Exact Conditional Moments In The Vasicek Model

From Mean Reversion to Conditional Distributions

The Vasicek model describes a rate that is pulled toward a long-run level while receiving continuous random shocks. Its explicit solution does more than identify the average future rate: it gives the entire conditional distribution at a fixed horizon. From that Gaussian distribution, conditional variances, second moments, and terminal threshold probabilities follow directly.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\), where \(B_t\) is a standard Brownian motion with respect to the filtration. For constants \(\kappa>0\), \(\theta\in\mathbb{R}\), and \(\sigma\in\mathbb{R}\), the Vasicek SDE is

Here \(\kappa\) is the speed of mean reversion, \(\theta\) is the long-run level, and \(\sigma\) is the diffusion coefficient. The drift is \(\kappa(\theta-r_t)\); it is positive below \(\theta\), negative above \(\theta\), and zero at \(\theta\). The coefficients are globally Lipschitz in the state and satisfy linear growth: the drift has Lipschitz constant \(\kappa\), and its absolute value is bounded by \(\kappa(|\theta|+|r|)\); the diffusion is constant and bounded. The standard Lipschitz existence-and-uniqueness theorem therefore applies. The calculations below use the explicit solution.

Fix \(s\ge0\) and \(h\ge0\). Since \(r_s\) is \(\mathcal{F}_s\)-measurable, conditional moments below are random variables determined by the information available at time \(s\). All stochastic integrals in the derivation are Itô integrals. In particular, the deterministic integrand \(u\mapsto e^{-\kappa(s+h-u)}\) on \([s,s+h]\) is progressively measurable and has finite squared integral, so it belongs to \(\mathcal{L}^2\).

The Conditional Gaussian Transition Law

Proof. Apply Itô’s formula to \(f(t,x)=e^{\kappa t}x\). Its partial derivatives are \(f_t(t,x)=\kappa e^{\kappa t}x\), \(f_x(t,x)=e^{\kappa t}\), and \(f_{xx}(t,x)=0\). Thus the \(\tfrac12 f_{xx}\sigma^2\,dt\) correction is zero, and

Integrate from \(s\) to \(s+h\), then multiply by \(e^{-\kappa(s+h)}\). This gives

The integrand in this last Itô integral is deterministic. Brownian increments after time \(s\) are independent of \(\mathcal{F}_s\), so the integral is independent of \(\mathcal{F}_s\) and is normally distributed with mean zero. Its variance, by the Itô isometry, is

Adding the \(\mathcal{F}_s\)-measurable term \(m_{s,h}\) gives the claimed conditional normal law. When \(h=0\) or \(\sigma=0\), the stochastic term has variance zero and the same representation gives the stated degenerate case. \(\square\)

The variance depends on the time gap and model parameters, not on the current rate. The conditional mean does depend on \(r_s\): its distance from \(\theta\) is reduced by the factor \(e^{-\kappa h}\). This distinction is central when using the transition law for calculations.

Conditional Moments and Terminal Thresholds

Proof. The transition-law theorem represents \(r_{s+h}\) as \(m_{s,h}+G\), where conditionally on \(\mathcal{F}_s\), \(G\) is centered normal with variance \(v_{s,h}^2\). Thus its conditional mean is zero and its conditional second moment is \(v_{s,h}^2\). Linearity of conditional expectation gives \(\mathbb{E}[r_{s+h}\mid\mathcal{F}_s]=m_{s,h}\). By the definition of conditional variance,

Finally, expanding the square gives \(\mathbb{E}[(m_{s,h}+G)^2\mid\mathcal{F}_s] =m_{s,h}^2+2m_{s,h}\mathbb{E}[G\mid\mathcal{F}_s]+\mathbb{E}[G^2\mid\mathcal{F}_s] =m_{s,h}^2+v_{s,h}^2\). All three identities hold almost surely. \(\square\)

Suppose \(h>0\) and \(\sigma\ne0\), so \(v_{s,h}>0\). Let \(\Phi\) denote the standard normal cumulative distribution function, \(\Phi(x)=\mathbb{P}(N\le x)\) for \(N\sim N(0,1)\). For any \(\mathcal{F}_s\)-measurable threshold \(L\), standardizing the conditional normal variable yields

The normal distribution has no atoms when its variance is positive, so replacing \(>\) by \(\ge\), or \(\le\) by \(<\), does not change these probabilities. If \(h=0\) or \(\sigma=0\), do not standardize by a zero standard deviation: the rate is \(m_{s,h}\) with conditional probability one, and the probability of \(r_{s+h}>L\) is \(\mathbf{1}_{\{m_{s,h}>L\}}\).

What “Crossing a Level” Means

The transition formulas calculate a terminal threshold event, such as \(r_{s+h}>L\). That is not in general the same event as the rate touching or crossing \(L\) at some time between \(s\) and \(s+h\). The latter is a pathwise event involving the whole trajectory, for example \(\{\sup_{s\le u\le s+h}r_u\ge L\}\); a one-time transition distribution does not determine its probability by the standardization above. Such a first-passage or boundary-crossing calculation requires a separate analysis. Keep the event being calculated explicit.

There is also a measure choice. The displayed probabilities and conditional expectations are under \(\mathbb{P}\), because the SDE was stated with a Brownian motion under \(\mathbb{P}\). In a pricing application, if the Vasicek SDE and its parameters are specified under the risk-neutral measure \(\mathbb{Q}\), the same derivation applies under \(\mathbb{Q}\), and the resulting probability is a \(\mathbb{Q}\)-conditional probability. One must not silently use parameters or a Brownian motion from one measure while interpreting the expectation under the other.

A Common Calculation Error

A frequent mistake is to mis-evaluate the factor \(\sigma^2/(2\kappa)\), often by squaring or dividing incorrectly. In the first worked example, the coefficient is \(0.02^2/(2\cdot2)\), which is \(0.0004/4=0.0001\), not \(0.000025\). The correct variance is therefore \(0.0001(1-e^{-2})\), approximately \(0.00008647\), and the standard deviation is approximately \(0.009299\). Using \(0.000025(1-e^{-2})\) would make the variance four times too small and would also give an incorrect standard deviation, second moment, and threshold probability.

A reliable check is to keep the variance in the second-moment identity: \(\mathbb{E}[r_{s+h}^2\mid\mathcal{F}_s]-\mathbb{E}[r_{s+h}\mid\mathcal{F}_s]^2=v_{s,h}^2\). This also clarifies why the second moment is not just the square of the conditional mean.