= Solution
A <one-factor short-rate model> uses a single scalar state, normally the <short rate> itself. Under the physical probability measure write
$$
dr_t=\mu(t,r_t)dt+\eta(t,r_t)dW_t,\qquad B_t=\exp\left(\int_0^t r_sds\right).
$$
The 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 measure>
$$
dr_t=\mu_Q(t,r_t)dt+\eta(t,r_t)dW_t^Q,\qquad\mu_Q=\mu-\eta\lambda.
$$
Changing 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 price
$$
\boxed{P(t,r;T)=\mathbb E^Q\left[\left.\exp\left(-\int_t^T r_sds\right)\right|r_t=r\right].}
$$
The 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 equation>
$$
P_t+\mu_QP_r+\frac12\eta^2P_{rr}-rP=0,\qquad P(T,r;T)=1.
$$
Conversely, 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 $P_t+\mu P_r+\eta^2P_{rr}/2-rP$. 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 $\eta P_r\ne0$, 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> are
$$
y(t,T)=-\frac{\log P(t,r_t;T)}{T-t},\qquad f(t,T)=-\partial_T\log P(t,r_t;T),\qquad f(t,t)=r_t.
$$
These 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 dynamics
$$
dr_t=a(b-r_t)dt+\eta\,dW_t^Q,\qquad a,\eta>0.
$$
Solving the linear equation gives $r_{t+u}=b+(r_t-b)e^{-au}+\eta\int_t^{t+u}e^{-a(t+u-v)}dW_v^Q$. Consequently, with $\tau=T-t$ and $B(\tau)=(1-e^{-a\tau})/a$,
$$
\int_t^T r_sds=b\tau+(r_t-b)B(\tau)+\eta\int_t^T B(T-v)dW_v^Q.
$$
This conditional normal random variable has <variance> $\eta^2\int_0^\tau B(u)^2du$. Its normal exponential moment yields
$$
P(t,T)=A(\tau)e^{-B(\tau)r_t},\qquad\log A=-b(\tau-B)+\frac{\eta^2}{2}\int_0^\tau B(u)^2du.
$$
Evaluating the elementary integral gives
$$
\boxed{B=\frac{1-e^{-a\tau}}a,\qquad\log A=\left(b-\frac{\eta^2}{2a^2}\right)(B-\tau)-\frac{\eta^2}{4a}B^2.}
$$
This 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:
$$
dr_t=a(b-r_t)dt+\eta\sqrt{r_t}\,dW_t^Q,\qquad a,b,\eta>0.
$$
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 gives
$$
B'=1-aB-\frac12\eta^2B^2,\qquad(\log A)'=-abB,\qquad B(0)=0,\ A(0)=1.
$$
Writing $d=\sqrt{a^2+2\eta^2}$, factor the quadratic in the first equation and integrate from $B=0$; integrating the second equation then gives the <CIR bond pricing> formulas
$$
\boxed{B(\tau)=\frac{2(e^{d\tau}-1)}{(d+a)(e^{d\tau}-1)+2d},\qquad A(\tau)=\left[\frac{2d\,e^{(a+d)\tau/2}}{(d+a)(e^{d\tau}-1)+2d}\right]^{2ab/\eta^2}.}
$$
The 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:
$$
dr_t=(\beta(t)-ar_t)dt+\eta\,dW_t^Q.
$$
For a smooth desired initial forward curve $f_0(t)$, its <exact forward-curve fit in the Hull-White model> is obtained by taking
$$
\boxed{r_0=f_0(0),\qquad\beta(t)=f_0'(t)+af_0(t)+\frac{\eta^2}{2a}(1-e^{-2at}).}
$$
To derive this rather than merely assert calibration, write $r_t=x_t+\phi(t)$, where $dx_t=-ax_tdt+\eta dW_t^Q$, $x_0=0$. The conditional Gaussian-integral calculation at time zero gives
$$
\log P(0,T)=-\int_0^T\phi(u)du+\frac{\eta^2}{2}\int_0^T B(u)^2du,
$$
so its forward curve is $\phi(T)-\eta^2B(T)^2/2$. Choose $\phi(T)=f_0(T)+\eta^2B(T)^2/2$. Differentiating this shift and adding $a\phi(t)$ 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> $vv^T$, 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. \b[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.]
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