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The unit zero-coupon bond pays one at maturity. Its risk-neutral valuation therefore isNo particular rate coefficients were specified, so this expectation is the general answer; evaluating it further requires a chosen one-factor short-rate model.

For the alternative derivation let $D_t=\exp (-{\int }_0^t{r}_{s}ds)$ be the bank discount factor. For an integrable discounted payoff, pricing giveswhich is a martingale by the tower property of conditional expectation. For a payoff determined by the Markov rate state, this conditional expectation is represented by the indicated pricing function. A path-dependent claim requires an appropriately enlarged state, as in part (c). Now $d{D}_t=-r_t{D}_{t}dt$ has finite variation. Applying Itô formula and the product rule under the risk-neutral measure, whose rate drift is $b-\sigma \theta$, yieldsUnder the smoothness and integrability conditions used above, the martingale has zero finite-variation drift. The bracket therefore vanishes, giving againFor the bond the terminal condition is $P(T,T)=1$. This argument also shows why stochastic discounting remains inside the conditional expectation.

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.

An instantaneous forward rate curve determines the unit-face-value zero-coupon bond through $f(t,T)=-{\partial }_T\log P(t,T)$ and $P(t,t)=1$. Integrating in maturity givesThe diagonal is the short rate, $\boxed{r_t=f(t,t)}$, and the continuous-time bank account has $B_t=\exp ({\int }_0^t{r}_{s}ds)$ when $B_0=1$. Therefore the discounted bond price martingale has valueUnder the risk-neutral measure of the Heath-Jarrow-Morton model, sufficiently integrable discounted bond prices are true martingales; without that integrability their stochastic dynamics initially establish a local martingale.

An interest rate model must describe a whole term structure of interest rates, not just one future scalar rate. Let $P(t,T)$ be the time-$t$ price of a unit zero-coupon bond maturing at $T$. For a sufficiently regular maturity curve, define the instantaneous forward rate and short rate bySince $P(t,t)=1$, integration givesHere $B$ is the continuous-time bank account. Thus modelling the two-parameter field $f(t,T)$ determines both the bonds and the discounting account. A Gaussian random field means that every finite collection of its values is jointly Gaussian; merely making each individual forward rate normally distributed is insufficient.

A useful arbitrage-free construction works under a money-market risk-neutral measure $Q$. Take independent Brownian motions $W^j$, deterministic coefficient sequences ${\sigma }_j(t,T)$, and a deterministic initial forward curve $f(0,T)$. With the coefficients viewed as vectors in a separable Hilbert space $H$, writeFor infinitely many factors, the last term means the $L^2$ limit of ${\sum }_j{\sigma }_j(t,T)d{W}_t^j$, not an assumption that the infinite vector of Brownian motions is itself $H$-valued. Require ${\int }_0^t\Vert \sigma (s,T){\Vert }_H^{2}ds<\infty$ and the maturity/time regularity needed for stochastic integration and exchange of integrals. Deterministic drift and volatility make this a Gaussian forward-rate field.

The field covariance is determined byFor every finite set of maturities, $(c_t(T_i,T_j))$ must be positive semidefinite. A one-factor model has rank-one instantaneous covariance, while additional or infinitely many factors permit richer correlations along the yield curve. Equivalently one can drive the curve with a maturity-correlated Brownian field $Z_t(T)$ satisfying $d⟨Z(T),Z(U){⟩}_t=c(T,U)dt$. A Brownian sheet gives the exampleMultiplication by deterministic maturity-dependent amplitudes, or deterministic increasing changes of the two sheet coordinates, gives further Gaussian covariance structures. A covariance specification alone is not sufficient: the forward drift must also obey absence of arbitrage.

To derive that drift restriction, putDifferentiate the moving lower endpoint in $\log P(t,T)=-{\int }_t^{T}f(t,u)du$ to obtainThe Itô formula therefore givesDiscounted bond prices must be martingales under $Q$. Their drift vanishes precisely when ${\int }_t^T\alpha (t,u)du=\frac{1}{2}\Vert \Sigma (t,T){\Vert }_H^2$. Differentiating in maturity gives the Heath-Jarrow-Morton model restrictionFor scalar amplitudes $v(t,T)$ multiplying a field with covariance kernel $c(T,U)$, this becomes $\alpha (t,T)=v(t,T){\int }_t^{T}v(t,u)c(T,u)du$. The extra maturity correlation factor cannot in general be dropped.

