Show that the price of a LIBOR rate paid in advance is a linear combination of caplets

Question

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?

Answer 1

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.