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For a non-dividend-paying stock and nonnegative deterministic interest, the European call lower bound is $C_E\ge S-XB$, where $B\le 1$ is the zero-coupon bond price. This is at least the in-the-money exercise payoff $S-X$; the call also preserves the possibility of avoiding payment of the strike. Positive interest makes the comparison strict when exercise would have positive payoff. At zero interest there can be indifference, and with negative interest the statement can fail: a deterministic stock $S_u=S_0{e}^{ru}$ with $r<0$ can give a larger payoff by immediate exercise.

A one-factor short-rate model specifies $d{r}_t=\mu (t,r_t)dt+\eta (t,r_t)d{W}_t$ with deterministic coefficient functions. All zero-coupon bond prices are functions of time, maturity and this single scalar state. The physical drift and pricing drift differ by $\eta \lambda$, where $\lambda$ is the market price of risk.

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.

For strike $K>0$ and maturity $T$, a European call option and a European put option have boundary conditionsThe Black-Scholes equation is solved backward from these conditions. At zero stock price, the zero state is absorbing in geometric Brownian motion, soAt the other boundary, with $\tau =T-t>0$,These conditions and a suitable growth bound select the financial solution and provide boundary data for numerical truncations. In the constant-coefficient model, under the risk-neutral measure,For $d_1=[\log (S/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$, the exercise event is $Z>-d_2$. Completing the square in $e^{\sigma \sqrt{\tau }Z}$ times the normal density gives the truncated first moment and hencewhere $\Phi$ is the standard normal cumulative distribution function. These are the Black-Scholes formula values and satisfy the specified terminal and boundary conditions.

Put-call parity is the relationA call minus a put has terminal financial payoff $S_T-K$. A portfolio of one share minus a zero-coupon bond paying $K$ has the same financial payoff, so absence of arbitrage and law of one price equate their current values. The formula here assumes the same strike and maturity and no dividends; known dividends modify the stock-side value.

The instantaneous forward rate is defined by $f(t,T)=-{\partial }_T\log P(t,T)$, with $P(t,t)=1$. Integrating in maturity gives the zero-coupon bond price and its diagonal short rate:The solution of the continuous-time bank account equation $d{B}_t=r_t{B}_{t}dt$, $B_0=1$, is $B_t=\exp ({\int }_0^t{r}_{s}ds)$; the integration variable is time. Its discounted bond value isAll these are adapted processes when the Heath-Jarrow-Morton model coefficients are adapted and the integrals exist.

For market completeness, stochastic integration over maturity givesThe Itô formula for the exponential then yieldsConsequently under a risk-neutral measure the no-arbitrage drift restriction isThe last equality needs sufficient maturity regularity and has a positive sign despite the negative definition of $\Sigma$.

Under the specified risk-neutral measure, the Itô product rule and $d(B_t^{-1})=-r_t{B}_t^{-1}dt$ giveThus discounted traded zero-coupon bonds have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true martingales, not merely local martingales. If $X$ is an attainable financial payoff with ${\mathbb{E}}_Q|B_T^{-1}X|<\infty$, its self-financing portfolio value $V_t$ has terminal value $V_T=X$ and satisfiesMultiplying by $B_t$ yieldsThis proves the requested conditional pricing formula. In a Brownian filtration, the Martingale representation theorem realizes the discounted conditional expectation as a stochastic integral. If an available bond has nonzero Itô diffusion exposure, its holdings can match that integral, with the remaining value in the bank account, giving replication under the usual square-integrability conditions.

These attainability or market completeness hypotheses matter for a completely arbitrary contingent claim: existence of a risk-neutral measure alone need not imply a unique price for every unspanned financial payoff, and a zero-drift local martingale alone need not satisfy the displayed terminal expectation identity. In the general path-dependent Heath-Jarrow-Morton model, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current short rate is a sufficient state for the financial payoff and dynamics.

