Module 31 · Computing credit risk
Convertible credit risk — the entangled spread
Module 28 showed a convertible as a bond floor plus an equity option, and Module 23 put credit risk in the floor. This module is about actually computing that credit risk — and two things make it harder than a straight bond.
Problem 1: identification. A convert's price depends on both equity volatility and the credit spread, and one market price cannot pin down two unknowns. Higher vol and tighter spread produce the same price as lower vol and wider spread. In practice you pin one and solve for the other: take vol from the listed options surface and back out the implied credit spread, or take spread from CDS or a comparable straight bond and imply vol. Many convert issuers have neither CDS nor straight debt outstanding — then you're proxying, and honest risk systems flag it.
Problem 2: the credit moves with the stock.The stock price isn't just the option input — it's a live read on the issuer's health. When the stock halves, the market doesn't hold the credit spread fixed; it widens it. Desk models capture this with an equity-linked hazard rate:
λ(S) = λ₀ · (S / S₀)^(−α)
where α (typically 1–2) controls how violently credit deteriorates as the equity falls. The consequence is visible in the lab below: the bond floor is not flat. As the stock slides, the spread widens and the floor itself sinks — exactly when you were counting on it. That equity-credit coupling, not the option math, is what makes convert credit risk its own subject.
The risk numbers inherit the state-dependence. CS01 depends on where the stock is: deep in the money the convert is basically equity and a spread move barely registers; busted, it's a distressed bond and CS01 is the whole story. And on a delta-hedged convert-arb position, jump-to-default cuts both ways: the convert collapses to recovery (a loss), but the short stock gaps down with the default (a gain) — the net JTD can even be positive.
🎛 Convertible credit lab
Implied spread
457bp
spread₀ · (S/S₀)^−α
Bond floor
$732
per 1,000 face
Convert value
$964
parity $800
Regime
hybrid
Credit weight
46%
share of value that is bond
Position CS01
$1,467
per +1bp, credit-weighted
Jump-to-default — long convert, short 10.8 sh/bond (delta-hedged)
−$5,644,175
convert → recovery × face
+$3,470,619
short stock gains 80% gap
−$2,173,557
net JTD on the hedged book
The hazard is tied to the stock — spread(S) = spread₀ · (S/S₀)^−α— so as the stock falls the spread widens and the bond floor itself sinks: the floor is not flat. CS01 is weighted by how much of the convert is still “bond” (1 − N(d₁)), so credit risk melts away as the stock rallies. Educational tool — not investment advice.
How the lab computes it
spread(S) = spread₀ · (S/S₀)^(−α) equity-linked hazard bond floor = Σ coupon·DF(r + spread(S)) + face·DF value ≈ bond floor + conversion option (Black–Scholes) CS01 = credit weight × Δfloor per 1bp, weight = 1 − N(d₁) JTD (hedged) = short-stock gain − (value − recovery × face)
The credit weight1 − N(d₁) is the model's estimate of how much of the convert is still behaving as a bond. It goes to zero as the convert goes deep in the money — taking the CS01 with it — and to one as the convert busts. Production models solve the same economics on a two-factor lattice (stock × default); the shape of the answer is what you see here.
Why the cross-gamma matters
On a straight bond, equity risk and credit risk live in different books. On a convert they are the same trade: a falling stock simultaneously cuts the option value, widens the spread, sinks the floor, and raises the credit weight of the position. That equity-spread cross-gamma is a first-order effect, not a refinement — a risk report that bumps equity and credit separately and adds the results will systematically understate the loss in the scenario that actually happens: both at once, in the same direction, for the same reason.
Things to try
- • Set α = 0 (no equity-credit link) — the floor goes flat, the naive textbook picture. Now raise α to 2 and drag the stock down: the floor sinks as you approach it. That gap is what the simple model misses.
- • Watch position CS01 as you move the stock from $20 to $90 — it decays toward zero as the convert goes equity-like. One instrument, and its credit sensitivity depends on where the stock is.
- • With the stock around $35, check the net JTD on the hedged position — the short stock gain can exceed the convert's loss. Then set “stock drop on default” to 40% (partial gap) and watch the net flip negative.
- • Push vol up 10 points with everything else fixed — the convert value rises. If the market price hadn't moved, you'd conclude the implied spread had widened. That's the identification problem live.