Comparing LIBOR Contract Strikes with Different Payment Dates
Summary
The document compares fair strikes for two LIBOR contracts that reference the same rate fixed at the first maturity but pay at different dates. It derives each strike from discounted expected payoff under the bank account numeraire, then uses a change to the later-maturity forward measure to simplify the strike for the contract paid at that later date. For a LIBOR fixing, that strike reduces to a ratio of the two relevant zero-coupon bond prices, expressed as a forward rate.
For the earlier payment, the expectation involves the inverse of the intervening bond price. Unlike the later-payment case, it does not simplify to a model-free expression from the information given. The document shows how to express the expectation under the later forward measure, where a specified LIBOR dynamics can be used to evaluate it. Thus, comparing the strikes depends on the rate model or volatility assumptions; the derivation does not claim a universal ordering.
Key ideas
- The two contracts have identical rate exposure but differ in when their payoff is discounted.
- The later-payment fair strike follows from zero-coupon bond prices and equals the corresponding forward LIBOR rate.
- The earlier-payment strike contains an expectation of the inverse bond price that requires a model specification.
- A forward-measure change expresses that expectation using the distribution of LIBOR under the later bond measure.
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Full text
# We have a two LIBOR contracts, how to compare their values by change of change of numeraire
# We have a two LIBOR contracts, how to compare their values by change of change of numeraire
We have two LIBOR contracts:
contract 1 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{1}$
contract 2 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{2}$.
Now, $F_{1}$ is the par strike such that if $K=F_{1}$ then contract one has 0 value at today $t$
$F_{2}$ is the par strike such that if $K=F_{2}$ then contract one has 0 value at today $t$.
Using change of numeraire, how to compare which one is bigger, $F_{1}$ or $F_{2}$? What is the intuition?
## Answer by ir7 (score 1)
https://quant.stackexchange.com/a/57634
Let $X_{T_1}$ be a random quantity known (fixed) at $T_1$ (measurable wrt $T_1$-information), $B$ be the standard bank account and $P$ standard zero-coupon bond price.
From standard pricing, for the second contract:
$$E_t\left[B_t B_{T_2}^{-1} \left(X_{T_1} - F_2\right) \right] =0 $$
implies
$$ F_2= E_t\left[B_t B_{T_2}^{-1} X_{T_1}\right] P(t,T_2)^{-1},$$ which can be written also as (based on tower property of conditional expectation)
$$ F_2= E_t\left[B_t B_{T_1}^{-1} P(T_1,T_2)X_{T_1}\right] P(t,T_2)^{-1} $$
For the first contract:
$$E_t\left[B_t^{-1}B_{T_1} \left(X_{T_1} - F_1\right) \right] =0 $$
implies
$$ F_1= E_t\left[B_t B_{T_1}^{-1} X_{T_1}\right] P(t,T_1)^{-1}.$$
If $X_{T_1}=L(T_1,T_2)$, the expectation in $F_2$ formula further simplifies
$$ E_t\left[B_t B_{T_1}^{-1} P(T_1,T_2)\tau^{-1}(P(T_1,T_2)^{-1}-1)\right] $$
$$=\tau^{-1}E_t\left[B_t B_{T_1}^{-1} (1-P(T_1,T_2))\right] = \tau^{-1} (P(t,T_1) - P(t,T_2)),$$
(deflated $T_2$-maturity bond is a martingale), leading to $$ F_2 = \tau^{-1} (P(t,T_1) - P(t,T_2))P(t,T_2)^{-1}. $$
We can also attempt to further understand the expectation in $F_1$ formula:
$$ E_t\left[B_t B_{T_1}^{-1} \tau^{-1}(P(T_1,T_2)^{-1}-1) \right] $$
$$ = \tau^{-1}E_t\left[B_t B_{T_1}^{-1} P(T_1,T_2)^{-1} \right] - \tau^{-1} P(t,T_1). $$
Unfortunately the expectation in the first term needs a model (dynamics of bond price or Libor rate).
Note that we can switch to $T_2$-forward measure
$$ E_t\left[B_t B_{T_1}^{-1} P(T_1,T_2)^{-1} \right] = E_t\left[B_t B_{T_2}^{-1} P(T_1,T_2)^{-2} \right]$$ $$ = P(t,T_2) E_t^{T_2}\left[ P(T_1,T_2)^{-2} \right] $$ $$ = P(t,T_2) E_t^{T_2}\left[ (1+\tau L(T_1,T_2))^{2} \right] $$
and one can use a driftless dynamics (say lognormal, only volatility specification needed) for $L(\cdot, T_1, T_2)$ (as it is a martingale under $T_2$-forward measure).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.