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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.

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.

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

The instantaneous continuously compounded interest rate. The continuous-time bank account grows at this rate; it equals the instantaneous forward rate at current maturity.

The maturity-indexed curve of interest rates inferred from prices of zero-coupon bonds. Continuously compounded yields $y(t,T)=-\log P(t,T)/(T-t)$ average the instantaneous forward rate over $[t,T]$. Modelling a curve requires both dependence across maturities and consistent evolution through observation time; specifying unrelated one-dimensional rate distributions is insufficient.

The continuously compounded yield of a unit-face zero-coupon bond is the constant rate producing its current discount factor over the remaining maturity. It equals the maturity average of the instantaneous forward rate. Differentiation gives $f(t,T)=R(t,T)+(T-t)R_T(t,T)$, and both rates tend to the short rate as $T\downarrow t$. This definition uses continuous compounding; quoted coupon-bond yields require a different cash-flow equation.