Comparative Statics for a Perpetual Coupon Bond Price

Summary

The document presents a perpetual coupon bond valuation problem in a diffusion setting. Its pricing equation relates the required return times the bond price to the coupon and the price’s drift and curvature terms, with a boundary condition at a stopping threshold and an asymptotic price equal to the coupon-to-rate ratio. The underlying asset is modeled with geometric Brownian motion.

The question asks how to determine the bond price’s sensitivity to the coupon-to-rate ratio when the closed-form solution is difficult to interpret because the interest rate also affects the characteristic roots. It provides the setup but no derivation, comparative-statics result, or numerical evidence. Any sensitivity analysis would therefore need to account for the dependence of both the equation and its solution’s roots on the interest rate; the document itself leaves that work unresolved.

Key ideas

• The bond price is specified by a differential asset-pricing equation with drift and volatility terms.
• The setup includes a threshold boundary and an asymptotic value equal to the coupon divided by the interest rate.
• The question concerns price sensitivity to the coupon-to-rate ratio.
• The closed-form solution is difficult to interpret because the interest rate also enters the characteristic roots.
• No comparative-statics derivation or sensitivity result is supplied.

Tags

# Comparative statics on $c/r$ using fundamental asset pricing equation


# Comparative statics on $c/r$ using fundamental asset pricing equation












Consider the fundamental asset pricing equation for a perpetual coupon bond:

$$rP = c + \mu P' + \sigma^2/2 P''$$ with standard boundary conditions $P(\bar x) = \bar x$ and $\underset{x\rightarrow \infty}{\lim} P(x) = c/r$. $r$ is the interest rate and $c$ is the coupon rate, the other parameters are drift and volatility of the GBM of the underlying asset.

How can I derive how does the price change as a function of $c/r$? Notice that the closed-form solution does not help much, as $r$ enters the roots of the characteristic equation.

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