Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 32 / 6

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.

The Hull-White model replaces the drift by $\alpha (t)-\kappa r_t$ with deterministic $\alpha$. The same calculation givesThis retains the one-factor Gaussian and Markov structure while allowing an exact initial-curve fit through $\alpha$. If $f_0(t)$ is the initial instantaneous forward rate and $r_0=f_0(0)$, differentiating the bond formula shows thatThus calibration flexibility is separated from stochastic mean reversion. Time-dependent calibration generally destroys time stationarity even though the state remains Markov. In the time-homogeneous Vasicek case, a stationary initialization is $r_0\sim N(\bar{r},{\eta }^2/(2\kappa ))$ independent of future increments; its covariance is ${\eta }^2{e}^{-\kappa |t-s|}/(2\kappa )$. A fixed initial short rate does not give a stationary process.

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.

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.

The correct state in a general Gaussian forward-rate field is the whole current curve. In time-to-maturity coordinates $g_t(x)=f(t,t+x)$, assume deterministic time-homogeneous volatility $\sigma (x)\in H$. Set $A(x)=⟨\sigma (x),{\int }_0^x\sigma (u)du⟩$. The Musiela forward-curve equation isUnder the appropriate maturity differentiability, or in its mild interpretation on a suitable function space, its solution isFor any future interval, the shifted current curve determines the deterministic part and future Brownian increments are independent of past information. This proves that the curve-valued process is time-homogeneous Markov. General observation-time-dependent deterministic volatility instead gives a time-inhomogeneous Markov curve. A low-dimensional state requires more: the maturity functions carrying the randomness must close under the shift evolution. Finitely many Brownian drivers by itself does not ensure that the short rate is Markov.

For a precise obstruction, take two independent stationary Ornstein-Uhlenbeck processes $Z^1,Z^2$ with distinct positive decay parameters $a_1,a_2$, and let $r_t=m+Z_t^1+Z_t^2$. Its centered covariance is $C(h)=c_1{e}^{-a_1|h|}+c_2{e}^{-a_2|h|}$ with $c_1,c_2>0$. A stationary scalar Gaussian Markov process must satisfy the covariance criterion for a stationary Gaussian Markov process:Indeed Gaussian conditioning gives $\mathbb{E}[r_{s+t}-m\mid r_s]=C(t)(r_s-m)/C(0)$; if the present is a Markov state, conditioning the future also on the earlier $r_0$ cannot change that mean. Multiplication by $r_0-m$ and taking expectations yields the equation. For the two-factor covariance the difference of its two sides isSo the short rate alone is not Markov, although the two-dimensional factor process is. Distinct exponentially decaying HJM volatility components realize such a multi-factor Gaussian curve model.

Stationarity is an additional condition, not a consequence of Gaussianity or time-homogeneity. A stationary Gaussian forward curve for the time-homogeneous model can be constructed whenis finite and the deterministic mean satisfies $m^{\prime }(x)+A(x)=0$. Initialize $g_0$ with mean $m$ and covariance $K$, independently of future noise. The mild solution has the same mean, since $m(x+t)+{\int }_0^{t}A(x+s)ds=m(x)$. Its covariance isThus its one-time Gaussian curve distribution is invariant, and the time-homogeneous transition law makes all time-shifted finite-dimensional distributions invariant as well. The initial field can explicitly be built from an independent noise on negative times using the same square-integrable kernels.

For example, ${\sigma }_j(x)={\eta }_j{e}^{-a_{j}x}$ with $a_j>0$ giveswhere $d{Z}_t^j=-a_j{Z}_t^{j}dt+{\eta }_{j}d{W}_t^j$ and the factors start in their stationary normal laws. This is a finite-dimensional Markov realization when the sum is finite. In contrast, a nonzero constant maturity volatility makes $K(x,x)$ infinite, excluding a finite-variance stationary curve of this form. A deterministic initial curve is generally nonstationary even when a stationary distribution exists. These distinctions separate Gaussianity, Markov state sufficiency, and stationarity.