Write $T=t_0$, $B_t=e^{-\rho (T-t)}$, and let $V_t=f(S_t,t)$ with $f(x,t)=xg(x,t)+B_{t}h(x,t)$. The portfolio holds $g(S_t,t)$ shares of stock and $h(S_t,t)$ units of the zero-coupon bond. It is self-financing when all changes of holdings are funded within it, which is the gain identityUnder the physical probability measure, write $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, where $\sigma >0$ and $d{B}_t=\rho B_{t}dt$. The Itô formula givesMatching the Brownian motion coefficients forces $f_x=g$. Since $f_x=g+x{g}_x+B_t{h}_x$, this is preciselyDifferentiating $f_x=g$ gives $f_{xx}=g_x$. Since $f_t=x{g}_t+\rho B_{t}h+B_t{h}_t$, matching the drift coefficients now givesThese are the self-financing conditions for smooth stock and bond holdings. Conversely, substituting both identities into the Itô formula gives the gain identity, so they are sufficient as well. Smoothness and the positive lognormal distribution density of $S_t$ extend the identities along the price process to every $x>0$, with boundary times obtained by continuity.

The portfolio with constant stock value in bond units must have $xg(x,t)=\theta B_t$, so $g(x,t)=\theta B_t/x$. The first condition then gives $h_x=\theta /x$, henceThe second condition reduces to $c^{\prime }(t)=\theta ({\sigma }^2/2-\rho )$. Initial portfolio wealth requires $h(S_0,0)=w_0/B_0-\theta =w_0{e}^{\rho T}-\theta$. Therefore the holdings areThe corresponding portfolio wealth isUnder the risk-neutral measure, $\log (S_t/S_0)=(\rho -{\sigma }^2/2)t+\sigma W_t^Q$, so its value in zero-coupon bond units is $V_t/B_t=w_0{e}^{\rho T}+\theta \sigma W_t^Q$. This also verifies the self-financing gain equation after using the bond as numéraire.

Let $T=t_0$, $\tau =T-t$, and let $N$ have the standard normal distribution. For a twice differentiable payoff with enough growth control for the following expectations and derivatives, risk-neutral pricing in the dividend-free Black-Scholes model givesThe conditional discounted payoff is a martingale under the risk-neutral measure. Applying the Itô formula to $e^{-\rho s}V(S_s,s)$ and setting its drift to zero therefore gives the Black-Scholes equationThe delta hedge holds $\Delta =V_x$ shares; the remaining bond value is $\beta =V-x\Delta$, or $h=\beta /e^{-\rho (T-t)}$ units of the zero-coupon bond. Substitution in the Itô formula and the displayed equation gives $dV=\Delta dS+hdB$, proving replication and self-financing.

Differentiating the pricing expectation in the current stock price gives its option delta and option gamma:Thus a nondecreasing payoff has a nondecreasing price; a nonincreasing payoff has a nonincreasing price. A convex payoff has $\Gamma \ge 0$, and a concave payoff has $\Gamma \le 0$. This is convexity preservation in Black-Scholes pricing.

The Black-Scholes parameter sensitivities follow directly from the same expectation. Differentiation in the interest rate gives the option rhoFor spot volatility, differentiation first givesThe Gaussian integration by parts identity $\mathbb{E}[N\psi (N)]=\mathbb{E}[{\psi }^{\prime }(N)]$, applied to $\psi (N)=f^{\prime }(xM)xM$, cancels the term containing $f^{\prime }$ and gives the option vegaConsequently increasing spot volatility increases a convex payoff's value and decreases a concave payoff's value, with weak inequalities in affine or degenerate cases. A short bond position gives positive option rho, while a long bond position gives negative option rho. The physical stock drift $\mu$ does not enter these prices: the risk-neutral measure replaces it by $\rho$.

The option theta is the calendar-time derivative at fixed current stock price. The Black-Scholes equation givesFor $\rho \ge 0$, a convex payoff with $\beta <0$ therefore has $V_t\le 0$, while a concave payoff with $\beta >0$ has $V_t\ge 0$. The signs are strict when $\rho >0$. At $\rho =0$, an affine payoff can give equality. Remaining-maturity sensitivity is $V_T=-V_t$ when the payoff is independent of $T$. All these time comparisons hold at fixed current stock price.

The stated time-monotonicity conclusion needs the nonnegative-rate condition. For example, the convex affine payoff $f(x)=x-K$ hasThus with negative interest rates it provides an explicit counterexample even though the replica is short in bonds.

