(Correlation between long rate and short rate in the one-factor Vasicek model). The one-factor Vasicek model is the one-factor Hull-White model of Example 6.5.1 with constant parameters,
$$
d R(t)=(a-b R(t)) d t+\sigma d \widetilde{W}(t),
$$
where $a, b$, and $\sigma$ are positive constants and $\widetilde{W}(t)$ is a one-dimensional Brownian motion. In this model, the price at time $t \in[0, T]$ of the zero-coupon bond maturing at time $T$ is
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10 Term-Structure Models
$$
B(t, T)=e^{-C(t, T) R(t)-A(t, T)},
$$
where $C(t, T)$ and $A(t, T)$ are given by (6.5.10) and (6.5.11):
$$
\begin{aligned}
C(t, T) & =\int_t^T e^{-\int_t^s b d v} d s=\frac{1}{b}\left(1-e^{-b(T-t)}\right), \\
A(t, T) & =\int_t^T\left(a C(s, T)-\frac{1}{2} \sigma^2 C^2(s, T)\right) d s \\
& =\frac{2 a b-\sigma^2}{2 b^2}(T-t)+\frac{\sigma^2-a b}{b^3}\left(1-e^{-b(T-t)}\right)-\frac{\sigma^2}{4 b^3}\left(1-e^{-2 b(T-t)}\right) .
\end{aligned}
$$
In the spirit of the discussion of the short rate and the long rate in Subsection 10.2.1, we fix a positive relative maturity $\bar{\tau}$ and define the long rate $L(t)$ at time $t$ by $(10.2 .30)$ :
$$
L(t)=-\frac{1}{\bar{\tau}} \log B(t, t+\bar{\tau}) .
$$
Show that changes in $L(t)$ and $R(t)$ are perfectly correlated (i.e., for any $0 \leq t_1<t_2$, the correlation coefficient between $L\left(t_2\right)-L\left(t_1\right)$ and $R\left(t_2\right)-R\left(t_1\right)$ is one). This characteristic of one-factor models caused the development of models with more than one factor.