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.

Not really. Delta means the price movement in the option compared to the underlying. Delta of 50 means stock Changes price by one buck. Your position changes 50 cents. So it's like being long or short 50 shares. If your long a call, the closer to the money your position becomes, the closer will the movement in price of the option will match the underlying. Also delta of 50 means you would need to be long 2 calls if you want to match the price movement of being long the underlying, if delta drops buy more calls to reach delta 100 of it increases you can unload some calls if you want to keep a delta parity with holding 100 shares. Anyhow it's a useful value for hedge calculations. Excuse the poor post, it's via iPad

The definition is correct but the practical use of the Delta is to determine either your equivalent share position for risk purposes or to determine the hedge ratio if you choose to hedge it rather than close the position.

Strictly speaking, no, and there's a lot of discussions of this subject out there. In practice, it's probably a good enough heuristic in most cases.

Delta as nothing to do with probabilities. Take a look at the delta of a future that could be 1 or 100%. Does it mean the future would have 100% probability finishing 'in the money' (higher) ? No.

http://en.wikipedia.org/wiki/Greeks_(finance) [edit]As a proxy for probability Some option traders also use the absolute value of delta as the probability that the option will expire in-the-money (if the market moves under Brownian motion).[5] For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has appropriately a 15% chance of expiring in-the-money. Similarly, if a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money. At-the-money puts and calls have a delta of approximately[6] 0.5 and −0.5 respectively (however, this approximation rapidly goes out the window when looking at a term of just a few years, with the ATM call commonly having a delta over 0.60 or 0.70), or each will have a 50% chance of expiring in-the-money. The correct, exact calculation for the probability of an option finishing in the money is its Dual Delta, which is the first derivative of option price with respect to strike.

Need to know the difference between risk neutral probabilities and the real ones. That's dual delta. There is no "correct, exact calculation for the probability of an option finishing in the money" in the real world. I hope that your 'business advisory' is not based on wikipedia !

I'm using sources which are easy to understand to those new to options asking questions. I was an Options Market Maker for 25 years and have been in the business from OTC trading desks, to wire clerk, to floor broker, then trader. My 'business advisory' is based on over 30 years of experience in equities and options on both the buy and sell side for both retail, institutions and for my own trading account.

Where on earth does it mean you can tell them something that is clearly not true using sources which are far from being reliable ?

well, the futures contract is in essense a call with a zero strike... so guess what - yes, it does have a 100% of finishing in the money