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Lets assume the price of the underlying equals the strike at some point prior to expiry. Then the probability of the price being still greater or equal the strike at expiry is 0.5. So the probability of the European option paying out is exactly half of the probability for the American option.


A relevant point here is that the european digital pays out only at expiry. Intuitively a decrease in the risk reversal (more negative) implies that spot will be more volatile at the strike level. If spot does in fact cross the strike level then the vega of the option will have flipped where higher vols will mean lower price. If you check the price the of a ...

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