Worked Examples — Pricing a Soft-Callable Convertible

A model discount becomes a trade only after the desk can reproduce the price, explain the call and put contribution, reconcile the stock hedge and survive a credit shock. This capstone carries the running contract from event map to valuation and hedge. It reports a complete numerical case while keeping model comparison separate from market calibration.

Learning objectives. Reproduce the three prices, attribute contractual optionality, size a hedge in shares and distinguish a finite model discount from expected trading profit.

Setup 1 (The running contract). All amounts are dollars per bond. Face is $1{,}000$, conversion ratio is 16 shares, spot is $50$, maturity is five years, and annual coupons are $60$ at years 1–5. Conversion is available at every lattice date. The issuer can call for $1{,}000$ from year 2 inclusive to maturity exclusive, only when spot is at least $81.25=1.30(1000/16)$. The holder has a $1{,}020$ put at year 3. Rates are continuously compounded: $r=7\%$, dividend yield $q=0$, volatility $30\%$, and credit spread $s=4\%$.

Coupons are paid before same-date exercise, including conversion; call and put prices exclude that coupon. Between dates there is no accrued-interest payment on conversion. At maturity the payoff is $60+\max(1000,16S_T)$. This explicit teaching convention differs from contracts that forfeit the final coupon on conversion. The spot trigger is a simplified soft call: it does not claim to implement a rolling 20-of-30-day test.