Skip to content
All library documents

Replicating an Advance-Paid LIBOR Payoff with Caplets

Article Quant Q&A · Author: Ronnie268
Read the original

Summary

The document explains how to value a payoff equal to a forward LIBOR rate fixed and paid at the start of its accrual period by translating it into a payoff at the period's end. Under the assumed risk-neutral framework, the rate known at the start can be carried forward at the fair LIBOR rate, turning the payoff into a nonlinear function of the rate at maturity.

It then invokes Carr–Madan static replication: a portfolio of caplets across strikes can replicate a terminal payoff when caplet prices reveal the rate's marginal distribution. The reasoning depends on the payment being known at the earlier date and on LIBOR serving as the relevant fair growth rate for deferral. The answer warns that this condition may fail for other rates, where static replication by the same argument need not work. No numerical example or market calibration is provided.

Key ideas

  • A LIBOR payoff fixed at the start of a period can be shifted to the end by applying the fair LIBOR growth factor.
  • The shifted payoff becomes nonlinear in the terminal LIBOR rate.
  • A sufficiently rich set of caplets across strikes can statically replicate terminal payoffs through the rate's marginal distribution.
  • The argument relies on the payment being known at the earlier date and LIBOR being the applicable fair discount or growth rate.
  • The same replication logic may fail for rates that do not satisfy that growth assumption.

Tags

Fixed incomeOptionsDerivatives pricing
Full text
# Show that the price of a LIBOR rate paid in advance is a linear combination of caplets


# 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?

## Answer by Arshdeep (score 3, accepted)

https://quant.stackexchange.com/a/64360

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

- 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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.