Stage 2 repeats the same closed-form move once per forecast tenor, except the float leg and annuity of those swaps discount on Stage 1's own oisDfByYear map — read directly in source, not assumed from the module's name, this is where a forecast zero rate's dependence on OIS quotes actually enters the tape, pillar by pillar, as the recursion runs. Every quote becomes a named rec.input(inst.label(), inst.quote()), and every zero rate becomes a named output (out.put("z:" + inst.label(), ...)) — one recording, both curves, all ten quotes and ten zero rates on the same tape.

One Jacobian row per reverse sweep

MultiCurveBootstrap.Kernel wraps that recording in a MultiOutput and keeps the compiled tape open, so the whole stack can be re-evaluated at shifted quotes — a bump cross-check, a scenario — without re-recording:

this.mo = MultiOutput.of(rec -> record(rec)).on("cpu").build();
// ...
public Result evaluate(Map<String, Double> quoteOverrides) {
  MultiOutput.Result r = mo.run(quoteOverrides, 1L, 0L);
  // one r.gradient(zeroLabel) per zero rate -> one Jacobian row each

Each of the ten zero rates gets one reverse sweep, and each sweep's gradient is that zero rate's full row against all ten quotes — read back by name, the same mechanism 2.2 first traced. Because an OIS zero rate's recording never touches a forecast quote at all, its row is exactly zero there by construction, not by a later filter: the Jacobian is block lower-triangular because the tape itself only has edges running one way.

The real run

MultiCurveBootstrapShowcase bootstraps five OIS par swaps against five 3M forecast par swaps, 1y through 5y, at a real ~25bp starting tenor basis:

yr	OIS zero	3M zero	basis (bp)
1	2.9559%	3.2080%	25.2
2	3.1037%	3.3653%	26.2
3	3.2031%	3.4743%	27.1
4	3.2732%	3.5543%	28.1
5	3.3340%	3.6251%	29.1

The basis widens from 25.2 to 29.1 bp across the curve — a real, computed number, not an assumption from the ~25bp input quotes, since the zero rate is a nonlinear transform of the bootstrapped discount factors. The 5y 3M forecast zero's Jacobian row is near zero against every OIS pillar except the closest ones, +0.0031 against OIS:swap:5, and +1.0322 against its own quote 3M:swap:5 — a diagonal entry just over 1.0, because a swap's own quote moves its own zero rate slightly more than one-for-one once the annuity effect is included. Wrote and ran a standalone probe (javac+java against the built classes, same approach as every prior module) taking an independent central bump (h=1e-6) of that same row: relative error 3.8×10⁻¹³ against the OIS entry, 2.1×10⁻⁶ against the diagonal one — both comfortably inside the 1e-5 the deeper doc claims, measured directly rather than taken on faith. Every one of the ten calibrating instruments reprices to its own input quote to 8 decimal places.

Try it yourself

The showcase reprices a 4y receive-fixed swap struck at 3.30% (par 3.6100%, annuity 3.6963, PV -0.01146). Change the maturity from 4 to 5 and predict the new PV before you run it — both the par rate and the annuity move. (par 3.6800%, annuity 4.5428, PV -0.01726: further underwater, because the 5y par rate has risen further above the fixed 3.30% strike than the 4y par rate had.)

▶️ Run it

mvn -o -q -pl nablatensor-examples exec:java \
  -Dexec.mainClass=com.nablatensor.examples.MultiCurveBootstrapShowcase

No engine string to pass — MultiCurveBootstrap.Kernel hardcodes "cpu" internally, for the reason this page's second sidenote traces.

⚠️ What this doesn't do

This is the stylised annual construction CurveSet's own doc comment names directly: fixed and floating legs share one annual grid, so every swap adds exactly one pillar and every stage solves in closed form. Real desks need sub-annual float frequencies, interpolation inside the solve rather than only on the finished curve, and cross-currency basis, none of which are here. The deeper doc also flags that an analytic block Jacobian is a possible faster alternative to the recorded-and-replayed one this page walks through — YieldCurve.java's own doc comment even points at CurveBootstrap (the single-curve version) as the place that already uses one; MultiCurveBootstrap deliberately doesn't, favoring one general mechanism over a bespoke analytic derivation per curve shape.

What's next

→ Deeper: Multi-curve (OIS) bootstrap with an adjoint Jacobian has the full builder API and the pinned test tolerances this page's numbers are checked against. → Next: Hull-White: swaptions from a root-find, not a tape — Module 7.5 prices a short-rate model whose own formula contains a solver, so it comes off the tape entirely rather than staying on it.