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Let stock A traded at 109/111, let's assume I want to simultaneously buy and sell stock A and recorded buy at the best ask and sell at the best bid, the resulting portfolio will incur a loss due to the spread. (i.e. buy at 111 and sell at 109 therefore loss of 2). But it is more realistic that since I bought at 111 then the latest traded price will be 111 ...

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I am actually getting a slightly different convexity adjustment to Antoine's: For clarity of notation, I use: $T_s=T_1$, $T_e=T_2$ and $yearfrac(T_1,T_2)=\tau$. We then have (by definition of Radon-Nikodym derivative): \mathbb{E}^{Q_{T_1}}_{t_0}\left[L(T_1, T_1, T_2)\right]=\mathbb{E}^{Q_{T_2}}_{t_0}\left[\frac{\partial Q_{T_1}}{\partial Q_{T_2}}L(T_1, T_1,...

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I think that your question can be solved easier. You may ask me why. Here is my answer: First of all the LIBOR forward rate $L(t, t, T)$ is $\mathbb{Q}^{T}$-martingale, where $\mathbb{Q}^{T}$ is a $T-$forward measure defined with the following Ranon-Nikidym derivative structure: \displaystyle\frac{d\mathbb{Q}^T}{d\mathbb{P}} = \frac{e^{-\...

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