Hull states that option prices increase with an increase in volatility.
I think that statement could be false in a specific scenario: when we are considering a deeply in-the-money European put option.
Since we are deeply in-the-money, the price of the underlying would be close to zero. Since the price of the underlying can't be negative, the effect of volatility would be asymetric: it would be more likely for the share price to recover than to fall anymore, simply because there isn't a lot of scope for a fall to happen from an already near-zero share price.
So a higher volatility is more likely to lead to a recovery of the share price, reducing the payoff, in turn reducing the price of the European put.
Is my reasoning wrong? Thanks in advance!