The complete Gaussian specification is consequentlyIts mean and covariance follow immediately from deterministic integration and the Itô isometry. The initial yield curve is an input, so fitting initial bond prices requires choosing $f(0,T)$ rather than changing a few scalar parameters. With the displayed drift, the discounted bond obeysOn finite horizons, deterministic square-integrable bond volatility gives an expectation-one stochastic exponential, hence a true discounted bond price martingale. This supplies an arbitrage-free pricing model for admissible portfolios in any finite collection of these bonds. Under a physical measure, a market price of risk vector changes the drift: if $d{W}_t^Q=d{W}_t^P+{\lambda }_{t}dt$, then the physical forward drift is $\alpha (t,T)+⟨\sigma (t,T),{\lambda }_t{⟩}_H$. Historical mean changes must therefore not be inserted in place of the risk-neutral drift restriction. Gaussianity under both measures additionally requires suitable deterministic drift changes; state-dependent risk premiums need not preserve Gaussianity under the physical measure.

There is also a useful general consistency test for a proposed Gaussian field beyond this independent-increment construction. DefineIn the filtration generated by a Gaussian field, conditional distributions of such linear functionals are Gaussian. For $s\le t\le T$, the martingale requirement is therefore exactlyThis follows by taking the conditional exponential moment and equating it to $e^{-L_s^T}$. It constrains both the deterministic mean and the covariance-induced conditional mean. Matching only unconditional discounted bond expectations does not suffice. For the Brownian-field construction, the Heath-Jarrow-Morton model restriction realizes this consistency condition through local drift cancellation.

Gaussian structure makes risk-neutral pricing particularly tractable. As a finite-factor example, take the Vasicek model under $Q$:Solving this Ornstein-Uhlenbeck process gives, for $u\ge t$,Hence, with $\tau =T-t$ and $B(\tau )=(1-e^{-a\tau })/a$, the conditional Gaussian integral $I={\int }_t^T{r}_{u}du$ hasThe exponential moment identity for a normal variable now gives the zero-coupon bond price $P(t,T)={\mathbb{E}}_Q[e^{-I}\mid {\mathcal{F}}_t]=A(\tau )e^{-B(\tau )r_t}$, whereThis outlines the derivation of the familiar exponential-affine form without assuming a deterministic future discount rate. More general forward-field models need not reduce to a finite-dimensional Markov short-rate process.

For another application, a call at time $T$ on a unit maturity-$U$ bond, with $t<T<U$ and strike $K>0$, is priced under the T-forward measure, using $P(t,T)$ as numéraire. The bond-price ratio $F_s=P(s,U)/P(s,T)$ is a martingale under that measure and has deterministic volatility $-(\Sigma (s,U)-\Sigma (s,T))$. It is consequently lognormal, with integrated log-varianceIntegrating the lognormal payoff gives the Gaussian bond-option formulaAt $V=0$ use $(P(t,U)-KP(t,T){)}^{+}$. The same idea prices other suitable Gaussian-rate derivatives through forward measures and normal exponential moments.

The attractions are exact initial-curve fitting, explicit covariance control across maturities, and tractable bond and option prices. The limitations are also structural: Gaussian short rates and instantaneous forward rates generally can be negative; finite-factor specifications restrict maturity correlations; and deterministic volatilities cannot reproduce arbitrary stochastic changes in rate dispersion. An infinite-factor model allows richer covariance but typically leaves risks that cannot be spanned by finitely many traded bonds. Thus Gaussianity is a useful modelling choice, not by itself a guarantee of positivity, market completeness, or absence of arbitrage.

Use a risk-neutral measure $Q$, and let $P_{s,t}$ be the price at time $s$ of a zero-coupon bond paying one at its maturity $t$. The information available at $s$ is the filtration ${\mathcal{F}}_s$. For positive differentiable bond prices, the instantaneous forward rate iswith $P_{t,t}=1$. The short rate is $R_s=f(s,s)$. The continuous-time bank account $B_s=\exp ({\int }_0^s{R}_{u}du)$ is the discounting numéraire.

