Applying Itô’s Lemma to a Vasicek Bond Pricing Process
Summary
The document applies Itô’s lemma to an exponential function of an integrated short rate under the Vasicek model. It derives the stochastic differential of the intermediate process, then uses its drift and diffusion terms to obtain the dynamics of the exponential quantity. The resulting drift depends on the deterministic function a and the model parameters, while the diffusion term is proportional to the Brownian shock.
For the process to be a martingale, its drift must vanish, which gives a differential condition for a. The accepted response corrects the original calculation by including the differential of the time dependent coefficient in x and shows that its drift contribution cancels terms involving the current short rate. The excerpt states the resulting dynamics but does not solve the condition for a or discuss measure changes, boundary conditions, or martingale integrability assumptions.
Key ideas
- The Vasicek short rate follows mean reverting drift with constant volatility.
- Itô’s lemma gives the drift and diffusion of the exponential process from those of its exponent.
- Correctly differentiating the time dependent coefficient in the integrated rate expression is essential.
- The martingale condition is obtained by setting the process drift to zero.
- The document does not provide a full solution for the deterministic function or its boundary conditions.
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# Application of Ito's Lemma, finding the condition for the martingale
# Application of Ito's Lemma, finding the condition for the martingale
The Vasicek short rate model is $$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$ Define the processes $x_t$ and $f(x,t)$ $$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$ $$f(x,t)=e^{a(T-t)-x_t}$$ Note that $a(t)$ is a function.
Question 1: Find $df(x,t)$ using Ito's Lemma.
My attempt: \begin{align} dx_t&=\frac{1-e^{-\kappa(T-t)}}{\kappa}dr_t+\frac{r_t}{\kappa}(-e^{-\kappa(T-t)})\kappa dt\\ &=\frac{1-e^{-\kappa(T-t)}}{\kappa}(\kappa(\theta-r_t)dt+\sigma dW_t)-r_te^{-\kappa(T-t)}dt\\ &=(1-e^{-\kappa(T-t)})(\theta-r_t)dt+\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t-r_te^{-\kappa(T-t)}dt\\ &=\Big( (1-e^{-\kappa(T-t)})\theta-r_t+r_te^{-\kappa(T-t)}-r_te^{-\kappa(T-t)}\Big)dt+\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t\\ &=\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)dt+\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t \end{align} $$(dx_t)^2=\frac{\sigma^2}{\kappa^2}(1-e^{-\kappa(T-t)})^2dt$$
$$\ln{f(x,t)}=a(T-t)-x_t$$ $$d\ln{f(x,t)}=d_ta(T-t)-dx_t=d_ta(T-t)-\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)dt-\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t$$ $$(d\ln{f(x,t)})^2=(dx_t)^2=\frac{\sigma^2}{\kappa^2}(1-e^{-\kappa(T-t)})^2dt$$ Applying Ito's Lemma to $f(x,t)$ \begin{align} d(e^{\ln{f(x,t)}})&=f(x,t)d\ln{f(x,t)}+\frac{1}{2}f(x,t)(d\ln{f(x,t)})^2\\ &=f(x,t)\Bigg(d_ta(T-t)-\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)dt-\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t\Bigg)+\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2dt\\ &=f(x,t)\Bigg(\frac{d}{dt}a(T-t)-\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)+\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2\Bigg)dt-\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t\\ \end{align} Question 2: Find the dynamics of the function $a(t)$ such that the process $f(x,t)$ is a martingale.
My attempt: $$\frac{d}{dt}a(T-t)-\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)+\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2=0$$ $$\frac{d}{dt}a(T-t)=\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)-\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2$$ $$d_ta(T-t)=\Bigg(\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)-\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2\Bigg)dt$$
Can anyone confirm the correctness of my attempts? Any help is appreciated.
## Answer by user16651 (score 5, accepted)
https://quant.stackexchange.com/a/30864
Note $$dx_t=\frac{1-e^{-\kappa(T-t)}}{\kappa}dr_t-e^{-\kappa (T-t)}r_tdt+\frac{1}{\kappa}\underbrace{d\left[1-e^{-\kappa(T-t)}\,,\,r_t\right]}_{0}+r_tdt$$ thus $$dx_t=\frac{1-e^{-\kappa(T-t)}}{\kappa}dr_t+\left(1-e^{-\kappa (T-t)}\right)r_tdt\,.$$ In other words $$dx_t=\theta\left(1-e^{-\kappa(T-t)}\right)dt+\frac{1-e^{-\kappa(T-t)}}{\kappa}\sigma\,dW_t\,.\tag 1$$ By application of Ito's lemma, we have $$d\left(e^{-x_t}\right)=-e^{-x_t}dx_t+\frac{1}{2}e^{-x_t}\underbrace{d[x_t\,,x_t]}_{\frac{\left(1-e^{-\kappa(T-t)}\right)^2}{\kappa^2}\sigma^2dt}$$ therefore $$d\left(e^{-x_t}\right)=\left(-\theta\left(1-e^{-\kappa(T-t)}\right)+\frac{\left(1-e^{-\kappa(T-t)}\right)^2}{2\kappa^2}\sigma^2\right)e^{-x_t}dt-\sigma\frac{1-e^{-\kappa(T-t)}}{\kappa} e^{-x_t}dW_t\tag 2$$ as a result $$df(t,x)=e^{-x_t}d\left(e^{a(T-t)}\right)+e^{a(T-t)}d\left(e^{-x_t}\right)+\underbrace{d\left[e^{a(T-t)}\,,\,e^{-x_t}\right]}_{0}$$ in other words $$df(t,x)=\left(\frac{\left(1-e^{-\kappa(T-t)}\right)^2}{2\kappa^2}\sigma^2-\theta\left(1-e^{-\kappa(T-t)}\right)-a'(T-t)\right)e^{a(T-t)-x_t}dt-\sigma e^{a(T-t)-x_t}\frac{1-e^{-\kappa(T-t)}}{\kappa}dW_t$$ or $$df(t,x)=\mu(t,T)f(t,x_t)dt+\sigma(t,T)f(t,x_t)dW_t\tag 3$$ where $$\mu(t,T)=\left(\frac{\left(1-e^{-\kappa(T-t)}\right)^2}{2\kappa^2}\sigma^2-\theta\left(1-e^{-\kappa(T-t)}\right)-a'(T-t)\right)$$ and $$\sigma(t,T)=-\sigma \frac{1-e^{-\kappa(T-t)}}{\kappa}$$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.