I note <span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="Lt[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">L[Ts,Te]tLt[Ts,Te] the forward rate at time <span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="t" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">tt for the period <span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">[Ts,Te][Ts,Te]. Recall it is the strike making equal to <span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="0" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">00 the value at time <span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="t" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">tt of a forward contract for the period <span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">[Ts,Te][Ts,Te].
The strategy I am looking at is the following : I enter (for <span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="0" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">00) at time <span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="t" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">tt in a payer (I pay the strike) forward contract for the period <span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">[Ts,Te][Ts,Te] and at a later time <span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="t′" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">t′t′ I unwind my position by entering in a receiver (I receive the strike) forward contract for the period <span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">[Ts,Te][Ts,Te]. Noting <span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="X" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">XX the notional and <span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="δ" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">δδthe year fraction represented by the period <span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">[Ts,Te][Ts,Te], my payout at time <span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="Te" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">TeTe is
Xδ(L[Ts,Te]Ts−L[Ts,Te]t)−Xδ(L[Ts,Te]Ts−L[Ts,Te]t′)=Xδ(L[Ts,Te]t′−L[Ts,Te]t).Xδ(LTs[Ts,Te]−Lt[Ts,Te])−Xδ(LTs[Ts,Te]−Lt′[Ts,Te])=Xδ(Lt′[Ts,Te]−Lt[Ts,Te]).
(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 <span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="N" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">NN and an associated local martingale measure <span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="QN" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">QNQN, I basically want to apply the <span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="EQN[∙|Ft]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">EQN[∙|ℱt]EQN[∙|Ft] operator and conclude that <span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="Lt[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">L[Ts,Te]tLt[Ts,Te] is a martingale under <span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="QN" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">QNQN.
Dividing <span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="Xδ(Lt′[Ts,Te]−Lt[Ts,Te])" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">Xδ(L[Ts,Te]t′−L[Ts,Te]t)Xδ(Lt′[Ts,Te]−Lt[Ts,Te]) by <span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="Nt" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">NtNt leads by for the "<span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="Lt′[Ts,Te]" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">L[Ts,Te]t′Lt′[Ts,Te] part" with a <span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="Nt′Nt" role="presentation" style="border: 0px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">Nt′NtNt′Nt factor I can get rid of.
How can I do this properly ? (I voluntarily stay in a non-diffusive setting.)