# Application of Ito's lemma to consol price process

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

$$$$Y_t=E_t\bigg[\int_t^\infty \exp\bigg(-\int_t^sr_udu\bigg)ds\bigg].$$$$

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.

• I’m voting to close this question because OP does not respond. Commented May 9, 2023 at 8:31

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 $$1\,dt\,.$$
• @Snoop You are not mistaken. It was silently assumed when I said $M_t$ must be a martingale. Commented May 10, 2023 at 5:22