If the discounted zero-coupon bond price is a martingale, its expectation is $Z_{0,T}=e^{-A(0,T)}$. The Gaussian moment-generating function applied to the representation above gives $\mathbb{E}{Z}_{s,T}=e^{-A(s,T)+v(s,T)/2}$, soDifferentiate in maturity $T\ge s$. The two mean derivatives are ${\mu }_{s,T}$ and ${\mu }_{0,T}$. Symmetry of the covariance in its last two arguments makes the two boundary terms in the derivative of the double integral equal. HenceWe obtain the Gaussian forward-rate covariance drift restrictionwhich proves that the martingale condition implies the mean restriction. Only maturity differentiation is used; a time derivative of the covariance is unnecessary.

Let $D_s=\exp (-{\int }_0^s{R}_{u}du)$, where $R_u=F_{u,u}$ is the short rate. At maturity the zero-coupon bond pays one, so $Z_{T,T}=D_T$. If the discounted bond price is a martingale, its terminal conditional expectation isThe positive factor $D_s$ is measurable with respect to the current filtration. Dividing it out givesConversely, this identity says $Z_{s,T}=\mathbb{E}[D_T\mid {\mathcal{F}}_s]$. The tower property gives $\mathbb{E}[Z_{s,T}\mid {\mathcal{F}}_r]=Z_{r,T}$ for every $r\le s$, establishing the martingale condition. The integrated short rate is a Gaussian random variable with finite variance, so the Gaussian moment-generating function proves that $D_T$ is integrable. Together with the preceding subparts, this proves all three descriptions equivalent.

Use $T$ for the fixed maturity and $s\le T$ for the observation time. The original PDF's discounting uses the instantaneous forward rate over $[s,T]$ and the short rate over $[0,s]$. Let $X(s,u)=F_{s,u}-{\mu }_{s,u}$ be the centered Gaussian forward-rate field, and defineAssume the covariance and mean have the regularity needed for these mean-square integrals and the maturity differentiation below. Splitting $Y$ at $u=s$ shows that the discounted zero-coupon bond price isThis integrated Gaussian forward-rate process has deterministic varianceIts independent increments hold relative to the entire specified filtration. Indeed, if $r\le s$ and $z\le r$, the covariance between $Y(s,T)-Y(r,T)$ and any earlier $X(z,w)$ is the integral ofSince uncorrelated jointly Gaussian variables are independent, the increment is independent of every finite family of earlier observations and hence of their generated sigma-algebra. The increment has normal distribution with mean zero and variance $v(s,T)-v(r,T)$. Also $Y(0,T)=0$ because $c(0,u,w)=0$.

The Gaussian moment-generating function now givesTherefore the Gaussian forward-rate covariance drift restriction is exactly the deterministic conditionIt turns $Z_{s,T}/Z_{0,T}$ into a Gaussian exponential martingale with deterministic variance. The subparts establish the requested equivalent descriptions from this common calculation.

Use the usual frictionless option model with nonnegative deterministic interest rates and no stock dividends. Let $B(t,T)\le 1$ be the price of a unit zero-coupon bond, $C_E$ the European call price, and $C_A$ the American call price. A call payoff dominates $S_T-X$, so no-arbitrage pricing gives $C_E\ge S_t-XB(t,T)$, and $C_A\ge C_E$. If $S_t>X$, thenThe last term is the immediate exercise payoff. Waiting retains downside optionality and defers payment of the strike. For positive interest and $X>0$ the inequality is strict while exercise has positive payoff. If $S_t\le X$, exercising for a zero payoff provides no advantage over keeping the right. Thus early exercise of a non-dividend-paying call is never strictly beneficial under these assumptions, and $C_A=C_E$. For completeness, discounted European call value is a true martingale in the standard pricing model. For any exercise stopping time $\tau \le T$, its pointwise dominance of immediate payoff gives ${\mathbb{E}}_Q[B(t,\tau )(S_{\tau }-X{)}^{+}\mid {\mathcal{F}}_t]\le {\mathbb{E}}_Q[B(t,\tau )C_E(\tau )\mid {\mathcal{F}}_t]=C_E(t)$. Taking the supremum over exercise rules proves $C_A\le C_E$, and waiting to maturity gives the reverse inequality. This proves no early exercise of a call without dividends.

