Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 29 / 6

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.

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.

Assume the displayed Gaussian forward-rate covariance drift restriction. Its diagonal instance givesUsing it on the two parts of the deterministic discounted exponent yieldsThe last equality uses covariance symmetry to reflect the triangular integration region across its diagonal. The independent increments of the integrated Gaussian forward-rate process and its conditional Gaussian moment-generating function then give, for $r\le s\le T$,All these prices are integrable because their exponents are Gaussian random variables with finite variance. Thus the mean restriction implies the discounted bond martingale condition, completing the equivalence of the first two descriptions.

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.