Credit Spread Measures

Credit spread measures

Six numbers on one bond are all called "the spread", and a quote almost never says which. They differ by a reference curve, not by a credit view: G against one point of the government par curve, I against one point of the swap curve, Z against the whole zero curve, ASW against the swap annuity, DM against a projected index, OAS against a lattice you had to pick a volatility for. Nothing raises when you compare two of them.

Every figure below is printed by scripts/spreads.py (runs in 0.9 s; QuantLib optional, imported inside quantlib_cross_check). ✅ Measured means this file produced it on 2026-09-09 with QuantLib 1.43, numpy 2.2.6, scipy 1.13.0, Python 3.11.3.

The rule: a spread is a pair — the number and the curve it was measured against. "YTM minus the government zero rate" is not the G-spread, and a Z-spread on a callable bond is not an OAS.

1. ✅ One bond, six spreads

A 5-year 3% annual-coupon corporate on a steep government zero curve (3.00 / 3.50 / 4.00 / 4.50 / 5.00% at 1–5y, annual compounding), priced at a 150 bp Z-spread. Swap zeros are the same curve plus a widening 10 → 26 bp swap spread. ✅ Measured: price 85.700750, YTM 6.434845%, government 5y zero 5.000000% but 5y par 4.902625%, swap 5y par 5.149527%.

🚨 The same bond is 110.71 bp and 153.22 bp on the same afternoon — a 42.5 bp range with no change in credit. Two of those numbers are wrong only if you mislabel them; one of them (143.48) is wrong however you label it.

2. 🚨 "YTM minus the government zero rate" is not the G-spread

The G-spread is quoted against the benchmark yield — a par yield, the coupon that prices the on-the-run bond at 100. A zero rate is a different animal: on an upward-sloping curve the par yield sits below the zero rate of the same maturity, because the par bond's early coupons are discounted at lower short rates. ✅ Measured, the 5y zero is 5.000000% and the 5y par is 4.902625% — 9.74 bp apart on this curve, and the whole error lands in the spread.

✅ The error is the slope, not the credit. The 5-year zero is pinned at 5.00% in every row; only the shape changes:

On a flat curve the two agree exactly, which is why the mistake survives testing: it is invisible in the one case people check. The sign is systematic — an upward-sloping curve always makes the naive number too small, so credit looks tighter than it is.

Also note the G-spread and the Z-spread are not the same number either — 153.22 vs 150.00, −3.22 bp. Both are correct; G collapses the curve to one point and Z uses all of it. On a steep curve, or a bond far from par, the two separate. Quote which one you meant.

3. ✅ ASW is not the Z-spread over the swap curve

The par/par asset-swap spread is the margin over the floating index that turns a bond into a par floater. The investor pays 100 for the package while the bond costs 85.70, and the swap absorbs the 14.30-point difference — so the ASW amortises (100 − price) over the swap annuity linearly, where the Z-spread shifts a discount curve. ✅ Measured on the same bond, on the same swap curve: ASW 110.71 bp vs Z-over-swap 124.53 bp = −13.82 bp.

The gap is a function of distance from par, not of credit. A bond at 100 has ASW ≈ Z; this one is 14.30 points below par, and the gap is 13.82 bp. 🚨 Comparing a deep-discount bond's ASW with a near-par bond's Z-spread ranks them wrongly, and both numbers are "the spread over swaps".

asset_swap_spread(price_dirty, coupon, maturity, swap_zeros) implements (PV_swap(bond flows) − price) / swap annuity. ⚠️ This is the par/par convention; a market-value asset swap and a proceeds asset swap divide by different notionals and give different numbers again.

4. ✅ Discount margin: the quoted margin is the DM only at par

For a floater the coupon resets, so a Z-spread is replaced by the discount margin — the spread over the projected index that reprices the bond. The quoted margin printed on the security is a coupon formula, not a valuation. ✅ Measured, 5y quarterly FRN, forward index flat at 4.00%, quoted margin 120 bp:

✅ DM equals the quoted margin exactly at 100.00 and nowhere else. The straight-line shortcut margin + (100 − price)/maturity is 4.56 bp light at 98.50 and 17.57 bp light at 95.00 — it ignores discounting, so it always understates the DM on a discount bond.

