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On page 4 of this paper, the author states the following:

"Looking at the limit case, when the strike tends towards 0, we should have the price of a forward contract and it should not depend on the equity skew around 0."

I cannot interpret mathematically what the author stated. Could you please provide a mathematical explanation of what the author said in words?

Thank you!

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The value of a call option at expiry is $V=\mathrm{max}(0, S_t-K)$.

If you set $K=0$, then you have $V=\mathrm{max}(0, S_t)$, and since $S\geqslant0$, $\mathrm{max}(0, S_t) = S_t$ - i.e. ie's equivalent to holding the stock, which at expiry you'll expect to be worth the whatever the forward is.

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  • $\begingroup$ The price today of a contract that will pay off $S_T$ at maturity is, pretty much by definition, the forward price. $\endgroup$
    – nbbo2
    May 30, 2017 at 17:02
  • $\begingroup$ yes, exactly. Is it not clear that that's what i meant? $\endgroup$
    – will
    May 30, 2017 at 17:03
  • $\begingroup$ Same idea, different words. $\endgroup$
    – nbbo2
    May 30, 2017 at 17:31

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