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I have a question related to this paper https://www.jstor.org/stable/2245302. Given a process for the short rate $r$, the authors consider the price process $Y$ for a consol bond that satisfies

\begin{equation} Y_t=E_t\bigg[\int_t^\infty \exp\bigg(-\int_t^sr_udu\bigg)ds\bigg]. \end{equation}

They then derive the following SDE

$$dY_t=(r_tY_t-1)dt+A(r_t,Y_t)dW_t,$$

where $A$ is some measurable function. They write "The drift $r_tY_t-1$ shown for $Y$ is implied directly by $(1.1)$ (i.e. the first equation that I wrote) and Ito's formula".

Maybe I am missing something obvious, but I am not sure how to apply Ito's lemma to derive the drift as the authors do in the paper. Could someone explain this to me? Any help would be greatly appreciated.

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    $\begingroup$ I’m voting to close this question because OP does not respond. $\endgroup$
    – Kurt G.
    Commented May 9, 2023 at 8:31

1 Answer 1

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It should be stressed that the authors state that only the drift $r_tY_t-1$ is implied by the equation for $Y_t$ and not necessarily every diffusion coefficient $A(r_t,Y_t)$ is consistent with the equation for $Y_t$ and a certain SDE for the short rate $r_t\,.$

A way to see this is the following. The consol bond $Y_t$ is a perpetual annuity that pays continuously a coupon $ds=1\,ds\,.$ If this coupon is reinvested into the money market account $\exp(\int_0^tr_u\,du)$ then the value of that account plus $Y_t$ is a non-dividend paying asset which must be a martingale under the risk-neutral measure and in the numeraire $\exp(\int_0^tr_u\,du)\,.$ That is: with $$ \textstyle Y_t+C_t=Y_t+\int_0^t\exp(\int_s^tr_u\,du)\,ds $$ the process $$ M_t=\frac{Y_t+C_t}{\exp(\int_0^tr_u\,du)}=\textstyle Y_t\exp(-\int_0^tr_u\,du)+\int_0^t \exp(-\int_0^sr_u\,du)\,ds $$ must be a martingale. The integration-by-parts formula gives $$ dM_t=\textstyle\exp(-\int_0^tr_u\,du)\,dY_t-r_t\,Y_t\exp(-\int_0^tr_u\,du)\,dt+\exp(-\int_0^tr_u\,du)\,dt $$ which can be rewritten as $$ \textstyle\exp(\int_0^tr_u\,du)\,dM_t=dY_t-(r_t\,Y_t-1)\,dt $$ or as $$ dY_t=(r_t\,Y_t-1)\,dt+\textstyle\exp(\int_0^tr_u\,du)\,dM_t\,. $$ This is an SDE for $Y_t$ with the desired drift and an unspecified diffusion term.

Note that the SDE for a dividend paying stock can be written as $$ dS_t=(rS_t-\delta S_t)\,dt+\sigma\,S_t\,dW_t $$ which is fully consistent with the SDE for $Y_t\,.$ The difference is that the stock pays a dividend rate $\delta$ and not an absolute dividend $1\,dt\,.$

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  • $\begingroup$ .(+1) but a possible remark: iirc the last SDE is for the dividend paying stock under the risk neutral measure, and not in general, but I may be mistaken $\endgroup$
    – Snoop
    Commented May 9, 2023 at 22:31
  • $\begingroup$ @Snoop You are not mistaken. It was silently assumed when I said $M_t$ must be a martingale. $\endgroup$
    – Kurt G.
    Commented May 10, 2023 at 5:22

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