Forward Libor Martingales Under the Payment-Date Forward Measure

# Libor rate and martingales


# 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 $T_x \neq T_b$.

## Answer by ir7 (score 3, accepted)

https://quant.stackexchange.com/a/57815

We just need to use the tower property of conditional expectation ($t\leq s\leq T$):

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