Deriving Vasicek Zero-Coupon Bond Prices from the Market Price of Risk

# Pricing a zero with Vasicek model


# 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 by user16891 (score 6, accepted)

https://quant.stackexchange.com/a/18900

### 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 $$ A(t,T)= exp\left((b + \frac{\sigma\phi}{a}-\frac{\sigma^2}{a^2})(B(t,T) - T + t) + \frac{B(t,T)^2\sigma^2}{4a}\right)$$