1
$\begingroup$

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 $T_x \neq T_b$.

$\endgroup$
3
$\begingroup$

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

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

$\endgroup$
1
  • $\begingroup$ Perfect, thank you very much! So this is the reason why there is no need for convexity adjustments when the fixing date does not coincide with the accrual start date, but an adjustment is needed when the payment date do not coincide with the accrual end date. $\endgroup$ – rvignolo Sep 6 '20 at 15:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.