I got asked something similar to this in interview today.
The interviewer wanted to know what is the probability that an at-the-money option will end up in-the-money when volatility tends to infinity.
I said 0% because the normal distributions that underly the Black-Scholes model and the random walk hypothesis will have infinite variance. And so I figured the probability of all values will be zero.
My interviewer said the right answer is 50% because the normal distribution will still be symmetric and almost uniform. So when you integrate from mean to +infinity you get 50%.
I am still not convinced with his reasoning.
Who is right?