Perpetual Defaultable Bond Valuation with a First-Passage Boundary

Summary

The document sets out the valuation equation for a perpetual coupon-paying defaultable bond in a structural credit model. It gives the differential equation and a general solution containing unknown constants, then specifies boundary conditions: bond value at the default threshold reflects recovery, while value at very high firm value approaches the value of a perpetuity. It also states the discounted expected recovery cash flow at the threshold.

The question is how to express this boundary-value problem through a Feynman–Kac representation without a terminal condition. The proposed approach decomposes value into discounted coupons received before the firm value first reaches the default boundary and expected recovery at the hitting time, using the first-passage distribution. The document presents the setup and a candidate formula, but does not supply a derivation or confirm the formula. Its result depends on the stated model assumptions and boundary interpretation; details such as the process dynamics and boundary behavior require careful verification.

Key ideas

• The bond value is described by a differential equation with coupon payments and a default boundary.
• The general solution includes constants determined by recovery and high-firm-value boundary conditions.
• The proposed valuation separates coupons paid before default from recovery paid when the threshold is reached.
• A first-passage distribution can express the timing of default in the candidate expectation formula.
• The document poses the Feynman–Kac derivation but does not resolve or verify it.

Tags

# Derivation defaultable bond price in Leland 1994 (Merton)


# Derivation defaultable bond price in Leland 1994 (Merton)












Consider the model in Leland (Journal of Finance, 1994).

The partial differential equation that describes the price of the (perpetual coupon defaultable) bond is: $$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + C = 0$$

The equation has general solution: $$F(V) = A_0 + A_1 V + A_2 V^{-x}$$ where $x \equiv \frac{m + \sqrt{m^2 + 2 r \sigma^2}}{\sigma^2}; \; \; m \equiv \mu - \frac{\sigma^2}{2}$. The unknown constants can be found by imposing the two boundary conditions $F(V_b) = (1-\alpha) \frac{V_b}{r-\mu}$ and $\underset{V \rightarrow \infty}{\lim} F(V) = \frac{c}{r}$. Recall that $\mathbb{E} \left[ \int_0^\infty e^{-rs} (1-\alpha)V_s ds | V_0 = V_b \right]=(1-\alpha) \frac{V_b}{r - \mu}$.

Which is the Feynman-Kac representation of this specific problem? Notice that the difference with the ``usual'' representation comes from the fact that there is no terminal condition, differently from e.g. https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula.

My attempt so far is: $$F(V_t) = \mathbb{E}_t \Big\{ \int_t^\infty e^{-r (s-t)} c \mathbf{I}_{\{V_s > V_b \}} ds \Big\} + \int_t^\infty e^{-r(s-t)} \mathbb{E} \Big\{ \int_s^\infty e^{-r(m-s)} (1-\alpha) V_m dm \Big| V_s = V_b \Big\} Pr\{ V_s = V_b \} ds$$

Notice that if I'm not mistaken the result should be: $$F(V_t) = \int_t^\infty e^{-r(s-t)} c [1 - F(s; V_t, V_b)] ds + \int_t^\infty e^{-r(s-t)} (1-\alpha) \frac{V_b}{r - \mu} f(s; V_t, V_b)$$

where $F(s; V_t, V_b)$ and $f(s; V_t, V_b)$ are respectively the cumulative distribution and the density of the first passage time $s$ to $V_b$ from $V_t$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.