The contrast for a put is the interest earned on a strike received sooner. The European put has payoff at most $X$, so $P_E\le XB(t,T)$. When interest is positive and $S_t<X(1-B(t,T))$, immediate exercise gives $X-S_t>XB(t,T)\ge P_E$. Hence committing never to exercise before maturity is worse than an available early-exercise strategy in such a deep-in-the-money state. This does not say that every put should be exercised early, only that a general prohibition is not rational.

The interest-rate hypothesis is essential to the printed claim. If $r<0$ and the stock is deterministic, $S_u=S_t{e}^{r(u-t)}$ with no dividends, choose spot sufficiently large that it stays in the money through $T$. The European call price is $S_t-X{e}^{-r(T-t)}<S_t-X$, so immediate exercise is better. At zero interest, early exercise can be indifferent in degenerate cases; it is the usual positive-interest setting that makes the non-exercise comparison strict.

For European options with the same strike and maturity, the terminal payoff identity isThus long the call and short the put has the same terminal payoff as one stock less $X$ units of the maturity-$T$ zero-coupon bond. With no dividends, the stock strategy has no intervening payments. The law of one price gives put-call parity:Equivalently the two nonnegative terminal-payoff portfolios, call plus strike bond and put plus stock, agree in every state. The parity proof does not require lognormal stock prices or positive interest.

For American options the exercise times can differ, so equality of their maturity payoffs is not an identity between their full rights. Under the nonnegative-interest, non-dividend assumptions of part (a), $C_A=C_E$ while $P_A=P_E+{\Pi }_P$ with early-exercise premium ${\Pi }_P\ge 0$. Hencewhich generally differs from European parity. For a concrete limiting state, if the stock is zero and remains zero and interest is positive, then $C_A=C_E=0$, $P_A=X$ and $P_E=XB(t,T)<X$. European parity is not a general American-option equality. It holds in a particular case only when the relevant early-exercise premiums cancel, or both vanish.

The two binary terminal payoffs are complementary:in every state, including $S_T=X$. Long one of each is therefore exactly a unit zero-coupon bond. The law of one price gives digital put-call parity:With deterministic interest, the two separate values are $B(t,T)Q(S_T\ge X\mid {\mathcal{F}}_t)$ and $B(t,T)Q(S_T<X\mid {\mathcal{F}}_t)$. There is no need to assume a continuous terminal price distribution: the strict inequality for the put and weak inequality for the call allocate any atom at the strike exactly once.

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.

A Gaussian forward-rate field models the whole instantaneous forward rate curve. Let $H$ be a real separable Hilbert space, let $W$ be a cylindrical Brownian motion, and let deterministic $\sigma (t,T)\in H$ satisfy the maturity-integrability and square-integrability conditions needed below. For a deterministic initial curve, specify under $Q$Using an orthonormal basis, the noise is ${\sum }_j{\sigma }_j(t,T)d{W}_t^j$; an infinite-dimensional cylindrical noise need not exist as an $H$-valued random vector. Its covariation kernel across maturities is the positive semidefinite kernel $C_t(T,U)=⟨\sigma (t,T),\sigma (t,U)⟩$. Thus arbitrary suitable maturity-correlation kernels can be represented by Hilbert-space factors, while finitely many factors give finite-rank kernels.

No arbitrage requires a drift restriction. Define the zero-coupon bond price and integrated volatility byDifferentiation of the moving lower endpoint and the Itô formula giveThe discounted bond must have zero drift. Differentiating ${\int }_t^T\alpha (t,u)du=\frac{1}{2}\Vert \Sigma (t,T){\Vert }^2$ with respect to maturity yields the Heath-Jarrow-Morton model restrictionThis is why forward-rate drift and covariance cannot be selected independently in an arbitrage-free random-field model. With deterministic volatilities square-integrable on the finite horizon, the discounted bond stochastic exponential is a true martingale, not only a local martingale.

The solution is explicitly Gaussian:with mean given by the first two terms and covarianceThe entire Gaussian random field can therefore be calibrated through its initial curve and maturity covariance structure. Bond prices are exponential functionals of these Gaussian rates; conditional bond ratios have lognormal laws under appropriate forward measures. Negative rates are possible. A large or infinite volatility space provides rich maturity correlations, but Gaussianity alone supplies neither a low-dimensional Markov state nor positivity of rates.

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.