How to change the Libor rate to Forward Libor rate in Swap?

Question

The realised PV of a swap (notional is 1 ) is :

$Swap(t)=\sum^n_{i=1} \tau_i \times D(t,Ti) \times (L(Ti, Ti, Ti+ \tau_i) - K)$

How do we get the expression with forward rate :

$Swap(t)=\sum^n_{i=1} \tau_i \times D(t,Ti) (L(t,Ti,Ti+ \tau_i) - K)$

I know we are supposed to used forward measure but I don't see how ?

Thank you !

Answer 1

General fact:

From a mathematical standpoint, we can write the PV of a flow to be received at $T$ as the value of its expectation under the $T$-forward measure (which is also the value of the forward at $t$: $F(t, T)$) discounted using the zero-coupon bond. We can show this by changing measures from the risk-neutral measure $\mathbb{Q}$ to the $T$-forward measure $\mathbb{Q}_T$ associated to the zero-coupon bond $P(t,T)$ as numéraire.

Indeed, using the numéraire change formula, we can write: $$ \begin{aligned} PV(t) &= \mathbb{E}_t^\mathbb{Q} \left[ e^{-\int_t^T r(u) du } X(T)\right] \\ &= \mathbb{E}_t^\mathbb{Q} \left[ e^{-\int_t^T r(u) du } F(T, T)\right] \\ &= P(t, T) \mathbb{E}_t^{\mathbb{Q}_T} \left[F(T, T) \right]\\ &= P(t, T) F(t, T)\\ \end{aligned} $$

In the last step, we used the fact that the forward price is a martingale under the $T$-forward measure: $$ F(t, T) = \mathbb{E}_t^{\mathbb{Q}_T} \left[F(T, T) \right] = \mathbb{E}_t^{\mathbb{Q}_T} \left[X(T) \right] $$

Application to the floating leg of the swap:

Let us apply this to the floating leg. The payment date of each flow is $T_i + \tau_i$, sowe switch from the risk-neutral measure $\mathbb{Q}$ to the $(T_i + \tau_i)$-forward measure $\mathbb{Q}_{T_i + \tau_i}$:

$$ \begin{aligned} FloatingLeg(t) &= \sum^n_{i=1} \tau_i \times P(t, T_i + \tau_i) \times \mathbb{E}_t^{\mathbb{Q}_{T_i + \tau_i}} \left[L(T_i, T_i, T_i + \tau_i)\right] \end{aligned} $$

Under $\mathbb{Q}_{T_i + \tau_i}$, the Libor forward $\left(L(t, T_i, T_i + \tau_i)\right)_t$ is a martingale, and as a result:

$$ FloatingLeg(t) = \sum^n_{i=1} \tau_i \times P(t, T_i + \tau_i) \times L(t, T_i, T_i + \tau_i) $$