Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 5

Take a unit-face zero-coupon bond with $P(T,T)=1$. Its continuously compounded zero-coupon yield to maturity, instantaneous forward rate, and short rate areThe last equality assumes the appropriate maturity differentiability at the diagonal. Integration of the forward-rate definition and differentiation of the yield definition giveAt fixed observation time $t$, the yield curve plots this yield against maturity $T$; it describes today's discount prices at different horizons. Forward rates are extracted from these prices, not identified with the subsequent realized path of the short rate.

The common short-rate market price of risk follows from absence of arbitrage, with the securities driven by the same Brownian motion. Take two traded derivatives $V_1,V_2$ with drifts $m_1,m_2$ and diffusion coefficients $s_1,s_2$. At a given state hold $s_2$ units of the first and $-s_1$ of the second, and place $-s_2{V}_1+s_1{V}_2$ in the bank account. The portfolio has zero current value and zero diffusion exposure. Its instantaneous excess drift isIf this were nonzero, selecting its positive orientation and maintaining the local risk-cancelling self-financing hedge would generate a gain without risk from zero wealth. Under the usual trading and regularity assumptions this violates absence of arbitrage. Thus the expression vanishes, and at nonzero exposuresAt zero exposure the correct relation is $m-rV=\theta s=0$, without dividing by zero. This is a condition on traded securities: the physical drift of the nontraded short rate alone does not determine $\theta$.

For $V(t,r)$ sufficiently smooth, Itô formula under the physical measure givesHence $m=V_t+b{V}_r+{\sigma }^2{V}_{rr}/2$ and $s=\sigma V_r$. Substitute these into the common risk-price condition $m-rV=\theta s$ from part (b), obtaining the Cox-Ross short-rate pricing equationFor a terminal payoff $H(r_T)$ the final condition is $V(T,r)=H(r)$. Writing prices as functions of $(t,r)$ requires the relevant Markov state to be sufficient for the claim; a path-dependent payoff generally requires additional state variables. The negative sign in $b-\sigma \theta$ is the adjustment from physical to risk-neutral rate drift.

The unit zero-coupon bond pays one at maturity. Its risk-neutral valuation therefore isNo particular rate coefficients were specified, so this expectation is the general answer; evaluating it further requires a chosen one-factor short-rate model.

For the alternative derivation let $D_t=\exp (-{\int }_0^t{r}_{s}ds)$ be the bank discount factor. For an integrable discounted payoff, pricing giveswhich is a martingale by the tower property of conditional expectation. For a payoff determined by the Markov rate state, this conditional expectation is represented by the indicated pricing function. A path-dependent claim requires an appropriately enlarged state, as in part (c). Now $d{D}_t=-r_t{D}_{t}dt$ has finite variation. Applying Itô formula and the product rule under the risk-neutral measure, whose rate drift is $b-\sigma \theta$, yieldsUnder the smoothness and integrability conditions used above, the martingale has zero finite-variation drift. The bracket therefore vanishes, giving againFor the bond the terminal condition is $P(T,T)=1$. This argument also shows why stochastic discounting remains inside the conditional expectation.