 Solution

If default liquidates the firm as soon as its value crosses below $F$, equity is extinguished at that event. If default never occurs before maturity, equity receives $(V_T-F{)}_{+}$. It is therefore a no-rebate down-and-out option on firm value, with strike price and barrier both equal to $F$.

For a firm that has survived to $t$ and has $V_t>F$, part (b) givesIf default has already occurred, $E_t=0$. At maturity a surviving firm has equity $(V_T-F{)}_{+}$.

The source uses a strict crossing $V<F$. With nonzero volatility, continuity and immediate two-sided Brownian fluctuations at a first barrier hit make the pre-maturity absorbing value at $V=F$ zero as well: starting at the barrier, the process crosses below it immediately almost surely. At maturity the contingent claim payoff at equality is also zero. Thus this convention has the same pricing boundary as the usual barrier-hit convention in this nondegenerate model.