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

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 !

### 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:

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

• How did you go from the second to third line when calculating the PV formula. It looks like you changed the measure without changing anything inside the expectation Nov 19, 2019 at 3:50
• Sorry it is a typo (the line should be removed) :) Fixed now. Nov 19, 2019 at 13:21