Delta is, I have been told, a rough estimate, of the probability of an option expiring in the money. My question is ... how rough? 1-2 percent? Percentage points? 5-10? Depends on how far OTM?

In the money options have a delta of greater than 50 At the money have a delta of 50 Out of the money have a delta is less than 50 Say the stock is at 30, and you are looking at a 30 call. The delta is going to be very near to 50, if not at 50. That means there is a 50% chance the option will expire ITM. An out of the money 32.50 call in the same example might have a delta of 30. This means there is a 30% chance it will expire in the money. Hope that helps.

Say the option has a delta of 25, iow a 25% probability of expiring itm. The problem is 'how much itm' will it be at expiry plus there's still a 75% probability of expiring otm, the problem is also again how much otm. In summary if you've shorted options then expiring otm, no matter by how much, is great. If you're long options then expiring itm may not be helpful because your premium paid may have been more than the value of the option at expiry, iow you've still lost money. daddy's boy

Are you saying that delta precisely (withn 1 percentage point) = probability of being ITM? It is not a rough estimate? (If it is a rough estimate and not precise, then how imprecise is my question.)

Its because the Black Scholes model is risk neutral, that is the expected return of the stock is the risk free rate. So the delta is the probability in a risk neutral world, in reality the expected return of the stock could be 100% consider a buyout offer at 100% of the price. Also the model assumes that returns are Log normally distributed. In fact the log normal distribution underestimates the number of very small (approx 0%) moves and the number of very large moves.

Delta is the change in the option premium if spot changes. It is an approximate measure of the probability that the option will finish in the money. It measures the sensitivity of the premium ( or the value) of an option with respect to the underlying rate of the currency or stock at that moment. If delta is close to 100 then the right to own the currency or stock is nearly as good as actually having them. So an option with a delta of 100 will behave in a similar way as the spot market. Delta changes when prices change, so delta measures the probability at that moment, because if spot changes, the premium will change too and probably not in a 1 to 1 relation. Delta is very useful for the concept of having a delta neutral position. Professional option hedgers usually use it to be protected from spot risk. Delta is used to hedge the CURRENT exposure. So when delta changes the hedge needs to be corrected to the new delta to become delta neutral again.

Guys, to cut to the chase, I know what delta is, et-certa. What I want to know is what is the mathematial diffeence between the probability of an option landing ITM (@ expiration) at a given moment and the delta at that same. Foe example, if the delta is 20; that means that roughly the chances of that contract landing ITM at expiration is 20 percent. But with this same contract what is the statistical probablity? Does, for lack of a better term, delta-probability differ from the statistical probablity at a givven moment, and if so by how much, how often, et-cetera. Is the statistical probablity for the aforementioned option 22% versus the 20 delta? 18? Is this spread generally about the same? CAtch the drift of the original question?

It depends on the probability distribution you use. Black Scholes Merton relies on lognormal, but general statistics uses the normal. There will always be a difference. If you use the normal distribution, you are assuming that financial instruments in general have a very tight range, when in fact, the ranges are wider. You will think option prices are expensive, when they are not.

It is a probability, but in the world were the drift of the stock is the risk free rate. If we could estimate the real drift, we would have a money making machine. The Black Scholes price is the price at which any two people can agree in spite of the fact that one thinks the stock is likely to go down and the other that it is likely to go up. So yes it is a probability, it may be close to the real one or very far from it.