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 $V_t$.

  • 1
    $\begingroup$ 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. $\endgroup$
    – Kevin
    Apr 27 at 22:21
  • $\begingroup$ 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. $\endgroup$
    – Luca Gi
    Apr 27 at 22:29
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    $\begingroup$ 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). $\endgroup$
    – Kevin
    Apr 27 at 22:35

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