Deriving a Perpetual Bond Pricing PDE with Default Risk
Summary
The document asks how to derive a time-independent pricing equation for a perpetual coupon bond that pays until default, with recovery at the default time. It models the short rate and default intensity as separate square-root mean-reverting processes driven by independent Brownian motions, then questions the origin of the PDE terms for discounting and default losses.
The key modeling lesson is that a pricing equation must account for both continuous discounting at the short rate and the hazard of default, which replaces the continuation value with recovery when default occurs. The post includes a proposed PDE but later says the lecture notes contained a mistake and expresses doubt about the stated price formula. It does not provide a corrected derivation or establish the proper recovery convention, so the displayed equation and payoff expression should not be treated as verified. The topic is useful for understanding risk-neutral valuation with default, but the document is an unresolved question rather than a complete solution.
Key ideas
- A perpetual coupon bond pays coupons until a default event and may pay recovery at default.
- The model treats the short rate and default intensity as independent square-root mean-reverting processes.
- The pricing equation must reflect both discounting and the change in value when default occurs.
- The post flags errors in its source notes and questions its own payoff formula, leaving the derivation unresolved.
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# Feynman Kac: Perpetual Bond
# Feynman Kac: Perpetual Bond
I would like to derive a PDE for a perpetual bond.
Suppose we have a bond that will pay a coupon $C$ until there is a default event that occurs. Take the time of default as $\tau$ and consider the doubly-stochastic process: $$ \begin{align*} \mathrm{d} r_t &= a(m-r_t) \mathrm{d}t +\sigma \sqrt{r_t} \mathrm{d}W_t \\ \mathrm{d} \lambda_t &= \theta(\nu-\lambda_t) \mathrm{d}t +\gamma \sqrt{\lambda_t} \mathrm{d}\hat{W}_t \end{align*} $$
Where $r_t$, $\lambda_t$ are the rate and the default intensity respectively. The two Brownian Motions are assumed to be independent, and all other parameters are positive constants. The price of this bond is $$ \begin{align*} P &= \mathbb{E} \Big[\int_0^\tau C e^{r_t t}\mathrm{d}t + Ze^{r_\tau \tau}\Big] \\ \end{align*} $$
For some $Z \in [0,1]$. I have in my notes that the price satisfies the following PDE $$ \begin{align*} a(m-x) \frac{\partial u}{\partial x} + \theta(\nu-\lambda_t) \frac{\partial u}{\partial y} + \frac{1}{2}\sigma^2x^2 \frac{\partial^2 u}{\partial x^2} + \frac{1}{2}\gamma^2y^2 \frac{\partial^2 u}{\partial y^2} - xu + y(Z-u) = 0 \end{align*} $$
I am not sure where the last two terms came from. I understand that to derive these PDEs, you appeal to the martingale property to "enforce no drift". That said, the solution (understandably) has no time-dependence, so without the discount factor (as with option pricing) I fail to see where the last two terms come from.
Any help is greatly appreciated! Feel free to add/remove/edit tags.
Edit: There was a mistake in the lecture notes.
Edit 2: After closing inspection, I am not satisfied that the explanation that I have been given is correct. I believe the pricing function I have presented above is wrong.
Guidance is much appreciated!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.