Bond Pricing in Continuous Time
The Vasicek Model
The Vasicek (1977) model is one of the foundational equilibrium term-structure models in finance. The single state variable is the short rate r, which follows an Ornstein-Uhlenbeck process: dr = \kappa (\theta - r) \, dt + \sigma \, dB. The stochastic discount factor is then specified as \frac{d\Lambda}{\Lambda} = -r \, dt - (\lambda_0 + \lambda_1 r) \, dB. The parameter \kappa > 0 governs the speed of mean-reversion: the short rate is pulled back toward its long-run mean \theta at rate \kappa. The parameter \sigma controls the instantaneous volatility of the short rate, and \lambda_0 + \lambda_1 r is the market price of interest-rate risk, affine in r. The model is tractable because both the SDF and the short rate are driven by the same Brownian motion B, so interest-rate risk is the only source of uncertainty and bond prices can be derived in closed form.
The affine specification of the market price of risk is the defining feature of the essentially affine class of term-structure models introduced by Duffee (2002). Its chief advantage over a constant market price of risk is that it decouples the physical and risk-neutral dynamics: \kappa^* and \kappa can now differ freely, while in the constant case they are equal by construction. This distinction matters because yields are shaped by risk-neutral expectations — pinning down \kappa^* and \theta^* — whereas the time series of rates identifies the physical parameters \kappa and \theta. Dai and Singleton (2000) show that this overidentification is a serious constraint in completely affine models: fitting the cross-section of yields forces unrealistic physical dynamics, and vice versa. Allowing \lambda_1 \neq 0 resolves the tension. A further empirical motivation comes from the failure of the expectations hypothesis. Fama and Bliss (1987) and Cochrane and Piazzesi (2005) document that forward rates and forward-rate combinations forecast bond excess returns with significant predictive power — a pattern that requires the risk premium to vary over time with the state of the economy, exactly as \lambda_0 + \lambda_1 r implies.
Consider a zero-coupon bond Z(r, T) with face value $1 maturing in T periods. The fundamental pricing equation in continuous time states that the expected excess return of any asset equals minus the covariance of the SDF with the asset’s return: \operatorname{E}(dZ) - r Z \, dt = -\frac{d\Lambda}{\Lambda} \, dZ. The left-hand side is the expected capital gain in excess of the risk-free return, while the right-hand side captures the risk premium required by investors.
This equation has a direct interpretation in terms of a risk-neutral measure. By Girsanov’s theorem, define \operatorname{P}^* via the Radon–Nikodym derivative d\operatorname{P}^*/d\operatorname{P}= \Lambda_T/\Lambda_0, so that dB^* = dB + (\lambda_0 + \lambda_1 r)\,dt defines a Brownian motion under \operatorname{P}^*. Substituting dB = dB^* - (\lambda_0 + \lambda_1 r)\,dt into the short rate equation, the dynamics of r under \operatorname{P}^* become dr = \kappa^*(\theta^* - r)\,dt + \sigma\,dB^*, \qquad \kappa^* \equiv \kappa + \sigma\lambda_1, \quad \theta^* \equiv \frac{\kappa\theta - \sigma\lambda_0}{\kappa^*}. Under this change of measure, the fundamental pricing equation reduces to \operatorname{E}^*(dZ) = rZ\,dt: every asset earns exactly the risk-free rate in expectation under \operatorname{P}^*, so the discounted price process e^{-\int_0^t r_s\,ds}Z is a martingale under \operatorname{P}^*.
Solving the PDE
The martingale condition \operatorname{E}^*(dZ) = rZ\,dt translates directly into a PDE for Z(r,T). Applying Ito’s lemma under the risk-neutral dynamics: \begin{aligned} dZ &= Z_r \, dr + \tfrac{1}{2} Z_{rr} (dr)^2 - Z_T \, dt \\ &= \bigl(\kappa^*(\theta^* - r) Z_r + \tfrac{1}{2} \sigma^2 Z_{rr} - Z_T\bigr) dt + Z_r \sigma \, dB^*. \end{aligned} The minus sign before Z_T arises because T denotes time to maturity, which decreases as calendar time advances. Setting the drift equal to rZ yields the bond pricing PDE: \kappa^*(\theta^* - r) Z_r + \tfrac{1}{2} \sigma^2 Z_{rr} - Z_T - r Z = 0, \tag{1} subject to the boundary condition Z(r, 0) = 1. The parameters \lambda_0 and \lambda_1 jointly govern the compensation investors demand for interest-rate risk. When \kappa^* > 0, the risk-neutral dynamics remain mean-reverting, and \theta^* generally differs from \theta, embedding a risk premium in bond yields.
