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Suppose the short rate $r$ follows the diffusive process $$dr=\mu dt+\sigma dB$$ where $B$ is the standard Brownian motion. The price of a bond portfolio $P(r,t,T)$ at time $t$ maturing at time $T$ follows dP=\frac{\partial P}{\partial t} dt+\frac{\partial P}{\partial r}dr+\frac12\frac{\partial^2 P}{\partial r^2}dr^2=\Big(\frac{\partial P}{\partial t}+ ...

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You're almost right, I think. You shouldn't talk about L(Ti,Ti+1) being a martingale, since this is a RV not observed until Ti. Rather we should talk about LtT, the regular Libor forward rate, and L'tT, the fair rate for an arrears FRA. We have that LtT is a martingale under the ZCB(Ti+1) measure, as you say. But also, L'tT is a martingale under the ...

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OK, so I think I have this figured out in my head now in terms of martingale measure theory. Thanks dm63 for pointing me in the right direction! Just for my own peace of mind and perhaps to help others in the future, my understanding is as follows: Vanilla Swap: We observe the LIBOR $L(T_i, T_{i+1})$ at time $T_i$ and payment occurs at $T_{i+1}$. Therefore ...

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This is indeed just a convention, as you point out. It comes from the fact that zero coupon bonds, by convention, do not have any volatility exposure. Rather, it is assumed the prices of ZCBs are given. Now, you can replicate a regular fra with strike K exactly using ZCBs: Long one ZCB with maturity T(i) and short (1+alpha K) ZCBs with maturity ...

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No you are long convexity. The futures contract has no convexity (since its value is linear as the underlying rate varies, specifically it moves by \$25 per bp per contract). Meanwhile, the FRA has positive convexity (it's like a mini bond). The fact that you are long convexity overall is consistent with the fact that the rate on the futures contract is ...

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