If $P_{s,t}/B_s$ is a true martingale for each fixed maturity, then its terminal value is $B_t^{-1}$ andMultiplication by the ${\mathcal{F}}_s$-measurable $B_s$ givesConversely this conditional-expectation formula makes the discounted bond a martingale by the tower property of conditional expectation. The equivalence uses integrability and true martingales, rather than merely local martingales.

Forward-rate modelling specifies the stochastic evolution of $f(s,t)$ as $s$ varies at fixed maturity. For example, with one Brownian driver, write $df(s,t)=\alpha (s,t)ds+\beta (s,t)d{W}_s^Q$. Differentiation of the integral representation and Itô formula give the bond's relative volatility $-{\int }_s^t\beta (s,u)du$ and drift $R_s-{\int }_s^t\alpha (s,u)du+\frac{1}{2}({\int }_s^t\beta (s,u)du{)}^2$. The discounted-bond condition therefore imposesor, after differentiation in maturity, $\alpha (s,t)=\beta (s,t){\int }_s^t\beta (s,u)du$. This illustrates how the no-arbitrage condition constrains forward-rate drift.

For the Gaussian short-rate model with constant coefficients, let $d{R}_s=\theta ds+\sigma d{W}_s^Q$ and fix $\Delta =t-s$. Future Brownian increments are independent of ${\mathcal{F}}_s$, givingThe last stochastic integral is Gaussian with mean zero and variance ${\Delta }^3/3$, by the Itô isometry. Its Gaussian moment-generating function therefore givesWhen $\sigma \ne 0$ and the initial rate is deterministic, the Brownian and short-rate filtrations coincide because $W_s^Q=(R_s-R_0-\theta s)/\sigma$. If $\sigma =0$, the same bond formula holds deterministically, while the asserted filtration equivalence is unnecessary.

For a deterministic drift function ${\theta }_v$, the same calculation givesLet the observed initial curve be $P_{0,t}^{\mathrm{obs}}>0$, normalized by $P_{0,0}^{\mathrm{obs}}=1$, and define $f_0(t)=-\frac{d}{dt}\log P_{0,t}^{\mathrm{obs}}$. For a differentiable initial forward curve chooseIndeed the model's initial forward rate is $R_0+{\int }_0^t{\theta }_{v}dv-{\sigma }^2{t}^2/2=f_0(t)$, so integrating it gives exactly the observed bond curve. This is forward-curve calibration of a shifted Brownian short rate, with $R_t=f_0(t)+{\sigma }^2{t}^2/2+\sigma W_t^Q$. Every finite collection of positive bond quotes can first be fitted by a smooth positive curve; an arbitrary entire curve requires the stated smoothness or an appropriate weaker integral interpretation. The freedom is in the deterministic drift, and the initial short rate must match the initial forward rate.

Let $F_0(T)$ be the observed initial instantaneous forward rate curve. Since ${\hat{W}}_0=0$, part (i) gives $f_0(T)=g(T)-{\sigma }^2{T}^2/2$. Thus the forward-curve calibration of a shifted Brownian short rate is achieved by the deterministic choiceSubstitution gives $f_0(T)=F_0(T)$ at every maturity where the curve is defined classically. It also reproduces the corresponding initial zero-coupon bond curve:There is no shape restriction on the initial curve beyond the regularity needed to define these integrals and forward derivatives; for continuous curves the match is pointwise, and for locally integrable curves the forward identity has its almost-everywhere interpretation.

Let $Q$ denote the given unique equivalent martingale measure. A zero-coupon bond paying one at $T$ has risk-neutral pricing valueWrite $h=T-t$. Split the integrated short rate into its known and future parts:By integrating the future Brownian motion increments in the opposite order, the last integral equals ${\int }_t^T(T-u)d{\hat{W}}_u$. It is independent of ${\mathcal{F}}_t$ and is a Gaussian random variable of mean zero and varianceThe Gaussian exponential moment formula $\mathbb{E}[e^{aZ}]=e^{a^2\mathrm{Var}(Z)/2}$ for centered $Z$ therefore gives the bond price in this Gaussian short-rate model with a deterministic shift:The instantaneous forward rate is the negative maturity derivative of the logarithm of the bond price. ConsequentlyIn particular, $P_t(t)=1$ and $f_t(t)=g(t)+\sigma {\hat{W}}_t=r_t$, as required. The derivative formula is pointwise for continuous $g$ and almost everywhere for locally integrable $g$.

