Applying Itô’s Lemma to the Log of a Zero-Coupon Bond
Summary
The document examines the stochastic differential equation for a zero-coupon bond price and asks why a technical note’s expression for the change in its logarithm lacks an explicit time derivative. The answers apply Itô’s lemma to the logarithm as a function of the bond price. The first derivative contributes the proportional price change, while the second derivative adds the variance correction, producing a drift reduced by half the squared bond volatility and a diffusion term equal to that volatility.
The key distinction is how the function is written. For fixed maturity T, the notation can be read as the logarithm function applied to the stochastic variable P(t,T), with time dependence already represented through the bond-price process. In that formulation, the one-variable Itô calculation does not add a separate partial derivative of the logarithm with respect to time. The exchange gives symbolic reasoning rather than a broader term-structure model; a function explicitly dependent on time beyond its dependence through the price would require the corresponding time term.
Key ideas
- For fixed maturity, the logarithm is applied to the bond price as a stochastic variable.
- Itô’s lemma adds a drift correction equal to half the squared volatility with a negative sign.
- The resulting log-price diffusion coefficient is the bond’s volatility.
- An explicit time derivative is needed when the transformed function depends directly on time beyond the price process.
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# Question about Interest rate model and zero coupon bond
# Question about Interest rate model and zero coupon bond
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?
## Answer by Wei (score 1)
https://quant.stackexchange.com/a/80337
I think you are getting confused by the notation, as this is just an application of the univariate Ito's lemma. Write out the formula for $dX$ where $X=f(P)$ for some generic (suitably differentiable) $f$, and then substitute $f=\ln$.
## Answer by NC520 (score 1)
https://quant.stackexchange.com/a/80631
Applying Ito's lemma to $\ln P(t,T)$: \begin{aligned} d\ln P(t,T) &= \frac{\partial\ln P(t,T)}{\partial P(t,T)} dP(t,T) + \frac{1}{2} \frac{\partial^2\ln P(t,T)}{\partial P(t,T)^2} (dP(t,T))^2 \\ &= \frac{dP(t,T)}{P(t,T)} - \frac{1}{2} \left( \frac{dP(t,T)}{P(t,T)} \right)^2 \\ &= \left( r dt+v(t,T) dz(t) \right) - \frac{1}{2} \left( v^2 dt \right) \\ &= \left( r - \frac{1}{2} v(t,T)^2 \right) dt + v(t,T) dz(t) \end{aligned}
Notice that $\ln P(t,T)$ needs to be read as the function $\ln$ applied to $P(t,T)$. Therefore, the differentiation is with respect to $P(t,T)$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.