The price of a bond at time T, measured in terms of its yield, is G(yT). Assume geometric Browni

(a) To calculate the process for the forward bond price, we can use Itô's lemma. Itô's lemma sta

Question

The price of a bond at time $T$, measured in terms of its yield, is $G\left(y_T\right)$. Assume geometric Brownian motion for the forward bond yield $y$ in a world that is defined by a numeraire equal to a bond maturing at time $T$. Suppose that the growth rate of the forward bond yield is $\alpha$ and its volatility $\sigma_y$. (a) Use Itô's lemma to calculate the process for the forward bond price in terms of $\alpha$, $\sigma_y, y$, and $G(y)$. (b) The forward bond price should follow a martingale in the world considered. Use this fact to calculate an expression for $\alpha$. (c) Show that the expression for $\alpha$ is, to a first approximation, consistent with equation (30.1).

   The price of a bond at time $T$, measured in terms of its yield, is $G\left(y_T\right)$. Assume geometric Brownian motion for the forward bond yield $y$ in a world that is defined by a numeraire equal to a bond maturing at time $T$. Suppose that the growth rate of the forward bond yield is $\alpha$ and its volatility $\sigma_y$.
(a) Use Itô's lemma to calculate the process for the forward bond price in terms of $\alpha$, $\sigma_y, y$, and $G(y)$.
(b) The forward bond price should follow a martingale in the world considered. Use this fact to calculate an expression for $\alpha$.
(c) Show that the expression for $\alpha$ is, to a first approximation, consistent with equation (30.1).

Options, Futures, and Other Derivatives

John C. Hull 10th Edition

Chapter 30, Problem 6

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Step 1

Itô's lemma states that for a function $f(t, x)$ of a stochastic process $x(t)$, the differential of $f$ can be expressed as: $$df = \left(\frac{\partial f}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2}\right)dt + \sigma \frac{\partial Show more…

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The price of a bond at time $T$, measured in terms of its yield, is $G\left(y_T\right)$. Assume geometric Brownian motion for the forward bond yield $y$ in a world that is defined by a numeraire equal to a bond maturing at time $T$. Suppose that the growth rate of the forward bond yield is $\alpha$ and its volatility $\sigma_y$. (a) Use Itô's lemma to calculate the process for the forward bond price in terms of $\alpha$, $\sigma_y, y$, and $G(y)$. (b) The forward bond price should follow a martingale in the world considered. Use this fact to calculate an expression for $\alpha$. (c) Show that the expression for $\alpha$ is, to a first approximation, consistent with equation (30.1).

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