I am reading the section on Skew adjustment for binary options on wikipedia (https://en.wikipedia.org/wiki/Binary_option#Skew) and am trying to get my head around it and gain some intuition.
First question is regarding Binary Put options, following the same calculations as for the Binary call and constructing a put spread I get that the Skew adjustment in this case is instead + Vega of the vanilla put times the skew, meaning that the skew adjustment makes a binary put worth less (if Skew is negative). This also makes some sense intuitively as the Vega of the binary put is negative when SK, meaning that when prices drop (and volatility rises) we are gaining less (relatively speaking). Is this reasoning correct?
The second is about the magnitude of this adjustment, the Skew should be greater the further we are from ATM meaning the adjustment is larger, but the Vega of the vanilla should be decreasing as we get further from ATM so we have two competing effects. When is the skew adjustment necesarry to make, when is it neglibible? And are there more factors to consider than the moneyness of the option? On average, how large is the adjustment (in terms of % of the value of the non skew-adjusted option)?