European Options on Zero-Coupon Bonds in the Vasicek Model

From Vasicek Rates to Bond-Option Prices

A European option on a zero-coupon bond pays at a fixed expiry $S$, while the underlying bond pays one unit at a later maturity $T$, where $t\le S<T$. Under the Vasicek model, the short rate at expiry is conditionally Gaussian. Since a zero-coupon bond price is exponential-affine in the short rate, its value at expiry is lognormal. Discounting the option payoff to time $t$ then gives a closed-form price.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{Q})$ carrying a Brownian motion $B_t$ under the risk-neutral measure $\mathbb{Q}$. The short rate follows the Vasicek SDE $dr_t=\kappa(\theta-r_t)\,dt+\sigma\,dB_t$, with $\kappa>0$, $\sigma\ge0$, and constant $\theta$. The stochastic integral is an Itô integral: its integrand is progressively measurable and belongs to $\mathcal{L}^2$ on every finite horizon, meaning $\mathbb{E}^{\mathbb{Q}}\int_0^T H_s^2\,ds<\infty$. The drift coefficient is $\kappa(\theta-r_t)$, and the diffusion coefficient is $\sigma$. Option prices below are expectations under $\mathbb{Q}$, not under the physical measure $\mathbb{P}$.

For $u>0$, define $B(u)=(1-e^{-\kappa u})/\kappa$. The Vasicek zero-coupon bond price, obtained from the affine bond-pricing equation, is $P(t,U)=A(t,U)e^{-B(U-t)r_t}$, where

A(t,U)=\exp\left[\left(\theta-\frac{\sigma^2}{2\kappa^2}\right) \big(B(U-t)-(U-t)\big)-\frac{\sigma^2}{4\kappa}B(U-t)^2\right].

Here $P(t,U)$ is the time-$t$ price of one unit paid at $U$. A call with strike $K>0$ pays $(P(S,T)-K)^+$ at $S$; the corresponding put pays $(K-P(S,T))^+$. The risk-neutral pricing rule discounts these payoffs by the money-market account, giving the conditional discount factor $\exp(-\int_t^S r_s\,ds)$.

The Conditional Distribution at Option Expiry

Solving the linear short-rate equation from $t$ to $S$ gives, with $\tau=S-t$,

r_S=\theta+(r_t-\theta)e^{-\kappa\tau} +\sigma\int_t^S e^{-\kappa(S-s)}\,dB_s.

The integrand is deterministic and square-integrable, so the Itô integral is Gaussian with mean zero and variance given by the Itô isometry. Consequently, conditional on $\mathcal{F}_t$, $r_S$ is normal with variance $\sigma^2(1-e^{-2\kappa\tau})/(2\kappa)$. Because $\log P(S,T)=\log A(S,T)-B(T-S)r_S$, the log bond price is normal as well.

Proposition: Conditional on $\mathcal{F}_t$, the random variable $\log P(S,T)$ is normal with variance $$ \nu^2=\sigma^2 B(T-S)^2\frac{1-e^{-2\kappa(S-t)}}{2\kappa}. $$ In particular, the bond price $P(S,T)$ is lognormal if $\sigma>0$ and $t<S<T$. If $\sigma=0$, or $S=t$, or $T=S$, its conditional variance is zero.

Proof. The solution above writes $r_S$ as an $\mathcal{F}_t$-measurable quantity plus an Itô integral with deterministic integrand. Its conditional variance is $\sigma^2\int_t^S e^{-2\kappa(S-s)}\,ds =\sigma^2(1-e^{-2\kappa(S-t)})/(2\kappa)$. The bond formula at time $S$ gives $\log P(S,T)=\log A(S,T)-B(T-S)r_S$, so multiplying the variance of $r_S$ by $B(T-S)^2$ gives the stated variance. The variance vanishes in each listed degenerate case. $\square$

For option pricing, the relevant distribution is not the ordinary $\mathbb{Q}$-distribution of $P(S,T)$ alone. The payoff is discounted to $t$, so the discount factor and the bond price must be treated together. The next result makes that adjustment precise without requiring an explicit formula for the tilted mean.

Definition: For fixed $t<S$, define a probability measure on events in $\mathcal{F}_S$, conditional on $\mathcal{F}_t$, by weighting with $\exp(-\int_t^S r_s\,ds)/P(t,S)$. This is the $S$-forward measure over this interval. It is well-defined because $P(t,S)=\mathbb{E}^{\mathbb{Q}}[\exp(-\int_t^S r_s\,ds)\mid\mathcal{F}_t]>0$.

