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Let $f_0(t)$ be the initial instantaneous forward rate curve and choose $r_0=f_0(0)$. In the Hull-White model, take $\beta (t)=f_0^{\prime }(t)+a{f}_0(t)+{\eta }^2(1-e^{-2at})/(2a)$. To verify the fit, write $r_t=x_t+\varphi (t)$ with $d{x}_t=-a{x}_{t}dt+\eta d{W}_t^Q$, $x_0=0$, and $\varphi (t)=f_0(t)+{\eta }^2(1-e^{-at}{)}^2/(2a^2)$. The Gaussian exponential formula gives $-{\partial }_T\log P(0,T)=\varphi (T)-{\eta }^2(1-e^{-aT}{)}^2/(2a^2)=f_0(T)$.

The Gaussian Hull-White model has pricing dynamics $d{r}_t=(\beta (t)-a{r}_t)dt+\eta d{W}_t^Q$. Its deterministic drift function can fit the initial yield curve exactly, while its Ornstein-Uhlenbeck process part controls stochastic variation. Like the Vasicek model, it permits negative rates.

The entire term structure is naturally indexed by observation time and maturity. Work under a risk-neutral measure $Q$ and let $f(t,T)$ be the instantaneous forward rate, with $0\le t\le T$. Define the short rate $r_t=f(t,t)$, the bank account $B_t=\exp ({\int }_0^t{r}_{s}ds)$ and the zero-coupon bond priceAn arbitrary Gaussian mean and covariance need not make these bond prices consistent. Absence of arbitrage requires the discounted traded bonds to be martingales under a suitable equivalent measure. We derive the constraint explicitly.

Consider the Gaussian forward-rate field $f(t,T)=m(t,T)+X(t,T)$ with deterministic mean and centered Gaussian random field satisfyingThe maturity kernel is symmetric, and its increments in the first parameter must be positive semidefinite kernels. Assume continuity, separability and sufficient integrability for the following mean-square integrals and maturity derivatives. This covariance gives independent observation-time increments, while permitting correlated fluctuations across all maturities. The filtration contains the history of the whole observed forward curve.

For a fixed maturity $T$, collect the past short rate and the present curve into the integrated Gaussian forward-rate processThen $P(t,T)/B_t=e^{-A(t,T)-Y(t,T)}$. Its centered Gaussian variance isFor $s<t$ and an observed field coordinate $X(z,U)$ with $z\le s$, the covariance of $Y(t,T)-Y(s,T)$ with it is zero: the integrand isThe uncorrelated jointly Gaussian variables are independent principle therefore makes the increment independent of the earlier field filtration. Its variance is $v(t,T)-v(s,T)$, since the same covariance calculation gives $\mathrm{Cov}(Y(t,T),Y(s,T))=v(s,T)$. Conditional Gaussian exponential expectation now givesHence the exact bond-martingale condition isDifferentiating in maturity, using symmetry of the covariance, gives the Gaussian forward-rate covariance drift restrictionThis is necessary and sufficient. For sufficiency, the derivatives of the half-variance identity agree by the displayed restriction. Its integration constant also agrees at $T=t$, becauseThus the full identity follows for every $T\ge t$, and the conditional expectation calculation proves a true martingale, not merely zero formal drift. The initial mean $f(0,T)=-{\partial }_T\log P(0,T)$ fits the observed initial curve exactly; thereafter the mean adjustment is fixed by the covariance.

If $c(t;T,U)={\int }_0^t{k}_s(T,U)ds$ is time-differentiable, the forward drift is $a(t,T)={\int }_t^T{k}_t(T,u)du$. Writingwith deterministic finite- or Hilbert-space factor loadings gives the Heath-Jarrow-Morton modelWith $\Sigma (t,T)={\int }_t^T\sigma (t,u)du$, Itô formula independently verifies bond drift $r_t-{\int }_t^{T}a(t,u)du+\Vert \Sigma (t,T){\Vert }^2/2=r_t$. Finite-factor models give low-rank maturity correlations; a genuine random field or an infinite factor space permits a much richer covariance structure. Gaussian forward rates and short rates may be negative, although the exponential bond prices stay positive. Under a physical measure, a Girsanov theorem market-risk-premium change adds $⟨\sigma ,\lambda ⟩$ to the forward drift; deterministic premia preserve Gaussianity, whereas arbitrary adapted premia need not.

