Hmmm ... technically speaking ETO's are smoothed as an Underlying goes ex-div.
Where a discrete dividend is paid during the life of an ETO the dividend amount is usually deducted from the Underlying price 45 days out from Expiry and the Implied Volatility altered accordingly.
As the stock goes ex-div the dividend amount reverts to zero and the price drop of the Underlying factored into the equation ... this contra's or smoothes the ETO fair value.
This particularly relates to holders of Puts who might otherwise believe they can profit at this time.
Of course holders are subject to the price of the Underlying moving more or less than the dividend amount which will effect the ETO.
Another factor is Implied Volatility which could also alter the ETO price reflecting Market sentiment.
An example is today with TLS gone ex-div and the ATM Put (TLSZ7) closed 190304 at $0.09 and last today $0.095 reflecting the $0.15 drop in TLS today. (divvy $0.12 plust $0.03 drop).
I have posted previously on this subject ...
Cheers ... tight stops.
This is only my view ... read the red stuff.
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