interest rates - Pricing a zero with Vasicek model

I'm trying to understand bond pricing with the Vasicek interest rate model. I'm using McDonald's book for this purpose (not homework).

Recall that Vasicek dynamics are $$\begin{equation*} \mathrm{d}r_t = a(b - r_t) \mathrm{d}t + \sigma \mathrm{d}Z_t. \end{equation*}$$

Now, Macdonald introduces the exponential affine formulas to price a unit zero:

$$\begin{align*} P(r,t,T) &= A(t,T)\exp\left(rB(t,T)\right) \\ A(t,T) &= \exp\left(\bar{r} (B(t,T) - T + t) + \frac{B(t,T)^2\sigma^2}{4a}\right)\\ B(t,T) &= \frac{1 - e^{-a(T-t)}}{a}\\ \bar{r} &= b + \frac{\sigma\phi}{a} - \frac{\sigma^2}{a^2} \end{align*}$$

In the course of deriving these expressions, Macdonald asks us to assume that $\phi$, which is the Sharpe ratio for the motion, is constant. But we can see that it is only constant when $a = 0$.

Later, Macdonald talks about the Sharpe ratio for the "interest rate risk", a phrasing I find very obscure. Is that the bond price process? The Vasicek process? In either case, they're driven by the same Brownian motion and should have the same (non-constant) Sharpe ratio.

Can somebody explain how to apply these formulas? Just a sketch would do -- but I'm stymied by the presentation.

Answer

Bond Prices

Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation $$\begin{align} d{{r}_{t}}=\mu ({{r}_{t}},t)dt+\sigma ({{r}_{t}},t)d{{W}_{t}^{P}} \end{align}$$ we assume the bond price to be dependent on $r_t$ only, independent of default risk, liquidity and other factors. If we write the bond price as $P(r_t, t)=V(t,r_t,T)$ such that $V(t,r_t,t)=1$ then $$\begin{align} dV=({{V}_{t}}+\mu \,{{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}})dt+\sigma {{V}_{r}}d{{W}_{t}} \end{align}$$ for simplicity let $$\begin{align} & {{\mu }_{V}}=\frac{{{V}_{t}}+\mu {{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}}}{V} \\ & {{\sigma }_{V}}=\frac{\sigma {{V}_{r}}\,}{V} \\ \end{align}$$ thus we have $$\begin{align} dV={{\mu }_{V}}\,Vdt+{{\sigma }_{V}}\,V\,d{{W}_{t}} \end{align}$$ The following portfolio is constructed: we buy a bond of dollar value V1 with maturity $T_1$ and sell another bond of dollar value $V_2$ with maturity $T_2$. The portfolio value $\Pi$ is given by $$\begin{align} \Pi ={{V}_{1}}-{{V}_{2}} \end{align}$$ According to the bond price dynamics,we have $$\begin{align} \Pi =({{\mu }_{{{V}_{1}}}}{{V}_{1}}-{{\mu }_{{{V}_{2}}}}{{V}_{2}})\,dt+({{\sigma }_{{{V}_{1}}}}{{V}_{1}}-{{\sigma }_{{{V}_{2}}}}{{V}_{2}})\,d{{W}_{t}} \end{align}$$ Suppose $V_1$ and $V_2$ are chosen such that $$\begin{align} & {{V}_{1}}=\frac{{{\sigma }_{{{V}_{2}}}}}{{{\sigma }_{{{V}_{2}}}}-{{\sigma }_{{{V}_{1}}}}}\Pi \\ & {{V}_{2}}=\frac{{{\sigma }_{{{V}_{1}}}}}{{{\sigma }_{{{V}_{2}}}}-{{\sigma }_{{{V}_{1}}}}}\Pi \\ \end{align}$$ then the stochastic term in $d\Pi$ vanishes and the equation becomes $$d\Pi =\left( \frac{{{\mu }_{{{V}_{1}}}}{{\sigma }_{{{V}_{2}}}}-{{\mu }_{{{V}_{2}}}}{{\sigma }_{{{V}_{1}}}}}{{{\sigma }_{{{V}_{2}}}}-{{\sigma }_{{{V}_{1}}}}} \right)\Pi \,dt$$ Since the portfolio is instantaneously riskless, in order to avoid arbitrage opportunities,it must earn the riskless short rate so that $d\Pi =r(t)\Pi dt$ ,then $$\frac{{{\mu }_{{{V}_{1}}}}-r(t)}{{{\sigma }_{{{V}_{1}}}}}=\frac{{{\mu }_{{{V}_{2}}}}-r(t)}{{{\sigma }_{{{V}_{2}}}}}$$ The above relation is valid for arbitrary maturity dates $T_1$ and $T_2$, so the ratio should be independent of maturity $T$.Let the common ratio be defined by $\lambda$, that is, $$\frac{{{\mu }_{V}}-r(t)}{{{\sigma }_{V}}}=\lambda \,({{r}_{t}},t)$$ The quantity $\lambda$ is called the market price of risk of the short rate.If we substitute $μ_V(r, t)$ and $σ_V(r, t)$ into above Equation, we obtain the following governing differential equation for the price of a zero-coupon bond $${{V}_{t}}+(\mu -\lambda \sigma \,){{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}}-{{r}_{t}}\,V=0$$

