I am trying to think what is the sensitivity of american digitals (henceforth binary or one-touch (OT)) options to a) implied volatilities between atm and the barrier (henceforth ITM vols), b) implied vols on the other side (henceforth OTM vols) , c) the slope of the volatility smile (henceforth skew).
to me, intuitively, since implied vols are the average of local vols , then increasing the ITM vols should cause the chance of hitting the barrier to rise, thus should push the price up. And decreasing OTM vols too should push the price up. So increasing the slope of the vol surface, so that vols at strike=barrier go up and vols on the other side of the spot go down, should increase the price of the option.
Is this reasoning correct , or are there scenarios where it is not true?
I cannot see what is wrong with it, HOWEVER, i know that if the barrier is quite far away, then the binary price converges to the european digital (ED) price , and the ED price's sensitivity to itm or otm vols can be positive or negative depending on various things.
BUT, if the barrier is not far away, then the price can be more than twice the ED price and so it may not behave like an ED, so what can we say about it there?