# 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 $$V_t$$.

• I'm not sure you can apply the Feynman-Kac theorem to solve an ODE. Because the product is perpetual, it doesn't depend on time and you do not have a parabolic PDE. Apr 27 at 22:21
• The ODE is parabolic for sure, it's a result in the 1994 paper. In Leland Toft 1996 they derive the FK representation with maturity and then take $t \rightarrow \infty$, which ends up being the parabolic formula from 1994. But in that paper they start from the end, I'm trying to reverse engineer it. Apr 27 at 22:29
• I'm just saying that the stated differential equation, $\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + C = 0$, is an ordinary differential equation and thus neither elliptic/parabolic/hyperbolic and I doubt the FK theorem applies. It's simply a Cauchy-Euler ODE with known solution technique. That ODE always appears when studying perpetual options in a GBM context (e.g. real options). Apr 27 at 22:35