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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.

Since $d(B_t^{-1})=-r_t{B}_t^{-1}dt$, multiplying the given bond dynamics by $B_t^{-1}$ cancels the drift and gives $d{Z}_t=Z_t\Sigma (t,T)d{W}_t^Q$. For an integrable discounted terminal payoff, the risk-neutral value process is the discounted-payoff martingaleFor an attainable claim this also follows directly from its self-financing discounted wealth being a true martingale, with terminal value $B_T^{-1}X$. In a Brownian market, the Brownian martingale representation theorem provides $d{M}_t={\psi }_{t}d{W}_t^Q$; a traded bond with nonzero diffusion exposure can hedge it by holding ${\psi }_t/(Z_t{\Sigma }_t)$ bond units, with the remainder in the bank account. Thus, under the usual attainability and integrability assumptions, this construction gives the replicated price as well as the risk-neutral pricing formula.

Because $B_t$ is ${\mathcal{F}}_t$-measurable,In a general Heath-Jarrow-Morton model, writing this value as $V(t,r_t)$ requires an additional state-reduction assumption. The conditional expectation is valid without asserting that the short rate alone is the full state.

Assume the forward field is sufficiently differentiable in maturity to differentiate its deterministic and stochastic integrals. Moving along the diagonal adds the maturity derivative to the fixed-maturity dynamics:The Heath-Jarrow-Morton model no-arbitrage drift is ${\alpha }_Q(t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)du$, so ${\alpha }_Q(t,t)=0$. Equivalently, from the integrated forward equation,That drift can depend on the entire previous forward curve or driving path, rather than on the current scalar rate.

The short rate need not be Markov. For a concrete One Brownian factor does not imply a Markov short rate example, take the deterministic volatility $\sigma (t,T)=T-t$. Its admissible risk-neutral drift is $(T-t{)}^3/2$, andSubtracting the deterministic part leaves $I_t={\int }_0^t{W}_{s}ds$. The past of $I$ reveals its derivative $W_t$, and $\mathbb{E}[I_{t+h}-I_t\mid {\mathcal{F}}_t^I]=h{W}_t$. Yet $W_t$ is not determined by $I_t$: the Gaussian covariance calculation givesConsequently the conditional law of the future depends on more than $I_t$, proving failure of the Markov property for the scalar short rate. The enlarged pair $(I_t,W_t)$ is Markov. Special maturity-volatility structures, such as those producing a Vasicek-type short-rate diffusion, can instead admit a scalar Markov realization.

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 $Q$ 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 $g$.

On a fixed finite horizon, a self-financing strategy consists of predictable stock holdings $h^i$, integrable against the price processes, and a bank account holding $h^0$. Its wealth and gains satisfyAn admissible trading strategy has discounted wealth $X_t/B_t$ bounded below by a deterministic constant throughout the horizon. With the bounded short rate in this part, an undiscounted deterministic lower bound is equivalent after adjusting the constant. An arbitrage is an admissible zero-initial-cost strategy with $X_T\ge 0$ almost surely and $\mathbb{P}(X_T>0)>0$.

Put $1=(1,\dots ,1{)}^{\top }$ and define the market price of riskThe bounded coefficients and bounded inverse make $\lambda$ bounded. Define the stochastic exponentialThe Novikov condition states that $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T\Vert {\lambda }_s{\Vert }^{2}ds)<\infty$ makes this exponential a true martingale with expectation one. Boundedness verifies the condition. Therefore $dQ=L_{T}dP$ defines an equivalent probability measure. The Girsanov theorem states thatis a $d$-dimensional Brownian motion under $Q$. Substituting into the asset equations givesThus $Q$ is an equivalent local martingale measure. The sufficient direction of the fundamental theorem of asset pricing says that such a measure excludes arbitrage among admissible self-financing strategies. Here the reason can be seen directly: $X/B$ is a local martingale under $Q$, and its lower bound makes it a supermartingale. Consequently $X_0=0$ implies ${\mathbb{E}}_Q[X_T/B_T]\le 0$, incompatible with the nonnegative terminal payoff of an arbitrage. Therefore the bounded-inverse market has no arbitrage.

For the requested law of one price counterexample, take a broader continuous-time local-martingale model on $[0,T]$. Let $X$ be a Brownian motion, let $\tau =\inf \{u\ge 0:X_u=-1\}$, and set $q(t)=t/(T-t)$ for $t<T$. DefineThe given hitting-time fact gives $\tau <\infty$ almost surely. The corresponding physical hitting time is $T\tau /(1+\tau )<T$, so $S$ reaches one and then stays one. In particular it is continuous at maturity and $S_t\ge 1$.

