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?


1 Answer 1


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:

$$E_t\left[B_t B_{T_2}^{-1} \left(X_{T_1} - F_2\right) \right] =0 $$


$$ F_2= E_t\left[B_t B_{T_2}^{-1} X_{T_1}\right] P(t,T_2)^{-1},$$ 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:

$$E_t\left[B_t^{-1}B_{T_1} \left(X_{T_1} - F_1\right) \right] =0 $$


$$ F_1= E_t\left[B_t B_{T_1}^{-1} X_{T_1}\right] P(t,T_1)^{-1}.$$

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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.