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In many papers and book I have found a reasoning that it is well summarized in this paper as

"The first proxy we use is an unadjusted Black-Scholes proxy in which the implied volatility of a short-maturity at-the-money option is used in place of the true instantaneous volatility state variable. The use of this proxy is justified in theory by the fact that the implied volatility of such an option converges to the instantaneous volatility of the logarithmic stock price as the maturity of the option goes to zero."

Where can I find a formal justification of this assertion?

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This is a result of Ledoit et al

Lemma proof in Appendix of http://www.ledoit.net/9-98.pdf

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    $\begingroup$ thanks it is quite useful. Nevertheless I do not see the point in taking a forward call, why not a simple call ? $\endgroup$
    – AlmostSureUser
    Commented Apr 29, 2016 at 14:20
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See also this dissertation by Durrleman.

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