To solve (1), we exploit the affine structure of the short rate dynamics and guess a solution of the form Z(r, T) = \exp\left(a(T) + r b(T)\right), so that the log bond price is affine in the short rate. Computing Z_r = b(T) Z, Z_{rr} = b(T)^2 Z, Z_T = (a' + rb')Z and substituting into (1), then collecting terms by powers of r, decouples the equation into two ODEs: \begin{aligned} a' &= \kappa^* \theta^* b + \tfrac{1}{2} \sigma^2 b^2, \\ b' &= -1 - \kappa^* b, \end{aligned} subject to a(0) = 0 and b(0) = 0. Solving explicitly: \begin{aligned} b(T) &= -\phi(T) \, T, \\ a(T) &= -\theta^* T \bigl(1 - \phi(T)\bigr) + \frac{\sigma^2}{2(\kappa^*)^2} T\bigl(1 - 2\phi(T) + \phi(2T)\bigr), \end{aligned} \tag{2} where \phi(T) = (1 - e^{-\kappa^* T})/(\kappa^* T) satisfies \phi(T) \to 1 as T \to 0 and \phi(T) \to 0 as T \to \infty.
Risk-Neutral Expectation
Under the risk-neutral measure \operatorname{P}^*, the bond price equals the expected discounted payoff: Z(r, T) = \operatorname{E}^*\!\left[\exp\!\left(-\int_0^T r_s \, ds\right) \bigg| r_0 = r\right]. Under \operatorname{P}^*, as established above, r follows an OU process with speed of mean reversion \kappa^* and long-run mean \theta^*: dr = \kappa^*(\theta^* - r) \, dt + \sigma \, dB^*. Solving this SDE explicitly gives r_t = \theta^* + (r - \theta^*)e^{-\kappa^* t} + \sigma \int_0^t e^{-\kappa^*(t-s)} \, dB^*_s. Since r_t is an affine function of B^*, the integrated short rate X_T \equiv \int_0^T r_s \, ds is normally distributed under \operatorname{P}^*. Its conditional mean and variance are \begin{aligned} \mu_T &\equiv \operatorname{E}^*[X_T \mid r_0 = r] = \theta^* T\bigl(1 - \phi(T)\bigr) + r\phi(T) T, \\ v_T^2 &\equiv \operatorname{V}^*(X_T) = \frac{\sigma^2 T}{(\kappa^*)^2}\bigl(1 - 2\phi(T) + \phi(2T)\bigr). \end{aligned} The mean follows directly from integrating \operatorname{E}^*[r_t] = \theta^* + (r - \theta^*)e^{-\kappa^* t}. The variance is obtained by integrating the autocovariance of the OU process, \operatorname{Cov}^*(r_s, r_t) = (\sigma^2/2\kappa^*)(e^{-\kappa^*|t-s|} - e^{-\kappa^*(t+s)}), over the square [0, T]^2.
For any X \sim N(\mu, v^2), the moment generating function gives \operatorname{E}[e^{-X}] = e^{-\mu + v^2/2}, so Z(r, T) = \exp\!\left(-\mu_T + \tfrac{1}{2}v_T^2\right) = \exp\!\bigl(a(T) + r b(T)\bigr), recovering exactly the affine form with a(T) and b(T) from (2).
The Yield Curve
Writing Z(r, T) = \exp(-y(T) \, T), the continuously-compounded zero-coupon yield is y(T) = \theta^*\bigl(1 - \phi(T)\bigr) - \frac{\sigma^2}{2(\kappa^*)^2}\bigl(1 - 2\phi(T) + \phi(2T)\bigr) + \phi(T) \, r, an affine function of r, which is the hallmark of affine term-structure models. At the short end, \phi(T) \to 1 so y(T) \to r; at the long end, \phi(T) \to 0 and the yield converges to \lim_{T \to \infty} y(T) = \theta^* - \frac{\sigma^2}{2(\kappa^*)^2}. The asymptotic long yield lies below \theta^* by the convexity correction \sigma^2 / (2(\kappa^*)^2). Because bond prices are a convex function of the short rate, they respond more to rate decreases than to rate increases of the same magnitude. This asymmetry pushes long-term bond prices above what discounting at \theta^* would imply, and correspondingly pulls long yields below \theta^*.