I've heard one way to interpret delta is that it is the probability of an option finishing in the money. For example, a delta of .50 means that it should have a 50% chance of finishing in the money. This makes intuitive sense as an option with a delta of .50 is probably at the money. I've always thought this was the case as this was mentioned in several books I've read and even a market maker said so; however, I'm reading through Cottle's book and he says this a poor interpretation as it is mathematically incorrect. So is this a correct way to interpret delta? Thanks.