Change Measure

we assume $Q$ be a martingale measure such that $$dW_{t}^{P}=-\lambda(r,t)dt+dW_{t}^{Q}$$ thus we have $$d{{r}_{t}}=\mu^*(r_t,t)dt+\sigma ({{r}_{t}},t)dW_{t}^{Q}$$ where $$\mu^*(r_t,t)=\mu({{r}_{t}},t)-\lambda ({{r}_{t}},t)\sigma ({{r}_{t}},t)$$

Affine Term Structure Models

A short rate model that generates the bond price solution of the form $$P(t\,,T)=V(t,r_t,T)={{e}^{A(t,T)\,-\,B(t,T){{r}_{t}}\,}}$$ Suppose the dynamics of the short rate $r_t$ under the risk neutral measure $Q$ is governed by $$\begin{align} d{{r}_{t}}=\mu^* ({{r}_{t}},t)dt+\sigma ({{r}_{t}},t)d{{W}_{t}^{Q}} \end{align}$$ where $$\begin{align} &\mu^* ({{r}_{t}},T)=\alpha (t)\,{{r}_{t}}+\beta (t) \\ &{{\sigma }^{2}}({{r}_{t}},T)=\gamma (t)\,{{r}_{t}}+\delta (t) \\ \end{align}$$ We show the governing equation for $P(t, T )=V(t,r,T)$ is given by $${{V}_{t}}+\mu ^*{{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}}-{{r}_{t}}\,V=0$$

Substituting the assumed affine solution of bond price into this equation, we obtain $$\begin{align} & {{B}_{t}}(t,T)+\alpha (t)B(t,T)-\frac{1}{2}\gamma (t){{B}^{2}}(t,T)=-1 \\ &B(T,T)=0 \\ \end{align}$$ and $$\begin{align} & {{A}_{t}}(t,T)=\beta (t)B(t,T)-\frac{1}{2}\delta (t){{B}^{2}}(t,T) \\ & A(T,T)=0 \\ \end{align}$$

Vasicek Model

Vasicek (1977) proposed the stochastic process for the short rate $r_t$ under the Martingle measure to be governed by the Ornstein–Uhlenbeck process: $$d{{r}_{t}}=a(b-r_t)dt+\sigma d{{W}_{t}^{Q}}$$ hence $$\begin{align} & \alpha (t)=-a\,\,\,\,\,\,,\,\,\,\,\,\,\,\beta (t)=ab \\ & \gamma (t)=\,0\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\delta (t)={{\sigma }^{2}} \\ \end{align}$$ thus we have $$B(t,T)= \frac{1 - e^{-a(T-t)}}{a}\\$$ and $$A(t,T)= exp\left((b + \frac{\sigma\phi}{a}-\frac{\sigma^2}{a^2})(B(t,T) - T + t) + \frac{B(t,T)^2\sigma^2}{4a}\right)$$