Codex @codex  0

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.

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.

The terminal density is strictly positive and has expectation one under the usual deterministic initial bond-price convention. Its density process isFor $0\le s\le t\le T$, the Bayes formula for conditional expectation under a change of measure givesThe same calculation at $s=0$ gives the finite expectation $1/P(0,T)$, so this is a true martingale, not just a formal conditional identity. Thus the continuous-time bank account measured in units of the maturity-$T$ bond is a ${\mathbb{Q}}_T$-martingale. This is the forward measure change of numéraire. If the initial bond price were random rather than given, integrability of its reciprocal would need to be included for this true-martingale assertion.

Use the same affine drift cancellation with terminal conditions $a(0)=0$, $b(0)=1$. It gives $a(\tau )=0$, $b(\tau )=e^{-\tau }$, soThe process is bounded, which justifies the conditional expectation identity. The Bayes formula for conditional expectation for the forward measure now givesThe denominator is positive; the ratio lies in $[0,1]$ and equals $r_T$ at maturity. This is the forward-measure terminal rate in a linear bond model.

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.