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Let $L(t, T_1, T_2)$ be the forward LIBOR rate at time $t$ for the period $T_1$ to $T_2$.

If a security pays some multiple of $L(T_1, T_1, T_2)$ at time $T_1$, how can we show that the price of this is a linear combination of caplets with different strikes?

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Assume that the payoff is $L(T1,T1,T2)=:X$ paid at $T_1$.

  1. This is equivalent to paying off $X(1+X)$ at time $T_2$.

You can do this because in the risk neutral setting, a certain payment known at time $T_1$ can be paid later at $T_2$ if the beneficiary were compensated with exactly the fair rate of growth present at $T1$, for the period between $T_1$ and $T_2$. More formally, you can arrive at this by change of measure between the ZCB at $T_1$ and $T_2$.

The payoff is now non-linear in $X$ maturing at $T_2$, so you can replicate using the Carr Madan formula. Intuitively this is possible because caplets determine completely the marginal distribution of $X$ at $T_2$, which is sufficient to price any terminal payoff at $T_2$.

For point 1, what's critical is that the payment is known at $T_1$. What's also critical is that your 'fair rate' (discount rate) is LIBOR, which is not true anymore in case of rates, so in that case static replication will fail.

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  • $\begingroup$ thanks for this. Please could you explain in a bit more detail why "caplets determine completely the marginal distribution of $X$ at $T_2$"? $\endgroup$ – Ronnie268 Jun 5 at 20:18
  • $\begingroup$ quant.stackexchange.com/questions/1621/… $\endgroup$ – Arshdeep Jun 5 at 22:59
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    $\begingroup$ You can compute the risk neutral density of $X$ from caplet prices by computing second derivatives. Caplet prices across all strikes can be used to completely determine (I.E. IMPLY) this density. Then any payoff at $T_2$ can be priced by integrating the payoff using this density. $\endgroup$ – Arshdeep Jun 5 at 23:00

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