# Libor rate and martingales

We know that the forward Libor rate $$L(t, T, T + \tau)$$, in the absence of arbitrage, is a martingale under the measure $$T + \tau$$, i.e. $$Q^{T+\tau}$$. In this context:

$$\tag{1}\label{1} L(t, T, T + \tau) = \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau \right],$$

with $$t \leq T$$. This means that the expectation under this specific measure of the spot Libor rate $$L(T, T, T + \tau) = \hat{L}(T, T + \tau)$$ can be computed analytically.

Now, briefly, what I would like to know if this property can be extended to the forward Libor rate:

$$\tag{2}\label{2} L(t, T, T + \tau) = \mathbb{E}_t^{T + \tau} \left[ L(s, T, T + \tau \right],$$

with $$t \leq s \leq T$$. If you can answer this, there is no need to keep reading.

The result in \eqref{1} is very useful for many computations. For example, it is used for a vanilla swap valuation, showing that its price depends only on the term structure of interest rates observed on the valuation date.

Let's analyze a simpler case. Imagine the following stream of payment:

    ⋅------x//////////x------>
|      |          |
t    Tx = T      Tp = T + τ


where Tx represents a fixing date and Tp a payment date. The value of this contract at time $$t \leq T_x$$ is given by:

\begin{align} V(t) &= \mathbb{E}_t^Q \left[ D(t, T + \tau) \cdot \tau \cdot L(T, T, T + \tau) \right]\\ V(t) &= P(t, T + \tau) \cdot \tau \cdot \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau) \right] \end{align}

where $$D(t, T)$$ represents the discount factor and $$P(t, T)$$ the discount bond or zero coupon bond. The previous equation yields to (using the property defined in \eqref{1}):

$$V(t) = P(t, T + \tau) \cdot \tau \cdot L(t, T, T + \tau)$$

So far, so good. Now I would like to compute the price of a generalized stream of payment, given by:

    ⋅------x----+//////////+----x------>
|      |    |          |    |
t      Tx   Tb         Te   Tp


where Tx represents a fixing date, Tb a beginning of accrual date, Te a end of accrual date and Tp a payment date.

The value of this contract at time $$t \leq T_x$$ is given by:

\begin{align} V(t) &= \mathbb{E}_t^Q \left[ D(t, T_p) \cdot \left(T_e - T_b \right) \cdot L(T_x, T_b, T_e) \right]\\ V(t) &= P(t, T_p) \cdot \left(T_e - T_b \right) \cdot \mathbb{E}_t^{T_p} \left[ L(T_x, T_b, T_e) \right] \end{align}

This last expectation seems to be not analytically tractable, right? What I would like to know is which restrictions I have to impose in order to solve it analytically. For example, is it sufficient to match only the end date and the payment date, i.e. $$T_e = T_p$$, which yields to:

\begin{align} V(t) &= P(t, T_p) \cdot \left(T_p - T_b \right) \cdot \mathbb{E}_t^{T_p} \left[ L(T_x, T_b, T_p) \right] \end{align}

This expectation could be solved if \eqref{2} is true. Maybe \eqref{2} can be demonstrated using the definition of the Libor forward rate:

$$L(t, T, T + \tau) = \frac{1}{\tau} \cdot \left( \frac{P(t, T)}{P(t, T + \tau)} - 1 \right)$$

Any ideas or this is not possible and both $$T_x = T_b$$ and $$T_e = T_p$$ should match in order to get an analytically tractable expectation?

Finally, just for completeness, I would like to point out that when $$T_e \neq T_p$$, the need of a convexity adjustment arises. This question is all about asking if a convexity adjustment or anything else is needed when $$T_x \neq T_b$$.

We just need to use the tower property of conditional expectation ($$t\leq s\leq T$$):
$$\mathbb{E}_t^{T + \tau} \left[ L(s, T, T + \tau)\right] =\mathbb{E}_t^{T + \tau} \left[ \mathbb{E}_s^{T + \tau} \left[ L(T, T, T + \tau)\right] \right]$$ $$= \mathbb{E}_t^{T + \tau} \left[ L(T, T, T + \tau)\right] = L(t, T, T + \tau)$$