Why Forward LIBOR Rates Are Not Automatically Martingales Under Any Numeraire

Summary

The document sets up a payer forward contract entered at zero value and later offset by a receiver contract on the same accrual period. Subtracting the two terminal cash flows yields a payoff proportional to the change in the forward rate between the entry and unwind times. The question asks whether this implies that the rate is a martingale under a measure associated with a general local martingale numeraire.

The central issue is that converting a payoff into a conditional expectation under a chosen numeraire measure introduces the numeraire ratio between the observation and settlement dates. That factor cannot generally be discarded. Forward rates are martingales under an appropriate measure associated with the relevant accrual-period discount bond, while a different numeraire typically changes their drift. The document poses the measure-theoretic question without supplying a resolution, proof, or assumptions on integrability and admissible trading. Its algebra motivates the issue but does not by itself establish an arbitrage or a martingale property.

Key ideas

• Offsetting payer and receiver forward contracts produces a payoff proportional to the change in the quoted forward rate.
• A numeraire change introduces a ratio of numeraire values into conditional expectation calculations.
• A forward rate is a martingale under a measure tied to the appropriate accrual-period bond numeraire.
• The payoff algebra alone does not prove a martingale property under an arbitrary numeraire measure.

Tags

# An arbitrage strategy involving forward contracts to show that LIBOR rates are martingales


# An arbitrage strategy involving forward contracts to show that LIBOR rates are martingales












I note $L_{t}^{[T_s, T_e]}$ the forward rate at time $t$ for the period $[T_s, T_e]$. Recall it is the strike making equal to $0$ the value at time $t$ of a forward contract for the period $[T_s, T_e]$.

The strategy I am looking at is the following : I enter (for $0$) at time $t$ in a payer (I pay the strike) forward contract for the period $[T_s, T_e]$ and at a later time $t'$ I unwind my position by entering in a receiver (I receive the strike) forward contract for the period $[T_s, T_e]$. Noting $X$ the notional and $\delta$ the year fraction represented by the period $[T_s, T_e]$, my payout at time $T_e$ is $$X\delta\left( L_{T_s}^{[T_s, T_e]} - L_{t}^{[T_s, T_e]}\right) - X\delta\left( L_{T_s}^{[T_s, T_e]} - L_{t'}^{[T_s, T_e]}\right) = X\delta \left(L_{t'}^{[T_s, T_e]} - L_{t}^{[T_s, T_e]}\right).$$

(From two times nothing I have generated a non zero P&L.) From there, assuming non arbitrage and therefore the existence of a local martingale numéraire $N$ and an associated local martingale measure $\mathbf{Q}^N$, I basically want to apply the $\mathbf{E}^{\mathbf{Q}^N} \left[ \bullet | \mathscr{F}_t\right]$ operator and conclude that $L_{t}^{[T_s, T_e]}$ is a martingale under $\mathbf{Q}^N$.

Dividing $X\delta \left(L_{t'}^{[T_s, T_e]} - L_{t}^{[T_s, T_e]}\right)$ by $N_t$ leads by for the "$L_{t'}^{[T_s, T_e]}$ part" with a $\frac{N_{t'}}{N_t}$ factor I can get rid of.

How can I do this properly ? (I voluntarily stay in a non-diffusive setting.)

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