# Regime-switching interest rate in Black-Cox model

Consider a modification of the time-homogeneous version of the model by Black and Cox (Journal of Finance, 1976).

The firm value $$V$$ follows a Geometric Brownian motion: $$dV/V = \mu dt + \sigma dZ$$ with absorbing default boundary $$V_b$$ and constant coupon C.

With a constant interest rate $$r$$, the fundamental differential equation is an ordinary differential equation with bond price $$F(V)$$ satisfying: $$\frac{1}{2} \sigma^2 V^2 F_{vv}(V) + \mu V F_v(V) - rF(V) + C =0$$

Consider now the following modification. The interest rate takes the following form $$$$r(V) = \begin{cases} r_1 \; \text{if } \; V \geq V_e \\ r_2 \; \text{otherwise} \end{cases}$$$$ with $$V_e > V_b$$. Is it possible to derive a close form solution for the bond?

A few additional and preliminary thoughts. I would proceed in the following way: price a "bond" with two absorbing barriers $$V_b$$ and $$V_e$$ and constant interest rate $$r_2$$ using Kunitomo Ikeda (Mathematical Finance, 1992) and another bond using one default boundary $$V_e$$ and constant interest rate $$r_1$$, then sum the two pieces. Does the intuition make sense?

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