Module 28
Convertible bonds
A convertibleis a corporate bond with a right bolted on: at your option, you can swap the bond for a fixed number of shares of the issuer's stock. That one clause changes everything about how it's priced and risk-managed, because the instrument decomposes into two pieces: a credit-sensitive bond floor — the value of the coupons and principal as plain corporate debt — plus an equity call optionon the issuer's stock. You're holding a bond and a stock option in one wrapper.
Where the stock sits relative to the conversion price puts the convert in one of three regimes:
- Busted — the stock is far below the conversion price. The option is nearly worthless, so the convert trades as distressed straight debt: credit-driven, delta near zero.
- Balanced (hybrid) — the stock is near the conversion price. This is maximum optionality: high gamma, positive convexity, and the sweet spot for convertible arbitrage.
- Equity-like — deep in the money. Delta heads toward 1 and the convert simply tracks the stock.
The greeks follow directly from the decomposition: fair value, implied vol, delta, gamma, vega (converts are long volatility), rho, and the credit sensitivity of the floor. The subtlety — and the reason converts are hard — is that equity risk and credit risk live in one instrumentand interact. A stock collapse doesn't just kill the option; it also widens the issuer's credit spread, so the floor drops just as the option dies. That's the “busted convert” double hit.
That interaction is also the trade. Convertible arbitrage: buy the convert, short delta × stock against it, and often buy CDS protection on the credit leg — then harvest gamma, vol and carry as the stock moves. For a risk system the lesson is structural: the convert, the equity hedge and the credit hedge must be tied together as one position, with the greeks stressed jointly — not booked as three unrelated lines.
A quick tour of the zoo before the lab: exchangeables convert into shares of a different company than the issuer; mandatories must convert at maturity, so they're mostly equity in a bond costume; warrants are the naked option without the bond; and AT1 CoCos are bank capital with regulatory write-down or conversion triggers — a different animal entirely, where conversion happens to you, not for you.
Dial the stock through the three regimes and watch the decomposition shift.
🎛 Convertible regimes lab
Per 100 face · conversion ratio = 100 / conversion price = 2.00 shares · risk-free rate fixed at 4%
Balanced (delta 65%, 30–70%)
Maximum optionality — high gamma and positive convexity. This is the convert-arb sweet spot.
Convert value
113.5
floor + option, per 100
Bond floor
79.5
straight-debt value
Option value
34.0
ratio × BS call
Delta
65%
equity sensitivity
Where the value sits
Conversion value = ratio × stock — what you would get by converting today. The convert is worth more than both the floor and the conversion value because the option lets you wait.
Simplified decomposition: we price the convert as bond floor + ratio × Black-Scholes call with independent credit and equity inputs. A real convert model solves the two jointly (the stock falling widens the credit spread, callable and putable features change the option, and conversion kills the remaining coupons). Educational tool — not investment advice.
How the lab computes it
Everything is per 100 face. The conversion ratio is how many shares one bond turns into, and the two legs are priced separately then added — a deliberate simplification:
ratio = 100 / conversion price bond floor = Σ coupon / (1+y)ᵗ + 100 / (1+y)ᵀ where y = risk-free + spread option = ratio × BScall(stock, K = conversion price, vol, T) convert ≈ bond floor + option delta = ratio × N(d₁) × stock / convert value
The floor is the convert priced as straight debt: coupons and principal discounted at the risk-free rate plusthe issuer's credit spread (annual compounding). The option is a Black-Scholes call on the stock struck at the conversion price, scaled by the ratio. Delta is expressed as an equity sensitivity — the percentage the convert moves for a 1% stock move — and the lab classifies the regime from it: below 30% is busted, 30–70% is balanced, above 70% is equity-like.
What the simplification leaves out is exactly the interesting part: in reality the spread you discount the floor at is a function of the same stock price that drives the option, so the two legs are not independent. Real convert models (lattice or PDE) solve them jointly — but the additive decomposition is how every trader thinks about the product.
Things to try
- • Drag the stock from $5 to $200 with the conversion price at $50 — watch the banner walk through busted → balanced → equity-like as delta climbs from ~0 to ~1.
- • Park the stock at $15 (busted) and widen the credit spread to 1000 bp — the floor crumbles while the option is already dead. That's the double hit in one slider.
- • Set the stock at the conversion price and raise the equity vol — the option value swells the most here. Converts are long vol, and balanced converts most of all.
- • Push the stock deep in the money and raise vol again — barely anything happens. Delta ≈ 1 means the optionality is spent.