Theorem: Let $t<S<T$, $\kappa>0$, and $\sigma>0$. Under the $S$-forward measure conditional on $\mathcal{F}_t$, the bond price $P(S,T)$ is lognormal with variance parameter $\nu^2$ as above and conditional mean $P(t,T)/P(t,S)$. Therefore, writing $$ h=\frac{\log\!\big(P(t,T)/(K P(t,S))\big)+\tfrac12\nu^2}{\nu}, $$ the European call and put prices are $$ C(t)=P(t,T)\Phi(h)-K P(t,S)\Phi(h-\nu), $$ $$ \operatorname{Put}(t)=K P(t,S)\Phi(-h+\nu)-P(t,T)\Phi(-h). $$

Proof. Set $D=\exp(-\int_t^S r_s\,ds)$. The Vasicek solution expresses both $r_S$ and $\int_t^S r_s\,ds$ as affine functions of Gaussian variables, so they are jointly Gaussian conditional on $\mathcal{F}_t$. Weighting a joint Gaussian distribution by the exponential of a linear function leaves its covariance unchanged and shifts its mean. This follows directly from the Gaussian moment-generating function: for a conditionally Gaussian vector $X$ with mean $m$ and covariance matrix $\Sigma$, weighting by $\exp(a^\mathsf{T}X)$ changes its normalized moment-generating function to $\exp(z^\mathsf{T}(m+\Sigma a)+\tfrac12 z^\mathsf{T}\Sigma z)$. Thus under the stated discount-factor weighting, $\log P(S,T)$ remains normal with the same variance $\nu^2$. Its mean is determined by its expected bond price. The bond-pricing identity gives $\mathbb{E}^{\mathbb{Q}}[D P(S,T)\mid\mathcal{F}_t]=P(t,T)$, while $\mathbb{E}^{\mathbb{Q}}[D\mid\mathcal{F}_t]=P(t,S)$. Hence its mean under the weighted measure is $P(t,T)/P(t,S)$.

To compute the call, write $Y=P(S,T)$. Under this measure, $\log Y$ is normal with variance $\nu^2$ and mean $\log(P(t,T)/P(t,S))-\tfrac12\nu^2$, since a lognormal variable with log-variance $\nu^2$ has mean equal to the exponential of its log-mean plus $\nu^2/2$. Integrating $(Y-K)^+$ over $Y>K$ gives $\mathbb{E}[(Y-K)^+\mid\mathcal{F}_t]= (P(t,T)/P(t,S))\Phi(h)-K\Phi(h-\nu)$. Multiplication by $P(t,S)$ gives the call formula. Integrating $(K-Y)^+$ over $Y<K$ gives the put formula. These integrations follow by completing the square in the normal density; the terms are respectively the truncated first moment of $Y$ and the probability of the truncation event. The pricing expectations are finite because the discounted first moment of $Y$ is $P(t,T)$, and the put payoff is bounded above by $K$. $\square$

Worked Prices and Checks

Worked Example: A Forward-At-The-Money Bond Option

Take $\kappa=1$, $\theta=0.04$, $\sigma=0.2$, $r_0=0.03$, $t=0$, $S=1$, and $T=2$. Choose $K=P(0,2)/P(0,1)$, so the strike equals the forward bond price.

First, $B(1)=1-e^{-1}=0.6321206$ and $B(2)=1-e^{-2}=0.8646647$. Since $\theta-\sigma^2/(2\kappa^2)=0.04-0.04/2=0.02$, the affine formula gives $\log P(0,1)=0.02(0.6321206-1)-0.01(0.6321206)^2-0.03(0.6321206)=-0.0303170$, hence $P(0,1)\approx0.9701380$. Likewise, $\log P(0,2)=0.02(0.8646647-2)-0.01(0.8646647)^2-0.03(0.8646647)=-0.0561231$, hence $P(0,2)\approx0.9454228$. Thus $K=P(0,2)/P(0,1)\approx0.974525$.

