Default can only occur at bond maturity $T$ at which principal $L$ is repaid. For example, the holder of a zero coupon bond can only default at maturity (as long as there are no other covenants as part of the overall debt contract). The pay-off debt say $D_T$
\begin{equation*}
D_T=
\begin{cases}
L \quad &\text{if}\quad V_T \geq L\\
V_T \quad &\text{if}\quad V_T< L
\end{cases}
\end{equation*}
Assuming Absolute Priority Rules, the pay-off debt $D_T$ to debt-holders in the event of a default at maturity is therefore
\begin{align}
D_T &=\text{min}\{L,V_T\}\end{align}
re-arrange the payoff by adding and subtracting $L$ on the RHS
\begin{align}
&= L + \text{min}\{0, V_T \; – L \}\qquad (0 = L – L)\end{align}and change the minimum to a maximum by placing a minus sign inside and outside the braces
\begin{align}
\therefore \quad D_T &= L \; – \underbrace{\text{max}\{L \; – V_T, 0 \}}_{\text{put option}}\quad \equiv \quad L\; – (L \; – V_T )^+
\end{align}Observe that this is the sum of a safe claim ($L$) and a short position in a put option on the firm’s assets $V_t$. That is,
Upper bound. This is a good place to note that $D_t$ can only go as high as $L$. This makes $L$ an upper bound to the debt value. Apart from the benefits in practice, the upper bound offers a key benefit in the model:
The upper bound $L$ is key to preserving the martingale property of the claim $D_t$. The martingale property allows us to value the claim as the expectation (measure Q) of its discounted payoff.[ref]What happens if the claim $D_t$ is not bounded? See remark 2.2.1 of Manuel Ammann, Credit Risk Valuation, second edition.[/ref]
Graphically the payoffs sum to risk neutrality,
Risky Debt + Put Option = Riskless Debt (Safe Claim)
Alternatively, we may consider what the equity holders receive at maturity. If the asset value $V_t$ exceeds the debt $L$, equity holders receive the positive difference, otherwise they receive nothing if the asset value $V_t$ is less than the debt $L$. Hence,
\begin{align}
E_T &=\text{max}\{V_T \; – L, 0 \}
\end{align}Observe that this resembles the payoff of a call option!
Risk-neutral price of Debt under the Q-measure.
We first return to the value of the asset under probability P
$$dV = \mu V dt + \sigma V dW$$
To obtain the price today of debt at future time $T$, we replace $\mu$ with the risk free interest rate $r$, discount the payoff form $T$ to the present time $t$, and then take conditional expectation under Q
$$D_t = E_Q\biggl[ \bigl[L – \text{max}\{L – V_T, 0 \} \bigl] e^{-r(T-t)} \biggl{|} \mathcal{F}_t \biggr]$$
$\sigma$ is the volatility of the asset $V_t$. By the martingale pricing approach,
\begin{align}
D_t & = Le^{-r(T-t)} – E_Q \biggl[ \text{max}\{L – V_T, 0 \} e^{-r(T-t)} \biggl{|} \mathcal{F}_t \biggl]\\
& = Le^{-r(T-t)} – E_Q \biggl[ e^{-r(T-t)}(L \; – V_T )^+ \biggl{|} \mathcal{F}_t \biggl]\\
& \; \; \vdots \\
&= Le^{-r(T-t)} – \biggl( Le^{-r(T-t)} \mathcal{N}(-d_2) – V_t\; \mathcal{N}(-d_1) \biggr)\\
&= Le^{-r(T-t)} \bigl(1 \; – \; \mathcal{N}(-d_2) \bigr) + V_t\; \mathcal{N}(-d_1) \end{align}and by symmetry of the normal distribution\begin{align}
&= Le^{-r(T-t)} \mathcal{N}(d_2) + V_t\; \mathcal{N}(-d_1) \end{align}where $\mathcal{N}(x) = \int_{-\infty}^x \text{exp} (- \frac{1}{2} z^2) dz$, the cummulative standard normal distribution function and,\begin{align}
d_1 &= \dfrac{\text{ln} \biggl(\dfrac{V_t}{L}\biggr) + \biggl(r + \dfrac{\sigma^2}{2} \biggr)(T-t)}{\sigma \sqrt{T – t}} = \dfrac{\text{ln} \biggl(\dfrac{V_t}{L} \cdot e^{r(T-t)}\biggr) + \biggl(\dfrac{\sigma^2}{2} \biggr)(T-t)}{\sigma \sqrt{T – t}}\\ \\
& = – \dfrac{\text{ln} \biggl(\dfrac{Le^{- r(T-t)}}{V_t}\biggr) + \biggl(\dfrac{\sigma^2}{2} \biggr)(T-t)}{\sigma \sqrt{T – t}}\\ \\
d_2 &= \dfrac{\text{ln} \biggl(\dfrac{V_t}{L}\biggr) + \biggl(r \; – \dfrac{\sigma^2}{2} \biggr)(T-t)}{\sigma \sqrt{T – t}} = d_1 – \sigma \sqrt{T-t}
\end{align}