Question

Suppose that $x$ is the yield on a perpetual government bond that pays interest at the rate of $$\$ 1$$ per annum. Assume that $x$ is expressed with continuous compounding, that interest is paid continuously on the bond, and that $x$ follows the process. $$ d x=a\left(x_0-x\right) d t+s x d z $$ where $a, x_0$, and $s$ are positive constants, and $d z$ is a Wiener process. What is the process followed by the bond price? What is the expected instantaneous return (including interest and capital gains) to the holder of the bond?

   Suppose that $x$ is the yield on a perpetual government bond that pays interest at the rate of $$\$ 1$$ per annum. Assume that $x$ is expressed with continuous compounding, that interest is paid continuously on the bond, and that $x$ follows the process.
$$
d x=a\left(x_0-x\right) d t+s x d z
$$
where $a, x_0$, and $s$ are positive constants, and $d z$ is a Wiener process. What is the process followed by the bond price? What is the expected instantaneous return (including interest and capital gains) to the holder of the bond?

Options, Futures, and Other Derivatives

John C. Hull 10th Edition

Chapter 14, Problem 15

Instant Answer

Step 1

The bond price is given by the formula $P(t) = e^{-\int_0^t x(s) ds}$, where $x(s)$ is the yield at time $s$. Taking the derivative of both sides with respect to $t$, we have $\frac{dP}{dt} = -x(t)P(t)$. Using Ito's lemma, we can find the differential of Show more…

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Suppose that $x$ is the yield on a perpetual government bond that pays interest at the rate of $$\$ 1$$ per annum. Assume that $x$ is expressed with continuous compounding, that interest is paid continuously on the bond, and that $x$ follows the process. $$ d x=a\left(x_0-x\right) d t+s x d z $$ where $a, x_0$, and $s$ are positive constants, and $d z$ is a Wiener process. What is the process followed by the bond price? What is the expected instantaneous return (including interest and capital gains) to the holder of the bond?

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