= Solution
{c}
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) gives
$$
\boxed{E_t=C^E(t,V_t;F)
-\left(\frac{V_t}{F}\right)^{1-2r/\sigma^2}
C^E(t,F^2/V_t;F).}
$$
If 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.
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