Why a One-Factor Vasicek Model Correlates Bond Returns

# One state variables implies perfect correlation of returns?


# One state variables implies perfect correlation of returns?












In Vasicek's seminal 1977 paper "An equilibrium characterization of the term structure", he states the bond price $P(t,s)$ is a function of the spot rate $r(t)$, $P(t,s) = P(t,s,r(t))$. He then states

> "Thus, the value of the spot rate is the only variable for the whole term structure.

Okay, so far so good. He continues,

> "Since there exists only one state variable, the instantaneous returns on bonds of different maturities are perfectly correlated.

I'm not sure why that would follow...could I get this translated to math, with a decent explanation? Finally, he states

> This means that the short bond and just one other bond completely span the whole of the term structure."

What exactly does this mean, and why would he need it?

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/18157

Under the Vasicek's model, the price of a zero-coupon bond is given by \begin{align*} P(t, T) = A(t, T)\exp\big(-B(t, T) r_t\big), \end{align*} where $A$ and $B$ are deterministic functions. In particular, $B$ is a positive increasing function (see any books on interest rate models). Then \begin{align*} \ln P(t, T) = \ln A(t, T) - B(t, T) r_t. \end{align*} For any four maturities $T_1$, $T_2$, $T_3$, and $T_4$, where $T_1 < T_2 <T_3 < T_4$, \begin{align*} \ln \frac{P(t, T_2)}{P(t, T_1)} = \ln \frac{A(t, T_2)}{A(t, T_1)} - [B(t, T_2)-B(t, T_1)] r_t, \end{align*} and \begin{align*} \ln \frac{P(t, T_4)}{P(t, T_3)} = \ln \frac{A(t, T_4)}{A(t, T_3)} - [B(t, T_4)-B(t, T_3)] r_t. \end{align*} It is now easy to check that \begin{align*} Var\bigg(\ln \frac{P(t, T_2)}{P(t, T_1)}\bigg) &= [B(t, T_2)-B(t, T_1)]^2 Var(r_t),\\ Var\bigg(\ln \frac{P(t, T_4)}{P(t, T_3)}\bigg) &= [B(t, T_4)-B(t, T_3)]^2 Var(r_t), \end{align*} and \begin{align*} Cov\bigg(\ln \frac{P(t, T_2)}{P(t, T_1)}, \ln \frac{P(t, T_4)}{P(t, T_3)}\bigg) &= [B(t, T_2)-B(t, T_1)] [B(t, T_4)-B(t, T_3)]Var(r_t). \end{align*} Then \begin{align*} Corr\bigg(\ln \frac{P(t, T_2)}{P(t, T_1)}, \ln \frac{P(t, T_4)}{P(t, T_3)}\bigg) &= 1. \end{align*}

As there are three model parameters, any two bonds can be used to calibrate them (there are two $A$s and two $B$s), and then all bond prices are known.

Updates based on poster's comments: Assuming that \begin{align*} dr_t = \alpha(r_t, t)dt + \beta(r_t, t) dW_t, \end{align*} where $\{W_t \mid t \geq 0\}$ is a standard Brownian motion. Moreover, assuming that the zero-coupon bond price is defined by \begin{align*} P(t, T) = P(t, T, r_t). \end{align*} Then \begin{align*} dP(t, T) &= \frac{\partial P(t, T)}{\partial t}dt + \frac{\partial P(t, T)}{\partial r_t}dr_t + \frac{1}{2}\frac{\partial^2 P(t, T)}{\partial r_t^2}\beta^2(r_t, t) dt\\ &=\bigg(\frac{\partial P(t, T)}{\partial t}+ \frac{\partial P(t, T)}{\partial r_t} \alpha(r_t, t) + \frac{1}{2}\frac{\partial^2 P(t, T)}{\partial r_t^2}\beta^2(r_t, t)\bigg)dt \\ &\qquad\qquad+ \frac{\partial P(t, T)}{\partial r_t}\beta(r_t, t)dW_t\\ &=P(t, T)\big[\mu(r_t, t, T)dt + \sigma(r_t, t, T)dW_t \big], \end{align*} for certain adapted functions $\mu(r_t, t, T)$ and $\sigma(r_t, t, T)$. Moreover, \begin{align*} Var\Big[dP(t, T)/P(t, T) \mid \mathcal{F}_t\Big] = \sigma^2(r_t, t, T) dt. \end{align*} Furthermore, for $T_1$, $T_2$, where $t < T_1 \leq T_2$, \begin{align*} Cov\Big[dP(t, T_1)/P(t, T_1),\, dP(t, T_2)/P(t, T_2) \mid \mathcal{F}_t\Big] = \sigma(r_t, t, T_1)\sigma(r_t, t, T_2) dt. \end{align*} That is, \begin{align*} Corr\Big[dP(t, T_1)/P(t, T_1),\, dP(t, T_2)/P(t, T_2) \mid \mathcal{F}_t\Big] = 1. \end{align*} In other words, the instantaneous returns on bonds of different maturities are perfectly correlated.