For a concrete two-parameter example, let $X$ be a standard Brownian sheet, so $c(t;T,U)=t\min (T,U)$. The restriction givesIndeed ${\int }_0^T\min (t,u)udu=t^3/3+t(T^2-t^2)/2$. This demonstrates directly how a Gaussian random field determines the compensating forward drift.

For the one-factor constant-volatility case, $\sigma (t,T)=\eta$, the Constant-coefficient Ho-Lee model hasandThese follow by integrating the explicit curve, not by assigning the short-rate drift independently. Exponentially decaying loadings $\eta e^{-\kappa (T-t)}$ instead give a Gaussian mean-reverting Hull-White model, with bond-rate loading $(1-e^{-\kappa (T-t)})/\kappa$.

A practical advantage is an explicit Gaussian bond-option formula. For a call of strike $K>0$ exercised at $T$ on a bond maturing at $U>T$, use the T-forward measure with numéraire $P(t,T)$. The forward bond ratio $F_t=P(t,U)/P(t,T)$ has a lognormal distribution under that measure, with remaining integrated varianceIts martingale property makes the conditional log mean $\log F_t-q/2$. Evaluating the positive-part lognormal integral givesFor $q=0$, the limit is $(P(t,U)-KP(t,T){)}^{+}$. The forward measure accounts for the stochastic short rate; discounting by one deterministic interest rate would generally be wrong. Calibration selects a positive-semidefinite covariance structure fitting observed rate co-movements and option prices, with the no-arbitrage mean restriction imposed afterward. With more independent factors than the available traded bond exposures span, the market is incomplete and additional claims need a specified pricing measure or risk-premium model. The Gaussian field is therefore a flexible curve model with explicit consistency and pricing equations, rather than a claim that all possible rate risks are hedgeable.

A Gaussian one-factor short-rate model is most naturally specified under a chosen risk-neutral measure $Q$, not by using a physical drift unaltered in a pricing formula. The Vasicek model hasIts explicit solution is $r_s=\bar{r}+(r_t-\bar{r})e^{-\kappa (s-t)}+\eta {\int }_t^s{e}^{-\kappa (s-u)}d{W}_u^Q$. It is a Markov process, and its conditional mean and variance are $\bar{r}+(r_t-\bar{r})e^{-\kappa (s-t)}$ and ${\eta }^2(1-e^{-2\kappa (s-t)})/(2\kappa )$. Hence all finite-dimensional distributions are Gaussian. The model permits negative rates; Gaussianity does not enforce their positivity.

The unit zero-coupon bond price is $P(t,T)={\mathbb{E}}_Q[e^{-{\int }_t^T{r}_{s}ds}\mid {\mathcal{F}}_t]$. Write $\tau =T-t$ and $B(\tau )=(1-e^{-\kappa \tau })/\kappa$. Stochastic integration of the explicit solution givesThis Gaussian variable has variance ${\eta }^2{\int }_0^{\tau }B(v{)}^{2}dv$. Evaluating its exponential moment gives exponential-affine bond pricingFor example, ${\int }_0^{\tau }B(v{)}^{2}dv={\kappa }^{-2}[\tau -2B(\tau )+(1-e^{-2\kappa \tau })/(2\kappa )]$, which gives the displayed expression by direct simplification. The term structure of interest rates is therefore analytically tractable, and bond-price changes are driven by the current short rate alone.

