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Deriving the Vasicek Zero-Coupon Bond Price

Original question

Can you help me derive the bond pricing formula under the Vasicek interest rate model for this assignment?
Solve the Vasicek short-rate process under the pricing measure, then find the mean and variance of the integral of the rate to maturity. That integral is Gaussian, so the zero-coupon bond price is the exponential of minus its mean plus half its variance. For a three-year bond with today's short rate 3%3\%, long-run level 4.5%4.5\%, mean-reversion speed 0.70.7, and diffusion volatility 1.5%1.5\%, the price per unit face value is about 0.890550.89055.

Integrate the short rate before taking the discount expectation

Measure time forward from today, fix maturity τ\tau, and assume

drs=κ(θ−rs) ds+σr dWsQ,κ>0.dr_s=\kappa(\theta-r_s)\,ds+\sigma_r\,dW_s^{\mathbb Q},\qquad \kappa>0.

An integrating factor gives rs=θ+(r0−θ)e−κs+σr∫0se−κ(s−u)dWuQr_s=\theta+(r_0-\theta)e^{-\kappa s}+\sigma_r\int_0^s e^{-\kappa(s-u)}dW_u^{\mathbb Q}. Define B(a)=(1−e−κa)/κB(a)=(1-e^{-\kappa a})/\kappa. Integrating the short rate over time and switching the deterministic and stochastic integrals yields

I=∫0τrs ds=θτ+(r0−θ)B(τ)+σr∫0τB(τ−u) dWuQ.I=\int_0^\tau r_s\,ds= \theta\tau+(r_0-\theta)B(\tau)+ \sigma_r\int_0^\tau B(\tau-u)\,dW_u^{\mathbb Q}.

The stochastic term is Gaussian with mean zero. The Itô isometry gives variance

v=σr2κ2[τ−2B(τ)+1−e−2κτ2κ].v=\frac{\sigma_r^2}{\kappa^2} \left[\tau-2B(\tau)+\frac{1-e^{-2\kappa\tau}}{2\kappa}\right].

With m=θτ+(r0−θ)B(τ)m=\theta\tau+(r_0-\theta)B(\tau), the normal moment-generating formula gives P(0,τ)=EQ[e−I]=e−m+v/2P(0,\tau)=\mathbb E^{\mathbb Q}[e^{-I}]=e^{-m+v/2}. Equivalently, P(0,τ)=A(τ)e−B(τ)r0P(0,\tau)=A(\tau)e^{-B(\tau)r_0}, where log⁡A=−θ(τ−B(τ))+v/2\log A=-\theta(\tau-B(\tau))+v/2. The current rate enters the log price linearly, which gives the model its affine bond-pricing form. The maturity-dependent coefficient B(τ)B(\tau) also gives the bond’s sensitivity to today's short rate. Differentiating the affine price gives ∂P/∂r0=−B(τ)P\partial P/\partial r_0=-B(\tau)P. As maturity grows, B(τ)B(\tau) approaches 1/κ1/\kappa, reflecting how the current short rate gradually loses influence on future rates.

The current-rate coefficient B(τ)B(\tau) comes from integrating the exponential decay of a one-unit increase in r0r_0. The random part requires integrating the effect of each Brownian shock over all subsequent rates up to maturity. After exchanging the integrations, a shock at time uu gets weight σr(1−e−κ(τ−u))/κ\sigma_r(1-e^{-\kappa(\tau-u)})/\kappa. This integrand is deterministic, so the accumulated random term is Gaussian and its variance is the integral of the squared weight. Only then apply the Gaussian exponential moment. The convexity adjustment is why discounting at the mean accumulated short rate alone gives a different price.

To check the variance algebra, write k=κk=\kappa and expand the squared kernel under the integral. After changing variables to the time from each shock to maturity, its integral is

σr2k2[τ−2(1−e−kτ)k+1−e−2kτ2k].\frac{\sigma_r^2}{k^2}\left[\tau-\frac{2(1-e^{-k\tau})}{k}+\frac{1-e^{-2k\tau}}{2k}\right].

The constant, first exponential and squared exponential each produce one of the three terms. The expression must be nonnegative because it came from integrating a square. When σr=0\sigma_r=0, it vanishes and the price is simply the discount factor along the deterministic mean-reverting rate path.

Translate a Gaussian integral into a bond price and yield

Set r0=0.03r_0=0.03, θ=0.045\theta=0.045, κ=0.7\kappa=0.7, σr=0.015\sigma_r=0.015, and τ=3\tau=3. Then B(3)≈1.253634B(3)\approx1.253634, m≈0.116195m\approx0.116195, and v≈0.0005493v\approx0.0005493. The bond price is e−0.116195+0.0005493/2≈0.890546e^{-0.116195+0.0005493/2}\approx0.890546. The continuously compounded three-year yield is −log⁡(0.890546)/3≈3.864%-\log(0.890546)/3\approx3.864\%. It lies above today's short rate in this exercise because the expected rate rises toward its long-run level over the investment horizon.

References and next steps