This seems intuitively true, but is not correct, if you are talking about Black-Scholes-Merton "delta". Try any BSM calculator and set volatility first to 20% and then to 200% (for example) and see how much the "deltas" for the call and put change. The higher the volatility, the higher the "delta" for the call and the lower the "delta" for the put. The reason for this is because the mathematics behind the formula in general and behind the "delta" in particular are completely wrong. This is all theory and I don't know if it's helping somehow. But the whole concept of "delta" is theoretical anyway.
I think that could be explained by the skew between put and call deltas in general but not sure (don't know theory behind it). However, the delta across calls or delta across puts with changing volatility should remain the same ATM, isn't it? Also, how much difference do you see in the put delta vs call delta?
No, and this was my point. "Delta" in BSM formula is calculated from several inputs, one of them is volatility. If volatility changes, delta changes too. I suppose that you look at "delta" as "probability of ending in the money" and from this you make the logical conclusion that it should be close to 0.5 irrespective of volatility. I don't know who came up with the idea that "delta" represents probability of ending in the money, but it's not true. Here is something relevant to the topic: http://en.wikipedia.org/wiki/Moneyness From the article: "Beware that (percentage) moneyness is close to but different from Delta". I'll give you an example about delta. Imagine we have this option: Stock (S) = 100 Strike (K) = 100 annual volatility (σ = 20% Riskless rate (r) = 0% Time to expiration (T-t) = 1 year According to BSM theory, first we have to calculate this d1: Then delta is the standard normal cumulative distribution function of d1. In excel this is =normsdist(d1). You can write the formula on excel and see how d1 and its standard normal cumulative distribution function (="delta") changes when you change some of the inputs. For example, in our example d1=0.1, but if volatility is set to 40%, d1 becomes 0.2. If volatility is 200%, then d1 becomes 1. Also when you increase time, d1 increases. This is because according to theory, prices follow random walk with constant volatility, and thus we can use stochastic calculus to calculate volatility of the stock during the time of the option. In the formula, "annual volatility" is multiplied by the square root of "time to expiration". This means that if you change volatility to 28.28%, you will get almost the same d1 and delta as if you change "Time to expiration" to 2 years. This is because 20%*sqrt(2)=~28.28%. So this probably answers the question in your first posting - why do ATM options with longer time to expiration have higher delta than same strike options with shorter time to expiration. This is true not only for BP, but also for the other stocks, SPY etc. The "delta" however doesn't tell us anything about probability of ending in the money.
"The "delta" however doesn't tell us anything about probability of ending in the money." Very good point . If you ever try to price a simple barrier, close to the trigger you'd get delta around 10000%. Hence I agree, it has nothing to do with a probability. It's just an hedge ratio. This is widely written by people who don't get it. The obvious example is a forward. For every initial level, a forward or a future got a delta around 100% (interest rate set to 0). Does it mean that the future has 100% probability to close above its initial level. Sure it doesn't. .
I am offended and injured by the last sentence. My desire to give is shaken, and my knowledge may have been erased by the shock.
Test it. Give me some hypothetical ATM deltas (no carry), stock price, keep the rest with you, and see if I can get the price of the option.
50 delta ATM does not exist (according to my specifications above). Now be real my friend, and give a real quote!