For a rigorous closed-horizon localization, stop additionally when $S$ hits $n+2$, or at $T$ if it never does. The stopped process is Brownian motion stopped on first exiting $(-1,n)$ and subjected to the deterministic clock; it is a bounded martingale, including at $T$. The stopping times increase to $T$, since every Brownian path on the finite random interval $[0,\tau ]$ has a finite maximum. Thus $S$ is a positive local martingale on the whole horizon. With the time-changed Brownian filtration, the original probability measure is an equivalent local martingale measure. The admissible-wealth supermartingale argument above proves no admissible arbitrage.

Nevertheless $S_T=B_T=1$ while $S_0=2\ne B_0=1$, so the law of one price fails. Shorting one share and buying two units of cash has zero initial cost and terminal gain one, but its wealth is $2-S_t$. It has no deterministic lower bound: for every $n$, the probability of Brownian motion hitting $n$ before $-1$ is $1/(n+1)>0$, obtained by optional stopping of the bounded stopped Brownian motion. That apparently profitable strategy is therefore inadmissible. The counterexample deliberately uses a strict local martingale; the bounded-coefficient class in the first part has true discounted asset martingales and does satisfy the law of one price.

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.

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 $r_0$.

For the quadratic Ornstein-Uhlenbeck state-price density, apply the Itô's formula under the reference probability measure. Since $f^{\prime }(x)=x$ and $f^{\prime \prime }(x)=1$,The stochastic integral is a true martingale on finite horizons by the finite moments of the Ornstein-Uhlenbeck process. A uniform sufficient condition is $\boxed{\alpha \ge {\sigma }^2/(2a)}$, making the finite-variation drift nonpositive. This is also the condition for nonpositive drift at every state.

Use the positive state-price density $\zeta$ to price a terminal payoff $H$ by ${\Pi }_t(H)=\mathbb{E}[{\zeta }_{T}H\mid {\mathcal{F}}_t]/{\zeta }_t$. The riskless account must cancel the drift of $\zeta$, giving the short rateThis pricing system is consistent with a money-market risk-neutral measure: with $M_t=e^{{\int }_0^t{r}_{u}du}$, the process ${\zeta }_t{M}_t/{\zeta }_0$ is a stochastic exponential whose diffusion coefficient $\sigma X_t/f(X_t)$ is bounded. The Novikov condition makes it a true density martingale. The original Brownian motion is for the reference measure, not automatically for this new pricing measure.

For $X_0=x_0$, the OU second moment is $\mathbb{E}{X}_T^2=x_0^2{e}^{-2\lambda T}+{\sigma }^2(1-e^{-2\lambda T})/(2\lambda )$. Thus the zero-coupon bond price isConditionally at time $t$, replace $T$ by $T-t$ and $x_0$ by $X_t$.

The short rate is nonnegative under the condition above, increases with $X_t^2$, and lies between $\alpha -{\sigma }^2/(2a)$ and $\alpha +2\lambda$. The model is tractable but imposes fixed rate bounds, identical rates for opposite values of $X$, and a restricted one-factor term structure. Its long-maturity yield is $\alpha$. These structural restrictions, including exclusion of negative rates under the supermartingale condition, may be unsuitable for a general interest-rate fit.

Write the discount factor as $D_t=B_t^{-1}=\exp (-{\int }_0^t{r}_{s}ds)$. Splitting the time integral at $t$ givesThe random variable $D_T$ lies in $(0,1]$ because the short rate is nonnegative and continuous on the finite maturity interval. A process of conditional expectations of an integrable terminal variable is a martingale, by the tower property of conditional expectation. Therefore

The short rate is the limiting instantaneous forward rate at the present maturity. The continuously compounded zero-coupon bond price is obtained by integrating the instantaneous forward rate curve in its maturity variable. Both requested relations areIn particular $P(T,T)=1$, and $f(t,T)=-{\partial }_T\log P(t,T)$. The short rate here is instantaneous, rather than the one-period rate used in a discrete-time bank account.

For fixed maturity $T$, define $b_t^T={\int }_t^T\sigma (t,u)du$ and $I_t^T={\int }_t^{T}f(t,u)du$. The stochastic Fubini theorem and the moving lower endpoint giveThe factor $1/2$ is the integral over one of the two triangles in the square $[t,T{]}^2$. Applying the Itô formula to $P(t,T)=e^{-I_t^T}$, its quadratic-variation correction cancels that factor:Set $D_t=e^{-{\int }_0^t{r}_{s}ds}$, the reciprocal of the continuous-time bank account. The Itô product rule then givesIf $|\sigma |\le M$, then $|b_t^T|\le MT$ on this finite horizon. The Novikov condition holds, so this stochastic exponential is a true martingale, not merely a local martingale. The discounted price is thereforeThe authoritative PDF discounts to $t$ in this part. The TeX transcription's upper endpoint $T$ would include future short rates and is incorrect here.