Derivation — Cash-Only vs Equity Split and the TF Critique

A cash/equity split can produce a reasonable convertible price while failing to describe the cash flows of a self-financing default hedge. The desk needs to understand that distinction before interpreting TF value as a complete credit model. This lesson derives the continuation equations and explains the AFV critique through the default term missing from the split.

Learning objectives. Derive the coupled TF equations, apply the exercise constraints, and explain why an explicit stock jump and recovery change the pricing operator.

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.