5. 🚨 A Z-spread on a callable bond is not an OAS — the gap IS the option cost

A Z-spread discounts the bond's promised cash flows. A callable bond will not pay them, and its price is depressed by the call, so the Z-spread absorbs the option value and reports it as credit. The OAS puts the cash flows on a calibrated short-rate lattice, exercises the call optimally at every node, and solves for the spread that is left over.

scripts/spreads.py builds a 10-step lognormal recombining lattice by forward induction. ✅ Calibration check: the lattice reprices every zero-coupon bond on the curve to 3.33e-16 — that is the only test a calibration needs.

✅ Measured, 10y 6.5% corporate callable at 100 from year 5, sigma = 20%, true OAS 80 bp:

🚨 40.04 bp of "credit spread" that is not credit. Rank this bond against a bullet on Z-spread and it looks 40 bp cheap; it is not.

✅ The identity that proves the lattice, and the volatility you had to choose

✅ Remove the call and the two measures must agree. On the same bond with no call option: Z-spread 80.1455 bp vs OAS 80.0000 bp, gap −0.1455 bp — the residual is the lattice's own convexity (discounting 1/(1+r+s) node by node is not the same operation as shifting an annually compounded zero curve), not a modelling error. If your OAS engine misses the bullet identity by more than a fraction of a basis point, the lattice is the problem.

🚨 The OAS is the only one of the six measures that depends on an assumption you made. ✅ Measured, same bond, same true 80 bp OAS, vol varied:

At sigma = 0 the option cost is 0.0049 bp — this bond's forward price never reaches the call price without volatility, so there is nothing to strip out and Z and OAS coincide. Every basis point of option cost above that is your vol assumption, and two desks quoting "the OAS" on the same bond with 10% and 30% vol are 45 bp apart. An OAS without its vol is not a number.

6. ✅ Cross-check against QuantLib 1.43

🚨 In QuantLib 1.43 both take a BondPrice, not a float. BondFunctions.zSpread(bond, 85.7007, curve, ...) raises TypeError: Wrong number or type of arguments for overloaded function, because the clean/dirty flag is part of the type: pass ql.BondPrice(price, ql.BondPrice.Clean). It fails loudly, which is the good case — unlike Settings.instance().evaluationDate, which fails silently.

7. What the script gives you

scripts/spreads.py — numpy and scipy (brentq) only; QuantLib imported inside one function.

Where this sits

• ../../../fin-models/skills/credit-risk-models/SKILL.md — where a spread turns into a default probability: Merton, N(−d2), the constant hazard, lambda(1−R), and the risk-neutral vs physical gap. This skill stops at the spread; that one starts there.
• ../cds-mechanics-and-upfront/SKILL.md — the other credit spread. A CDS par spread and a bond Z-spread are different instruments with different discounting, and the basis between them is a trade, not an error.
• ../../../fin-models/skills/term-structure-models/SKILL.md — every curve above is an input. 🚨 A zero rate is a triple (day count, compounding, instrument); §2 here is what happens when you also forget whether it is a par rate or a zero rate.
• ../corporate-bond-data-and-trace/SKILL.md — where the 85.70 came from, and why a TRACE print is not a mid. A spread computed off a one-sided last print inherits the bid-offer.
• ../ratings-transitions-and-migration/SKILL.md — the rating that put the bond in the index whose OAS you are comparing against.
• ../../../fin-libraries/skills/lib-quantlib/SKILL.md — 🚨 Settings.instance().evaluationDate is a global and a stale one gives NPV exactly 0.0; §6 sets it before building any curve. Also BondPrice, DiscountCurve and BondFunctions.
• ../../../fin-core/skills/market-data-sourcing/SKILL.md — the FRED ICE BofA OAS series (BAMLC0A0CM and friends) are index OAS: option-adjusted, market-value weighted, and not comparable to a single bond's G-spread.