The Hull-White model replaces the drift by $\alpha (t)-\kappa r_t$ with deterministic $\alpha$. The same calculation givesThis retains the one-factor Gaussian and Markov structure while allowing an exact initial-curve fit through $\alpha$. If $f_0(t)$ is the initial instantaneous forward rate and $r_0=f_0(0)$, differentiating the bond formula shows thatThus calibration flexibility is separated from stochastic mean reversion. Time-dependent calibration generally destroys time stationarity even though the state remains Markov. In the time-homogeneous Vasicek case, a stationary initialization is $r_0\sim N(\bar{r},{\eta }^2/(2\kappa ))$ independent of future increments; its covariance is ${\eta }^2{e}^{-\kappa |t-s|}/(2\kappa )$. A fixed initial short rate does not give a stationary process.

Let $P_{s,t}>0$ denote the time-$s$ price of a zero-coupon bond paying one at maturity $t$, with $P_{t,t}=1$. Information available at $s$ is the sigma-algebra ${\mathcal{F}}_s$ in the market filtration. For a curve differentiable in maturity, the instantaneous forward rate and its diagonal short rate areThe last identity follows by integrating the maturity derivative and using $P_{s,s}=1$. The continuous-time bank account reinvesting at the short rate is $B_s=\exp ({\int }_0^s{R}_{u}du)$, and its reciprocal is the discount factor. Modelling forward rates means specifying a random maturity-indexed curve with an evolution consistent with the same pricing measure for every bond. The expectation below is under that money-market risk-neutral measure, not an unspecified physical measure.

For fixed maturity $t$, if $P_{s,t}/B_s$ is a true martingale on $0\le s\le t$, its terminal value is $1/B_t$. ConsequentlyConversely, this representation gives $P_{s,t}/B_s={\mathbb{E}}_Q[B_t^{-1}\mid {\mathcal{F}}_s]$, a conditional-expectation martingale by the tower property. For example, for $r<s$, conditioning the right side on ${\mathcal{F}}_r$ returns ${\mathbb{E}}_Q[B_t^{-1}\mid {\mathcal{F}}_r]$. These claims assume ${\mathbb{E}}_Q{B}_t^{-1}<\infty$. A positive discounted local martingale alone need not equal the conditional expectation of its terminal value, so the word “martingale” cannot be weakened without an extra integrability argument.

In the Vasicek model, the Brownian motion $W$ and coefficients in $d{R}_s=\alpha (\beta -R_s)ds+\sigma d{W}_s$ must be understood under this pricing measure. Multiplying by $e^{\alpha s}$ and integrating gives, conditionally on ${\mathcal{F}}_s$,Assume first $\alpha \ne 0$, and put $\tau =t-s$ and $B(\tau )=(1-e^{-\alpha \tau })/\alpha$. Integrating the explicit solution over $s\le u\le t$ yieldsThe interchange of the stochastic integrals is justified by their deterministic square-integrable kernels on the finite triangular domain; equivalently approximate the kernels by step functions, interchange finite sums, and pass to the limit using the Itô isometry. The last integral conditionally has a centered normal distribution independent of ${\mathcal{F}}_s$, with varianceA deterministic Brownian integral is Gaussian because its step approximations are sums of independent normal increments; their variances converge by the Itô isometry. Completing the square in the normal density proves $\mathbb{E}{e}^{-Y}=e^{-m+v/2}$ for a normal variable of mean $m$ and variance $v$. Hence the zero-coupon bond price is exponential-affine:An equivalent familiar form isAt $t=s$, both coefficients are zero and the bond value is one. These formulas are valid for negative as well as positive nonzero $\alpha$ on finite horizons, though a genuinely mean-reverting Vasicek model has $\alpha >0$. For $\alpha =0$, direct integration gives $B(\tau )=\tau$, $J(\tau )={\tau }^3/3$ andAll the exponential moments used above are finite, so the prices defined by the conditional expectation do give true discounted martingales.

