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The fair rate calculated in the above example is the rate to be used for fixed rate leg to yield 0 NPV. You would need to reconstruct the swap with the float leg as is but a fixed leg with fair rate from above. That should yield a 0 NPV for the swap.


Option 1 is not a steepener trade. It is an outright bearish trade that the 5y5y forward rate will move upwards. Option 2 is a steepener trade, if the dv01 is equal on the 5yr and 10yr legs. Ignoring discounting , you would need to pay fixed on 50mm 10yr versus receiving on 100mm 5yr to make it duration neutral, and thus a curve trade.


We assume a single-curve environment. Let us recall that a floating LIBOR payment fixed at time $T$ and paid at time $T^\prime$ can be written in terms of zero-coupon bonds: $$L(t,T,T^\prime):=\frac{1}{T^\prime-T}\left(\frac{P(t,T)}{P(t,T^\prime)}-1\right)$$ Let $\mathcal{T}:=\{T_0,\dots,T_n,\dots,T_m\}$ be a schedule such that a spot 5y swap starts fixing ...


I am assuming that in option 1 you are entering into payer swap. If the curve is flat then option a) and b) are the same because you will get the same cashflows in both cases. Why? In option b) both the floating legs and fixed legs on 10Y swap and 5Y will cancel for the first 5 years i.e. the cashflows will be opposite sign, effectively making it 5y5y swap. ...

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