Convertible Greeks from the Pricing Engine

A hedge ticket needs shares, not an ambiguously labelled “55% delta”. A convertible engine must report both the derivative and its units, while the desk checks how much numerical noise comes from exercise boundaries. This lesson obtains price and Greeks from the same running contract and turns them into an executable stock hedge.

Learning objectives. Distinguish share delta from normalised delta and price elasticity, calculate finite-difference Greeks, and test bump and grid stability before using gamma.

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.