Yes
@totoschillaci not an exact science and
@Stefans you already answered in your follow up. If it is working for you, well done.
Let me say firstly i teach this to Uni students. I do not have all the answers but let me share my thoughts.
Firstly to assume probability matches the delta you need to assume:
1. data is normally distributed; and
2. follows random brownian motion
Both criteria have been proven over history as flawed. Also the delta is N(d1) in black-scholes’ formula while the probability of exercise is N(d2) - without including the price of the option. That is, probability of exercise would be N(d1)/ { E[ST | ST>K] }.
Let me give an example of what i think delta to probability is:
- Strike price = $20/share
- Time to expiration = 0.5 years (I used a 360-day year, so this is 180 days)
- Annual (effective) risk-free rate = 2%
- (Annual) volatility of (continuously compounded) returns of the underlying: 10%
Using
@risk in Excel to run the Monte Carlo simulations: 100,000 iterations for each delta you use solver to determine the (spot) price of the underlying for each delta. I compounded daily returns, with serial correlations of zero.
These are the results:
- Delta = 0.00, P(exercise) = 0.0%
- Delta = 0.10, P(exercise) = 0.0%
- Delta = 0.30, P(exercise) = 0.0%
- Delta = 0.30, P(exercise) = 0.0%
- Delta = 0.40, P(exercise) = 0.0%
- Delta = 0.41, P(exercise) = 0.0%
- Delta = 0.42, P(exercise) = 0.0%
- Delta = 0.43, P(exercise) = 0.0%
- Delta = 0.44, P(exercise) = 0.0%
- Delta = 0.45, P(exercise) = 0.1%
- Delta = 0.46, P(exercise) = 0.5%
- Delta = 0.47, P(exercise) = 1.8%
- Delta = 0.48, P(exercise) = 5.0%
- Delta = 0.49, P(exercise) = 12.4%
- Delta = 0.50, P(exercise) = 25.0%
- Delta = 0.51, P(exercise) = 42.3%
- Delta = 0.52, P(exercise) = 61.0%
- Delta = 0.53, P(exercise) = 77.6%
- Delta = 0.54, P(exercise) = 89.2%
- Delta = 0.55, P(exercise) = 95.8%
- Delta = 0.56, P(exercise) = 98.6%
- Delta = 0.57, P(exercise) = 99.6%
- Delta = 0.58, P(exercise) = 99.9%
- Delta = 0.59, P(exercise) = 100.0%
- Delta = 0.60, P(exercise) = 100.0%
- Delta = 0.70, P(exercise) = 100.0%
- Delta = 0.80, P(exercise) = 100.0%
- Delta = 0.90, P(exercise) = 100.0%
- Delta = 1.00, P(exercise) = 100.0%
If the delta were a good approximation to the probability that the option would be exercised, a graph of P(exercise) vs. delta would look like this:
/
In fact, the graph looks like this:
_/¯
I can run some more simulations, particularly with different volatilities on the underlying returns, but the results here are pretty clear: the option delta isn’t remotely a good approximation to the probability that the option will be exercised. By the way, another interesting result is that the delta of an at-the-money call option isn’t necessarily 0.50. Here, the delta for an at-the-money call option is 0.57. You get a 50-delta call when the spot price is $19.75 and the strike price is $20.00.
If you are really into Options i recommend Option Volatilty & Pricing by Sheldon Natenberg, or these three recent academic articles
1.
http://www.sciencedirect.com/science/article/pii/S0096300315010073?via=ihub
2.
http://www.sciencedirect.com/science/article/pii/S0165188916000026
3.
http://www.sciencedirect.com/science/article/pii/S0022247X1730104X
Personally Options are not for me as i do not believe in normally distributed data.
Goodluck in your journey and be nice to franky
(20min delay)