Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 6

Let $Q$ denote the given unique equivalent martingale measure. A zero-coupon bond paying one at $T$ has risk-neutral pricing valueWrite $h=T-t$. Split the integrated short rate into its known and future parts:By integrating the future Brownian motion increments in the opposite order, the last integral equals ${\int }_t^T(T-u)d{\hat{W}}_u$. It is independent of ${\mathcal{F}}_t$ and is a Gaussian random variable of mean zero and varianceThe Gaussian exponential moment formula $\mathbb{E}[e^{aZ}]=e^{a^2\mathrm{Var}(Z)/2}$ for centered $Z$ therefore gives the bond price in this Gaussian short-rate model with a deterministic shift:The instantaneous forward rate is the negative maturity derivative of the logarithm of the bond price. ConsequentlyIn particular, $P_t(t)=1$ and $f_t(t)=g(t)+\sigma {\hat{W}}_t=r_t$, as required. The derivative formula is pointwise for continuous $g$ and almost everywhere for locally integrable $g$.

Let $F_0(T)$ be the observed initial instantaneous forward rate curve. Since ${\hat{W}}_0=0$, part (i) gives $f_0(T)=g(T)-{\sigma }^2{T}^2/2$. Thus the forward-curve calibration of a shifted Brownian short rate is achieved by the deterministic choiceSubstitution gives $f_0(T)=F_0(T)$ at every maturity where the curve is defined classically. It also reproduces the corresponding initial zero-coupon bond curve:There is no shape restriction on the initial curve beyond the regularity needed to define these integrals and forward derivatives; for continuous curves the match is pointwise, and for locally integrable curves the forward identity has its almost-everywhere interpretation.