As volatility goes to infinity, the delta of a call option goes to 1. The delta approximates the probability that the option expires in the money. So it seems that the probability of expiring in the money is very close to 1.
However, the price of the put option approaches the constant function at the strike price. This is what one would see if the probability of the call expiring out of the money is close to 1.
This seems to suggest that the probability that the call option expires in the money and out of the money is both close to 1.
Put another way, as the volatility increases, the probability of a call expiring in the money increases as well. But so does the price of a put option. So it would seem that the price of a put increases, even though the probability of the put expiring in the money decreases.
How to make sense of this?