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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?

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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).

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