To fit an initial bond curve, let its observed prices $p_0(t)$ be positive, satisfy $p_0(0)=1$, and have a sufficiently smooth instantaneous forward rate $f_0(t)=-d\log p_0(t)/dt$. Set $R_0=f_0(0)$; this compatibility is forced by the infinitesimal-maturity bond price. Replace $\beta$ by a deterministic function $\theta (t)$, with $\alpha \ne 0$, so thatIts mean $m(t)={\mathbb{E}}_Q{R}_t$ satisfies $m^{\prime }=\alpha (\theta -m)$, $m(0)=R_0$. The variance of the integrated rate remains ${\sigma }^{2}J(t)$, since changing deterministic drift does not change the centered Gaussian process. ThereforeThe required mean is consequently $m(t)=f_0(t)+{\sigma }^{2}B(t{)}^2/2$. Solve its mean equation for $\theta$:Combining the last two terms gives the explicit Hull-White model calibration in the question's parameter convention:With $R_0=f_0(0)$, the specified mean solves the same linear differential equation and initial condition as the model mean, so it is the actual mean. The resulting model forward curve equals $f_0$; integrating and using $P_{0,0}=p_0(0)=1$ proves $P_{0,t}=p_0(t)$ for every maturity. If the drift is instead written $\tilde{\theta }(t)-\alpha R_t$, its calibration parameter is $\tilde{\theta }=\alpha \theta$, not $\theta$ itself.

The printed allowance of arbitrary constants needs a qualification for this final claim. If $\alpha =0$, replacing $\beta$ by $\theta (t)$ changes no coefficient at all: $R_t=R_0+\sigma W_t$ regardless of $\theta$. Its initial curve is fixed at $P_{0,t}=\exp (-R_{0}t+{\sigma }^2{t}^3/6)$. With $\sigma \ne 0$, the compatible smooth observed curve $p_0(t)=e^{-R_{0}t}$ is already a counterexample to the unconditional fitting claim. Thus arbitrary compatible smooth positive initial curves can be fitted by the stated replacement when $\alpha \ne 0$; it cannot fit arbitrary curves when $\alpha =0$. In that degenerate case one can instead introduce an independent deterministic drift $k(t)$ in $d{R}_t=k(t)dt+\sigma d{W}_t$, choosing $k(t)=f_0^{\prime }(t)+{\sigma }^{2}t$ to recover the desired initial curve. Likewise an externally fixed $R_0\ne f_0(0)$ is incompatible with exact fitting, and an arbitrary nonsmooth set of prices need not define the differentiable forward curve assumed in the model.

A one-factor short-rate model uses a single scalar state, normally the short rate itself. Under the physical probability measure writeThe second process is the continuous-time bank account. Specifying the physical diffusion alone does not determine bond prices: it must be accompanied by the market price of risk $\lambda (t,r)$, or equivalently by pricing-measure dynamics. If the Girsanov theorem applies to the likelihood $\mathcal{E}(-\int \lambda dW)$, then under the risk-neutral measureChanging measure alters drift, not quadratic variation. The coefficients need conditions for a nonexplosive solution and for the relevant discounted prices and measure-change density to be true martingales.

A unit zero-coupon bond maturing at $T$ has priceThe Markov property makes current time and rate sufficient state variables. Apply Itô formula to $P(t,r_t;T)/B_t$. Its drift must vanish, giving the short-rate bond pricing equationConversely, under the appropriate boundary and integrability conditions the Feynman-Kac formula identifies its solution with the displayed discounted expectation. For a terminal rate-dependent payoff $H(r_T)$, the same equation has terminal condition $H$ instead of one. A coupon bond is a sum of such unit-bond prices weighted by the coupon payments.

The common market price of risk can also be derived from trading, instead of being introduced arbitrarily for each maturity. Under the physical measure a bond's diffusion exposure is $\eta P_r$ and its excess drift above bank financing is $P_t+\mu P_r+{\eta }^2{P}_{rr}/2-rP$. For two nondegenerate traded bonds, combine their holdings to cancel their Brownian exposure. Absence of arbitrage requires that combination to earn the bank rate; hence their excess-drift-to-exposure ratios must coincide. Calling that common ratio $\lambda$ yields the pricing equation with ${\mu }_Q=\mu -\eta \lambda$ for every maturity. A derivative $V(t,r)$ is then hedged locally by holding $V_r/P_r$ units of a nondegenerate bond and financing the remaining value through the bank account. Their Brownian exposures agree, and substituting both pricing equations proves the drift gain identity. This is short-rate diffusion hedging; it requires $\eta P_r\ne 0$, rather than a bare assertion that one factor automatically gives completeness everywhere.

