# 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 – Slade Nov 19 '19 at 3:50
• Sorry it is a typo (the line should be removed) :) Fixed now. – byouness Nov 19 '19 at 13:21