Why Forward Libor Rates Are Martingales Under the Forward Measure

Summary

The document explains why the accrual factor associated with a forward Libor rate is a martingale under the forward measure whose numeraire is the bond maturing at the end of the accrual period. The factor is expressed as the ratio of the bond price maturing at the start date to the bond price maturing at the end date. By the definition of the forward measure, the price of a tradable asset divided by the numeraire bond is a martingale, so this bond-price ratio has the required property.

The forward Libor rate is the accrual factor minus one, divided by the accrual fraction. Since this is a linear transformation of the factor, it is also a martingale. The response gives a direct numeraire argument rather than deriving a change of measure from bond dynamics. Its conclusion applies to the stated forward-measure setup and relies on the bond prices being tradable; the document does not discuss technical conditions for a true martingale versus a local martingale.

Key ideas

• Under the forward measure, tradable asset prices divided by the numeraire bond are martingales.
• The accrual factor for forward Libor is a ratio of bond prices.
• The forward Libor rate is an affine transformation of that accrual factor.
• The response uses the numeraire definition rather than deriving a measure change from bond dynamics.

Tags

# libor rate - local martingale


# libor rate - local martingale












I am a newbie for Libor rates and all these questions...

Let be : $L(t,\delta)$ the Libor rate and $L_{t}(T,\delta)$ the forward Libor rate. Let's define : $Lb(T,\delta):=1+\delta L(T,\delta)=1/B(T,T+\delta)$ and $Lb_{t}(T,\delta):=1+\delta L_{t}(T,\delta)=B(t,T)/B(t,T+\delta)$. The question is to prove that under the forward measure of maturity $T+\delta$ that both $Lb_{t}(T,\delta)$ and $L_{t}(T,\delta)$ are local martingales. I began to define the forward measure of maturity $T+\delta$ (under which the numeraire is $B(T,T+\delta)$ ) :$Q^{T+\delta}$ but it's a lot of calculus. So how can we solve this ? Do we have to start from the model $dB(t,T)/B(t,T)=r_{t}dt+\Gamma (t,T)dW_{t}$ in order to define $dQ(t,T+\delta)=...dQ$ ?

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/22309

By definition, under the $T+\delta$-forward measure, the price of any tradable asset relative to the bond price $B(t, T+\delta)$ is a martingale. Since \begin{align*} Lb_t(T, \delta) = \frac{B(t, T)}{B(t, T+\delta)}, \end{align*} it is a martingale by definition. Regarding $L_t(T, \delta)$, since \begin{align*} L_t(T, \delta) = \frac{1}{\delta}\left(\frac{B(t, T)}{B(t, T+\delta)} -1 \right) \end{align*} is a linear combination of martingales, it is also a martingale.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.