The entire yield curve is generated consistently from these bond prices. The continuously compounded yield and the instantaneous forward rate areThese encode different summaries: the yield averages the forward rate over the remaining lifetime. Bond prices must satisfy $P(t,t)=1$ and be positive; they need not be below one in models permitting negative rates.

The Vasicek model has constant pricing dynamicsSolving the linear equation gives $r_{t+u}=b+(r_t-b)e^{-au}+\eta {\int }_t^{t+u}{e}^{-a(t+u-v)}d{W}_v^Q$. Consequently, with $\tau =T-t$ and $B(\tau )=(1-e^{-a\tau })/a$,This conditional normal random variable has variance ${\eta }^2{\int }_0^{\tau }B(u{)}^{2}du$. Its normal exponential moment yieldsEvaluating the elementary integral givesThis explicit form makes estimation and valuation convenient. The rate is Gaussian, its conditional mean reverts to $b$, and its long-run variance is ${\eta }^2/(2a)$. Its disadvantage, when nonnegative rates are desired, is a positive probability of negative rates.

The CIR model replaces constant volatility by square-root volatility:The standard solution stays nonnegative. The Feller positivity condition for the CIR model is $2ab\ge {\eta }^2$ in this parameterization; with a positive initial rate it makes zero inaccessible. Below this threshold zero can be reached, although the process does not cross into negative rates. For an affine diffusion bond pricing ansatz $P=A(\tau )e^{-B(\tau )r}$, substitution into the bond equation and matching its constant and rate coefficients givesWriting $d=\sqrt{a^2+2{\eta }^2}$, factor the quadratic in the first equation and integrate from $B=0$; integrating the second equation then gives the CIR bond pricing formulasThe square-root form produces an affine term structure while respecting nonnegative rates. These parameters are pricing-measure parameters; historical mean-reversion estimates alone do not determine them without a risk-premium specification.

Both preceding constant-parameter models constrain the family of initial curves. The Hull-White model allows a deterministic time-dependent drift:For a smooth desired initial forward curve $f_0(t)$, its exact forward-curve fit in the Hull-White model is obtained by takingTo derive this rather than merely assert calibration, write $r_t=x_t+\varphi (t)$, where $d{x}_t=-a{x}_{t}dt+\eta d{W}_t^Q$, $x_0=0$. The conditional Gaussian-integral calculation at time zero givesso its forward curve is $\varphi (T)-{\eta }^{2}B(T{)}^2/2$. Choose $\varphi (T)=f_0(T)+{\eta }^{2}B(T{)}^2/2$. Differentiating this shift and adding $a\varphi (t)$ gives exactly the displayed $\beta (t)$. Thus $P(0,T)=\exp (-{\int }_0^T{f}_0(u)du)$ for every maturity, while $a$ and $\eta$ govern the stochastic fluctuations. This Gaussian model still allows negative rates.

Finally, one-factor models are economical and often give explicit bond prices, but their single Brownian driver is restrictive. Bond diffusion exposures $v_i=\eta P_{i,r}$ give the instantaneous covariance matrix $v{v}^T$, of rank at most one. Nondegenerate maturity changes are instantaneously perfectly correlated, with correlation $+1$ or $-1$, though their finite-horizon correlations need not equal one. A single factor therefore cannot independently describe level, slope and curvature shocks to the whole yield curve. Multiple factors or a full forward-rate model offer greater flexibility. A valid one-factor model must specify the pricing drift, fit the initial curve to the required accuracy, and handle boundary and martingale conditions; naming a rate diffusion alone is not a complete pricing model.