Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 / 5

Construct the strictly positive rolling one-period bond account using successive one-period zero-coupon bonds:At time $t-1$, invest the entire account value in the bond maturing at $t$. This gives a positive self-financing portfolio usable as a numéraire.

We use the finite-discrete-time fundamental theorem of asset pricing: in a frictionless market with finitely many adapted assets and trading dates, no arbitrage is equivalent to existence of an equivalent martingale measure for asset prices, including dividends, expressed in a positive traded numéraire. Maturing bond payoffs are reinvested in that numéraire. Let $\mathbb{Q}$ be such a measure and $L_t=\mathbb{E}[d\mathbb{Q}/d\mathbb{P}\mid {\mathcal{F}}_t]$ its positive density process. The martingale pricing relation for a unit zero-coupon bond isDefine $Z_t=L_t/{\beta }_t$. The Bayes formula for conditional expectation then givesThe positive expectations are finite because these are the traded finite bond prices; in particular $\mathbb{E}{Z}_T=Z_0{P}_0(T)$ when initial information is trivial. Normalize $Z_0=1$ without changing any ratio. With nontrivial initial information the same identities are conditional on that information. The state-price density need not be unique when the bond market is incomplete; existence is sufficient.

A unit zero-coupon bond pays one at its maturity, so $P_t(t)=1$. Monotonicity in maturity gives $P_t(t+1)\le 1$. Using the state-price density representation with $T=t+1$,The process is positive and integrable, as noted in part (a), and adapted. Thus it is a supermartingale. Strictly decreasing maturity prices yield a strict one-step conditional inequality; weak decrease is already enough for the conclusion.

The spot interest rate $r_T$ is known at $T-1$. The state-price density price of its floating payment is, by the law of total expectation,Here $P_0(0)=1$ if $T=1$. All these terms are integrable: $Z_T{r}_T$ is bounded in absolute value by $Z_T/P_{T-1}(T)+Z_T$, whose expectation is finite from the one-step pricing relation. Subtracting the fixed payment $f$ givesThis also follows directly from a floating-rate payment bond replication. At time zero buy one unit of the bond maturing at $T-1$ and short $1+f$ units of the bond maturing at $T$. Their initial cost is the displayed ${\xi }_0$. Hold them until $T-1$. The first bond then pays one; spend that one to buy $1/P_{T-1}(T)$ units of the maturity-$T$ bond, leaving the earlier short position in place. This rebalance is self-financing. At maturity the net payment isFor $T=1$, the first unit is time-zero cash, and the same immediate rebalance gives the deterministic payoff. Thus the replication establishes the zero no-arbitrage price without requiring completeness of other claims.

Each floating payment in the interest rate swap has initial value $P_0(t-1)-P_0(t)$ by the previous replication. Their sum telescopes to $1-P_0(T)$. The fixed leg pays $s$ at each of the same dates, so its initial value is $s{\sum }_{t=1}^T{P}_0(t)$. Consequently the par swap rate isThe denominator is positive. This is for unit accrual periods and the printed floating-minus-fixed payments. No extra exchange of principal occurs in the contract; the principal-like terms appear only because the floating-leg replication telescopes.