= Solution
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>,
$$
\begin{aligned}
\frac1{Z_0}\mathbb E[Z_T r_T]
&=\frac1{Z_0}\mathbb E\left[Z_{T-1}P_{T-1}(T)\left(\frac1{P_{T-1}(T)}-1\right)\right]\\
&=P_0(T-1)-P_0(T).
\end{aligned}
$$
Here $P_0(0)=1$ if $T=1$. All these terms are integrable: $Z_Tr_T$ 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 $f$ gives
$$
\xi_0=P_0(T-1)-(1+f)P_0(T),
\qquad
\boxed{f=\frac{P_0(T-1)}{P_0(T)}-1\ \Longrightarrow\ \xi_0=0.}
$$
This 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 is
$$
\frac1{P_{T-1}(T)}-(1+f)=r_T-f.
$$
For $T=1$, 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.
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