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For a maturity-$T$ call on a unit maturity-$U$ zero-coupon bond, where $t<T<U$, deterministic forward-rate volatilities make the bond-price ratio lognormal under the T-forward measure. Its integrated variance is $V={\int }_t^T\Vert \Sigma (s,U)-\Sigma (s,T){\Vert }^{2}ds$. For $V>0$, put $d_1=[\log (P(t,U)/(KP(t,T)))+V/2]/\sqrt{V}$ and $d_2=d_1-\sqrt{V}$. Integrating the lognormal distribution above the strike gives the displayed price. At $V=0$, the price is $(P(t,U)-KP(t,T){)}^{+}$. The forward measure is essential: the raw bond price has a stochastic discount rate under the money-market measure.

For a Gaussian forward-rate field with deterministic volatility, put $F=P(t,U)/P(t,T)$, $K_b=(1+\delta K{)}^{-1}$, and $v^2={\int }_t^T\Vert \Sigma (s,U)-\Sigma (s,T){\Vert }^{2}ds$. Under the T-forward measure, $F$ is lognormal with that integrated variance. With $d_1=\log (F/K_b)/v+v/2$ and $d_2=d_1-v$, the displayed expression is the caplet price. At $v=0$ use the discounted intrinsic bond-put value. Gaussian instantaneous rates do not make the reset floating rate Gaussian; the exponential bond relation is what yields this bond-option formula.

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.

An interest-rate cap is the sum of caplets over successive reset and payment dates. Consider one accrual period $[T,U]$ with year fraction $\delta >0$ and strike $K\ge 0$. The reset rate is $L(T;T,U)=[P(T,U{)}^{-1}-1]/\delta$; the unit-notional payment at $U$ is $\delta (L-K{)}^{+}$. At reset its value isHence the caplet is equivalent before reset to $c=1+\delta K$ puts expiring at $T$ on the maturity-$U$ zero-coupon bond, with strike $K_b=1/c$. This algebra accounts for payment at $U$ rather than incorrectly treating the payment date as $T$.

Under the T-forward measure, use the bond-ratio process $F_s=P(s,U)/P(s,T)$, which is a martingale. From the Gaussian HJM dynamics its relative volatility is $-[\Sigma (s,U)-\Sigma (s,T)]$. With deterministic volatilities its logarithm at reset is normal with integrated varianceFor $v>0$, write $d_1=\log (F_t/K_b)/v+v/2$ and $d_2=d_1-v$. A normal integral gives ${\mathbb{Q}}^T(F_T<K_b\mid {\mathcal{F}}_t)=\Phi (-d_2)$ and ${\mathbb{E}}_{Q^T}[F_T{1}_{\{F_T<K_b\}}\mid {\mathcal{F}}_t]=F_t\Phi (-d_1)$. Therefore the Gaussian caplet bond-put formula isFor $v=0$, the value is $(P(t,T)-(1+\delta K)P(t,U){)}^{+}$. This is also obtained from the Gaussian bond-option formula by put-call parity. A cap price is the sum of its caplet prices, multiplied by the notional principal.

In the Vasicek model, the instantaneous forward-rate volatility is $\sigma (s,u)=\eta e^{-\kappa (u-s)}$. ThusIndeed, $\Sigma (s,U)-\Sigma (s,T)=\eta e^{-\kappa (T-s)}(1-e^{-\kappa (U-T)})/\kappa$, whose squared integral gives this expression without ambiguity. Gaussian short and instantaneous forward rates do not mean the reset floating rate $L$ is Gaussian: the latter is an inverse bond price minus one. The lognormal bond-ratio calculation, under the proper forward measure, is the reason for the normal-CDF pricing formula.

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.

Since $P(T,T)=1$, part (b) shows that the density process of the T-forward measure isIts terminal expectation is one and it is strictly positive, so the stated Radon-Nikodym derivative defines an equivalent probability measure. By the Girsanov theorem,is a Brownian motion under ${\mathbb{Q}}_T$. Substitution of $d{W}_t=d{W}_t^T-b_t^{T}dt$ cancels the entire instantaneous forward rate drift:Because the integrand is deterministic and bounded, its Itô integral is square-integrable on $[0,T]$. With the usual fixed initial instantaneous forward rate curve, the requested true martingale is

Use the Radon-Nikodym derivative at $T_1$ and the discounted bond price martingale from part (b):For the variance, put $b_t^i={\int }_t^{T_i}\sigma (t,u)du$ and $d_t=b_t^2-b_t^1={\int }_{T_1}^{T_2}\sigma (t,u)du$, for $t\le T_1$. The ratio $R_t=P(t,T_2)/P(t,T_1)$ has, by the Itô formula,Under the T-forward measure ${\mathbb{Q}}_{T_1}$, $d{W}_t=d{W}_t^{T_1}-b_t^{1}dt$, so its drift becomes $-d_t^2/2$. Since $R_{T_1}=P(T_1,T_2)$,The first two terms are deterministic; the Itô isometry supplies the variance of the last. The two requested answers areIn fact the terminal zero-coupon bond price has a log-normal distribution under this forward measure, with the deterministic negative half-variance correction ensuring the displayed mean.

A T-forward measure is a probability measure $Q^T$, equivalent to the physical measure, under which prices expressed in units of the positive maturity-$T$ bond are martingales. Equivalently, every attainable payoff $X_T$ has time-$t$ priceThe zero-coupon bond is the numéraire.

The forward contract initiated at $t$ has payoff $S_T-F_t^T$ and zero value. Pricing under the T-forward measure givesBecause $F_t^T$ is ${\mathcal{F}}_t$-measurable and $B_t^T>0$,The tower property of conditional expectation therefore makes $(F_t^T{)}_{t<T}$ a $Q^T$-martingale.

If $K_1\le K_2$, then pointwiseThe positive pricing formula under the T-forward measure givesTherefore $C_t^{T,K_1}\ge C_t^{T,K_2}$, so the call price is non-increasing in strike.