# 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.)

• It seems like I have confused you in the related question you posted some days ago, so I hope someone will better guide your understanding. But remember that the forward rate as you correctly mention, is not the value of a traded instrument but merely a scalar. Making the parallel with equity forwards, this is the forward price vs forward value paradigm: the latter comes from arbitrage pricing theory (numéraire induced EMM) but is linked to the first through a definition (forward value is zero at inception)... – Quantuple Apr 5 '17 at 8:27
• This is why you can't apply $F(t,T)/B_t$ is a $\Bbb{Q}^B$-martingale (hence $F(t,T)=\Bbb{E}^{\Bbb{Q}^B}[B_t/B_T F(T,T)]$) to determine the forward price, but rather, from the definition of the forward value, the forward price comptes as: $F(t,T) = \Bbb{E}^{\Bbb{Q}^B}[F(T,T)]$ where by AOA $F(T,T)=S_T\,\, \Bbb{P}-\text{a.s.}$. – Quantuple Apr 5 '17 at 8:30
• My problem is actually indeed still the same : I don't understand where the particular type (spot, forward, annuity) of measure and numéraire arises from. For my previous question, you told it was bank account numéraire becaus Bergomi is looking at discounted P&L. Why is it, in the LIBOR case, the $T_s$-forward measure ? – Olorin Apr 5 '17 at 10:17
• I am only confused by how to stick financial accounting reasoning to the maths, no problem with the maths for me etc, to precise. – Olorin Apr 5 '17 at 10:18