# 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?

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).