Convertible Bond

A convertible bond starts life as a normal corporate bond — fixed coupons, principal back at maturity — but carries a golden ticket: the holder may swap the bond for a fixed number of shares whenever that's more attractive.

If the stock soars, you convert and ride the equity. If it stagnates or falls, you keep collecting coupons and get your principal back like any bondholder. Downside cushioned, upside open — you pay for this via a lower coupon than a plain bond would offer.

Issuers — often growth companies with expensive debt and volatile shares — like converts because the embedded option lets them borrow cheaply, betting that conversion (dilution) is a happy problem to have.

The vocabulary

• Conversion ratio: shares received per bond. Conversion price = face / ratio.
• Parity = ratio × share price — the bond's value if converted right now.
• Bond floor = value as a straight bond (coupons + principal discounted at the issuer's credit spread).
• Premium = convert price − parity: what you pay for the option and the floor.

Issuer options complicate life

• Call provisions: after a period, the issuer may redeem early (usually if the stock trades above a trigger ≈ 130% of conversion price) — forcing holders to convert and capping the option's life.
• Puts: holders sometimes may sell back at par on set dates — a valuable floor-raiser.

Convertible arbitrage

The classic hedge-fund trade: buy the convert, short delta shares against it, and capture the embedded option cheaply (converts often issue "cheap" to vol). The book earns from gamma trading and coupon carry, and suffers when credit gaps or borrow disappears.

Pricing: a contingent claim on two risk factors

A convert depends on the share price (equity risk) and the issuer's survival (credit risk) — inseparably, since default crushes both bond and shares. The standard framework is a PDE/lattice with an equity-dependent default intensity \(\lambda(S)\):

with recovery \(R\), coupon flow \(c\), and free-boundary conditions at each node: \(V \ge \text{parity}\) (holder converts), \(V \le \max(\text{call price}, \text{parity})\) (issuer calls), \(V \ge \text{put price}\) on put dates. Typically \(\lambda(S) = \lambda_0 (S_0/S)^{\alpha}\) — spreads blow out as the stock falls, generating the realistic "credit cliff".

Greeks worth naming

• Delta rises from ~0 (busted convert) to ~ratio (deep ITM); gamma peaks near the conversion price.
• Rho/credit DV01: dominant when busted — the convert is then a distressed bond.
• Vega: converts are long equity vol; issuance waves cheapen listed vol via arb hedging.

Market conventions

Desks quote implied vol given a credit-spread assumption (or vice versa) — the "vol-credit smile" of the convert market. Documentation details (dividend protection via ratio adjustments, takeover ratchets) materially move value and are priced explicitly.