A unit-face-value zero-coupon bond pays one unit of currency at its maturity $T$ and makes no earlier payments. Write its time-$t$ price as $P(t,T)$, with $P(T,T)=1$. The instantaneous forward rate is related to the bond price byThe short rate is $r_t=f_t(t)$, and the money-market account is $B_t=\exp ({\int }_0^t{r}_{s}ds)$.

Apply Itô formula to $f_t(T)=F(T-t,r_t)$, holding maturity fixed:The supplied equation identifies its volatility and drift asThis is exactly the one-factor Heath-Jarrow-Morton model drift restriction with zero market price of risk. The initial condition also verifies $f_t(t)=F(0,r_t)=r_t$.

To verify absence of arbitrage directly, without merely naming the drift restriction, putIntegrate the equation in its first variable. Since $F(0,r)=r$ andwe obtainThe bond price from the forward curve is $P(t,T)=e^{-G(T-t,r_t)}$. Its logarithm therefore satisfiesA second application of Itô formula givesThus every discounted bond price is a positive local martingale under the original measure. The sufficient no-arbitrage theorem is: if a common equivalent probability measure makes all discounted traded prices local martingales, then there is no arbitrage using self-financing strategies whose discounted wealth has a deterministic lower bound. Its proof is the supermartingale bound on such wealth. Apply it to any finite collection of bond maturities, using the original measure itself. This establishes no admissible arbitrage in the bond market, on horizons where the stated short rate model and smooth forward curve are defined. It is the Forward-rate equation for a Markov short-rate diffusion construction.

For the constant-coefficient case, substitute $F(\theta ,r)=A(\theta )r+B(\theta )$. The initial condition requires $A(0)=1$ and $B(0)=0$. Since $F_r=A$ and $F_{rr}=0$, the equation becomesThe right-hand side does not depend on $r$, so $A^{\prime }=0$ and $A=1$. It follows that $B^{\prime }=a_0-b_0^2\theta$, givingThe corresponding zero-coupon bond price isIt has discounted volatility $-b_0(T-t)$, which is deterministic and square-integrable on finite horizons. Its discounted price is consequently a true martingale, so the no-arbitrage conclusion follows again. This is the Gaussian short-rate model with constant coefficients.

As a separate check of the signs, conditional on ${\mathcal{F}}_t$,The final integral is centered Gaussian with variance $(T-t{)}^3/3$, obtained by integrating $\mathrm{Cov}(W_s-W_t,W_u-W_t)=\min (s-t,u-t)$. Its negative exponential expectation gives exactly the positive $b_0^2(T-t{)}^3/6$ term in the bond-price exponent above.

Put $n=T-t$ and define the finite geometric seriesStarting the autoregressive process of order one at time $t$ gives $r_{t+k}={\beta }^k{r}_t+{\sum }_{j=1}^k{\beta }^{k-j}{\xi }_{t+j}$. ConsequentlyThe future innovations are independent and identically distributed random variables, independent of ${\mathcal{F}}_t$. Factorizing their exponential conditional expectation and using the cumulant-generating function $K$ yieldsThus the exponential-affine bond pricing coefficients areThey vanish at maturity, giving $P(T,T)=1$. Everywhere finiteness of $K$ ensures that each zero-coupon bond price is positive and finite, including when $\beta$ is zero, negative, or one.

Let $q(T)$ denote the original zero-coupon bond price $P(0,T)$ computed above. A deterministic shift $h_t$ of the short rate, with $h_0=0$, gives ${\hat{r}}_t=r_t+h_t$. Subtracting the two recursions shows that the required added drift isSince the shift is nonrandom, the new discount factor differs by $\exp (-{\sum }_{s=1}^T{h}_s)$, and hence $\hat{P}(0,T)=q(T)\exp (-{\sum }_{s=1}^T{h}_s)$.

For deterministic calibration of an autoregressive short rate, defineAll these quantities are finite. The telescoping sum ${\sum }_{s=1}^T{h}_s=d_T$ gives $\hat{P}(0,T)=p(T)$. Explicitly,These constants fit every maturity simultaneously. Each successive maturity fixes $h_T$ uniquely, so the deterministic calibration is also unique with the prescribed $r_0$.