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

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?

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.

• thanks for this. Please could you explain in a bit more detail why "caplets determine completely the marginal distribution of $X$ at $T_2$"? Commented Jun 5, 2021 at 20:18
• quant.stackexchange.com/questions/1621/… Commented Jun 5, 2021 at 22:59
• 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. Commented Jun 5, 2021 at 23:00