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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.