Applying Itô’s Lemma to a Yield-Dependent Bond Price

Summary

The document poses a fixed-income question about applying Itô’s lemma to a bond price defined as the integral of discounted cash flows under a yield. It first evaluates the integral as the reciprocal of the yield, then assumes the yield follows a stochastic process with mean reversion and diffusion. The author applies Itô’s lemma to this reciprocal price and derives a change in bond value containing both a drift term and a stochastic term. They then ask how to define the bond’s interest income per unit time and propose adding a discounted payment to the price change when calculating total return.

No answer or correction is included, so the proposed total-return expression is not established as valid. The document is useful as an example of the distinction between price changes and income in bond returns, and of how Itô’s lemma introduces a second-order term when the underlying yield is stochastic. It leaves important modeling details open, including the bond’s cash-flow interpretation and how coupon income should be represented over an infinitesimal interval.

Key ideas

• The example represents a yield-dependent bond price as the reciprocal of yield.
• Applying Itô’s lemma introduces drift, diffusion, and a second-order contribution to the price change.
• Total return combines price changes with income, whose definition depends on the bond’s cash flows.
• The document asks a question but does not confirm whether its proposed return formula is correct.

Tags

# Application of Ito's lemma relating to bond price


# Application of Ito's lemma relating to bond price












I'm interested in solving the following questions but I am confused on the second part because I do not know how to define/calculate the interest per "unit time", which I'm guessing is informal terminology for "dt". I've included my attempt after the picture.

(i) - this is trivial: $B(y) = \int_0^\infty e^{-yt} dt = y^{-1} e^{-yt}|_{t=0}^{t = \infty} = \frac{1}{y}$

(ii) - I'm not sure how to define the interest. Would this be the interest payment received from the bond payment? i.e. informally $\exp(-y_t t)dt$?

In this case we get $$\text{Exp. total return per unit time} = \frac{dB_t + e^{-y_t t}dt}{B_t}$$

and from Ito's lemma we obtain $$dB_t = \frac{-1}{y_t^2} dy_t + \frac{1}{2} \frac{2 d\langle y \rangle_t}{y_t^3} = -\frac{a(m-y_t) dt + by_t dZ_t}{y_t^2} + \frac{b^2}{y_t} dt = \frac{\bigg((b^2+a)y_t - am\bigg) dt + by_t dZ_t}{y_t^2}$$

so that

$$\text{Exp. total return per unit time} = \frac{\bigg(e^{-y_t t}y_t^2 + (b^2+a)y_t - am\bigg) dt - by_t dZ_t}{y_t} $$

Would this be the right idea? Thanks for any advice!

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