Consider the following Cauchy-Euler ODE, which is in particular the asset pricing equation for a (perpetual coupon defaultable) bond:

$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + C = 0$$

where $k \in \mathbb{R}_+$ and $$r = \begin{cases} r_1 \; \text{ if } \; V > k \\ r_2 \; \text{otherwise} \end{cases}$$

If $r$ was constant, the general solution has the form: $$F(V) = A_0 + A_1 V + A_2 V^{-x}$$ where $x \equiv \frac{m + \sqrt{m^2 + 2 r \sigma^2}}{\sigma^2}; \; \; m \equiv \mu - \frac{\sigma^2}{2}$ and the coefficients have to be determined using boundary conditions.

Is it possible to derive a similar general solution for the case of $r$ above? I'm hoping that one could solve separately the ODE over the two ranges separately and then paste the two solutions with a condition on the first derivative. Is this approach correct?


1 Answer 1


I solved it for the case $\mu = r_1$, the solution in $\mathbb{C}^1$ takes the guessed form $$F(V) = \begin{cases} A_0 + A_1 V + A_2 V^{-x} \; \text{ if } \; V>k \\ B_0 + B_1 V + B_2 V^{-y} \; \text{ else } \end{cases}$$ where the constants can be found by plugging the guess into the ODE and the remaining ones by imposing boundary conditions and smooth pasting at the discontinuity.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.