Pricing a Defaultable Coupon Bond with Correlated State Variables

Summary

The document sets up a pricing problem for a perpetual defaultable coupon bond whose value depends on an underlying asset value and a stochastic coupon rate. Both state variables are modeled as geometric Brownian motions, with potentially correlated shocks. A proposed partial differential equation includes each variable’s drift and variance terms, a mixed derivative for their correlation, a coupon payment term, and the risk-free discount rate.

The author also proposes a solution formed from separate powers of the asset value and coupon rate, plus a constant. The document is posed as a question and supplies no answer, derivation, boundary conditions, or validation. Consequently, it introduces a useful two-factor valuation setup but does not establish that the proposed equation or separable power-form solution is correct. In particular, a complete valuation would require specifying how default and coupon payments work and applying suitable boundary or terminal conditions.

Key ideas

• The bond price is modeled as a function of underlying asset value and a stochastic coupon rate.
• Correlated Brownian shocks produce a mixed partial derivative in the pricing equation.
• The proposed equation includes drift, diffusion, coupon income, and risk-free discounting.
• A sum of separate power functions is suggested as a solution, but the document does not verify it or give boundary conditions.

Tags

# Pricing equation with two correlated states


# Pricing equation with two correlated states












Consider the following asset pricing setting for a perpetual defaultable coupon bond with price $P(V,c)$, where $V$ is the value of the underlying asset and $c$ is a poisson payment that occurs with probability $\lambda$.

Both $V$ and $c$ evolve as geometric Brownian motions, potentially correlated with: $$\frac{dV_t}{V_t} = \mu_V dt + \sigma_V d Z^V_t$$ $$\frac{dc_t}{c_t} = \mu_c dt + \sigma_c d Z^c_t$$ and $Corr(d Z^V_t, d Z^c_t) = \rho dt$.

I believe the pricing equation for the bond looks like: $$\frac{\sigma_V^2}{2}V^2 P_{VV} + \rho \sigma_V \sigma_c V c P_{Vc} + \frac{\sigma_c^2}{2}c^2 P_{cc} + \mu_V V P_V + \mu_c c P_c + \lambda c= rP$$

I'm guessing that the solution for this equation is: $$P = A_0 + A_1 V^{\gamma_1} + A_2 V^{\gamma_2} + A_3 c^{\gamma_3} + A_4 c^{\gamma_4}$$

Is the above correct?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.