Why a Forward Libor Rate Is a Martingale Under Its Payment-Date Measure
Summary
The document examines confusion about whether Libor and a zero rate are martingales under a forward measure. The question defines a rate over the interval from the present time to a future date, then reasons from bond prices and numeraires to compare that rate with a forward rate derived from two bonds.
The response identifies the key issue as the tenor definition: a forward Libor fixing at a date applies to the subsequent accrual period, rather than the period from today to that date. For a rate accruing from T to T plus a tenor, the bond-price relation gives a martingale under the forward measure associated with the accrual end date. This distinction resolves the apparent contradiction. The explanation is limited to the stated bond and rate definitions and does not develop broader measure-change results or practical curve-construction details.
Key ideas
- A Libor rate’s fixing date and accrual period must be distinguished.
- A forward Libor fixed at T accrues over the period from T to T plus its tenor.
- The rate is obtained from the ratio of bond prices at the accrual start and end dates.
- Under the stated setup, the relevant martingale measure is associated with the accrual end date.
- Treating Libor as covering the period from today to T leads to the confusion described.
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Full text
# Is Libor a martingale under T-forward measure
# Is Libor a martingale under T-forward measure
We denote `discount factor` $D(t)$, and `zero coupon bond` $B(t,T)$ as: $$B(t,T) =\dfrac{1}{D(t)} E_t[D(T)]$$ here $E_t[X] = E[X|\mathcal{F}(t)].$
And we define
`Zero curve` $Z(t,T)$ $$B(t, T)\cdot e^{Z(t,T)(T-t)} = 1$$
`Libor` $L(t,T)$ $$B(t, T)\cdot (1 + (T-t) L(t, T)) = 1.$$ Denote $E^{T}[\ ]$ the $T$-forward measure i.e use $B(t,T)$ as numeraire.
I remember that Libor and zero curve should be `martingale` under the $T$-forward measure. But we see the representations, equivalently $\dfrac{1}{B(t,T)}$ should be martingale under the $T$-forward. This is the $T$-forward price of $1,$ but discounted value of $1$ is not martingale under original measure i.e $$E_t[D(T)\cdot 1] \neq D(t)\cdot1.$$ We can see that forward rate $$B(t,T-\delta) = (1 + (T-t)F(t,T-\delta,T))B(t,T)$$ is really $T$-forward martingale, since $\frac{B(t,T-\delta)}{B(t,T)}$ is $T$-forward martingale, equivalently $D(t)B(t,T-\delta)$ is martingale under the original measure. So, I really confuse here. Can anyone tell where is the mistake?
## Answer by Antoine Conze (score 3)
https://quant.stackexchange.com/a/34472
Your definition of Libor is invalid as you make it cover the period $t, T$.
A Libor with tenor $\delta$ that fixes on $T$ (or to be accurate usually 2 days before $T$) covers the period $T, T+\delta$. Thus the forward Libor rate $L(t, T, T+\delta)$ is computed as $$ B(t, T+\delta) (1 + \delta L(t, T, T+\delta)) = B(t, T) $$ hence $$ L(t, T, T+\delta) = \frac{1}{\delta}\left(\frac{B(t, T) }{B(t, T+\delta)} -1 \right) $$ is a martingale under the $T+\delta$ forward measure.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.