I've bought Gatheral's book on Local Volatility and I have troubles with understanding a part where he shows that local variance is a conditional expectation of instantaneous variance.

Why in the second equation from the bottom he just skips the term $\theta (S_T-K)dS_T$? He says that it's because $F_{t,T}$ is a martingale. I see that $F_{t,T}$ is a martingale, but don't know how this helps. Also , what "condiational expectations" is he talking about? The notation looks a bit sloppy. Thanks for help.

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  • $\begingroup$ From equation (9) $dF_{TT} = \sqrt{v_T} F_{T,T} dZ$ with $Z$ a standard Brownian motion. Now since $dF_{TT} = dS_T$ by absence of arbitrage, you see that when you take the expectation of the penultimate equation, the first term is zero by the properties of the Brownian motion (the integrated version would be to say that that an Itô integral has zero expectation). $\endgroup$
    – Quantuple
    Apr 16 '18 at 9:46

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