Here $B(T-S)=B(1)=0.6321206$, and $\sqrt{(1-e^{-2})/2}=\sqrt{(1-0.1353353)/2}\approx0.657519$. Therefore $\nu=0.2(0.6321206)(0.657519)\approx0.0831263$. The forward-at-the-money choice makes the logarithm in $h$ zero, so $h=\nu/2\approx0.0415632$. Substitution in the standard normal CDF gives $\Phi(h)\approx0.5165766$ and $\Phi(h-\nu)=\Phi(-h)\approx0.4834234$. The call price is $P(0,2)\Phi(h)-K P(0,1)\Phi(-h) =P(0,2)(2\Phi(h)-1)\approx0.9454228(0.0331532)\approx0.0313440$. The put price is the same because this strike makes the forward call and put values equal.

Check: put-call parity gives $C-\operatorname{Put}=P(0,2)-K P(0,1)=0$ by the definition of $K$, agreeing with the equal computed prices.

Worked Example: Zero Volatility

Set $\sigma=0$, $\theta=r_0=0.03$, and keep $\kappa=1$. Then the rate is constant: $r_s=0.03$, and $P(0,U)=e^{-0.03U}$. For $S=1$, $T=2$, and $K=0.95$, the bond price at expiry is $P(1,2)=e^{-0.03}\approx0.9704455$. The call payoff is $0.9704455-0.95=0.0204455$, and the put payoff is zero. Discounting the call payoff by $P(0,1)=e^{-0.03}$ gives $C(0)=e^{-0.03}(e^{-0.03}-0.95)=e^{-0.06}-0.95e^{-0.03}\approx0.0198413$.

Check: $P(0,2)-K P(0,1)=e^{-0.06}-0.95e^{-0.03}\approx0.0198413=C(0)-\operatorname{Put}(0)$. The general formula using $h$ is not evaluated by dividing by $\nu$ here, because $\nu=0$; the deterministic payoff is the appropriate limiting case.

Worked Example: Bond Maturity Equals Option Expiry

Suppose $T=S$, so the underlying bond pays one unit exactly when the option expires. At expiry its value is $P(S,S)=1$, with no uncertainty. For $K=1.05$, $S=1$, and the constant rate $r=0.03$, the call payoff is $(1-1.05)^+=0$ and the put payoff is $(1.05-1)^+=0.05$. Since $P(0,1)=e^{-0.03}\approx0.9704455$, the put price is $0.05e^{-0.03}\approx0.0485223$.

Check: parity gives $C-\operatorname{Put}=0-0.0485223$. Its right side is $P(0,S)-K P(0,S)=(1-1.05)e^{-0.03}=-0.05(0.9704455)\approx-0.0485223$, as required.

Put-Call Parity and Common Pitfalls

Subtracting the put formula from the call formula cancels the normal-CDF terms using $\Phi(x)+\Phi(-x)=1$. This gives the bond-option parity relation $C(t)-\operatorname{Put}(t)=P(t,T)-K P(t,S)$. It is also obtained directly from the payoff identity $(x-K)^+-(K-x)^+=x-K$, followed by risk-neutral discounting. The parity relation is therefore a useful independent check on calculations.

A common error is to discount the option payoff by $e^{-r_t(S-t)}$, treating the current short rate as constant. Under Vasicek, the discount factor is instead $\exp(-\int_t^S r_s\,ds)$; its conditional expectation is $P(t,S)$. A second error is to use the variance of $r_S$ without multiplying by $B(T-S)^2$. The option is written on the bond price, whose logarithm scales rate uncertainty by that bond’s maturity loading.

The formula assumes $t<S<T$ and $\nu>0$. At $S=t$, the payoff is already known and is valued directly. At $T=S$, the underlying bond pays one at expiry and its value there is deterministic. At $\sigma=0$, the entire rate path is deterministic conditional on the current rate, so direct payoff evaluation replaces the division by $\nu$. These cases preserve put-call parity.

Check Your Understanding

Use the stated maturity ordering and distinguish the bond’s maturity from the option’s expiry in each answer.

For $t<S<T$, derive the conditional variance of $r_S$ from the explicit Vasicek solution.
Explain why the log bond price at $S$ has variance $\sigma^2B(T-S)^2(1-e^{-2\kappa(S-t)})/(2\kappa)$.
Show directly from the payoff identity that $C(t)-\operatorname{Put}(t)=P(t,T)-KP(t,S)$.
What should replace the formula involving $h$ when $\sigma=0$, and why?
If $T=S$, what is the bond’s value at option expiry, and what does that imply for the option payoff?

European Options on Zero-Coupon Bonds in the Vasicek Model

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