I am reading Technical Note No.31 of John Hull's book "Options, Furutes, and Other Derivatives". At the beginning, the PDE of a zero-coupon bond price is given by \begin{equation} dP(t,T)=rP(t,T)dt+v(t,T)P(t,T)dz \end{equation} where $r$ is the short rate, $v(t,T)$ is the volatolity. Then it says that: from Ito's lemma for any $T_1$ and $T_2$ with $T_2>T_1$ \begin{equation} \begin{aligned} d\ln P(t,T_1)= \left[r - \frac{v(t,T_1)^2}{2}\right] dt + v(t,T_1)dz(t)\\ d\ln P(t,T_2)= \left[r - \frac{v(t,T_2)^2}{2}\right] dt + v(t,T_2)dz(t) \end{aligned} \end{equation} However, according to Ito's lemma, \begin{equation} \begin{aligned} d\ln P(t,T) &= \left[\frac{\partial\ln P}{\partial t} + rP\frac{\partial\ln P}{\partial P} + \frac{1}{2}(vP)^2 \frac{\partial^2\ln P}{\partial P^2}\right] dt + vP\frac{\partial\ln P}{\partial P} dz(t)\\ & = \left[\frac{\partial\ln P}{\partial t} + r - \frac{1}{2}v^2 \right] dt + v dz(t) \end{aligned} \end{equation} My question is why there is no term $\frac{\partial\ln P}{\partial t}$ in John Hull's technical note?
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1Because $\ln x$ does not directly depend on $t,.$ – Kurt G. Aug 17 '24 at 12:34
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Thank you for your replying, why zero-coupon bond price $\ln P(t,T)$ does not directly depend on $t$? – Stephen Ge Aug 17 '24 at 14:50
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$P(t,T)$ depends directly on $t,.$ Ignoring $T$ let's write $X_t=P(t,T),.$ Ito's lemma is about $f(\color{red}{t},X_t),.$ What is that function $f(t,x)$ in your case? – Kurt G. Aug 17 '24 at 15:12
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After reading the techinical note, my understand is $f(t,x) = \ln P(t,T)$ – Stephen Ge Aug 17 '24 at 15:14
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We do not have to read a technical note to answer this question. $f(t,x)=\ln P(t,T)$ is obvious nonsense. What do you mean by $x$ on the left hand side? – Kurt G. Aug 17 '24 at 15:22
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sorry, I thought that $x = P(t,T)$ and $f(t,x) = \log(x) = \log(P(t,T))$ – Stephen Ge Aug 17 '24 at 16:06
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Ok. Now apply Ito's lemma to this $f.$ – Kurt G. Aug 17 '24 at 16:13