Well, a reasonable approximation that will include the skew but not the term structure is to price a tight call or put spread and multiply it by two
Option math question
- Thread starter levijean
- Start date
Let me give you a simple, non expert, mom and pop retail trader's approach:
The calculation at expiration: The premium of $0.50 means the underlying will have to be at $20.5, so you go to the option probability curve (your broker should provide or go to CBOE site to get it) and find the probability that the underlying will be at $20.5 and you are done. Of course the probability calculator assumes Black Scholes is correct. If it is not, the probability will be off but it should still give you a good feel of your odds.
At any other time to expiration: I think you can get the probability, again assume volatility, interest rate and dividend rate are all fixed and Black Scholes is correct, by running a Black Scholes with remaining time to expiration at different strike prices and your given underlying ($20), find the strike that gives a premium of $0.50. Now you can look at the probability curve to find the probability that the underlying could be at that strike price or above. The probability will be higher than at expiration since you have positive time premium. If you have to vary the volatility and interest rate, it will be a complex calculation.
I assume you are trying to make an assessment at the time you place the trade and not at any other time since at any other time the underlying, volatility and time to expiration all change and you have to use the then values for the calculation.
I may be wrong if so someone correct me.
Good luck.
The calculation at expiration: The premium of $0.50 means the underlying will have to be at $20.5, so you go to the option probability curve (your broker should provide or go to CBOE site to get it) and find the probability that the underlying will be at $20.5 and you are done. Of course the probability calculator assumes Black Scholes is correct. If it is not, the probability will be off but it should still give you a good feel of your odds.
At any other time to expiration: I think you can get the probability, again assume volatility, interest rate and dividend rate are all fixed and Black Scholes is correct, by running a Black Scholes with remaining time to expiration at different strike prices and your given underlying ($20), find the strike that gives a premium of $0.50. Now you can look at the probability curve to find the probability that the underlying could be at that strike price or above. The probability will be higher than at expiration since you have positive time premium. If you have to vary the volatility and interest rate, it will be a complex calculation.
I assume you are trying to make an assessment at the time you place the trade and not at any other time since at any other time the underlying, volatility and time to expiration all change and you have to use the then values for the calculation.
I may be wrong if so someone correct me.
Good luck.
The mathematical formula for the probability of a touch is:
e-qt * N(d1) x 2
So the probability of a touch is approximately twice the delta of the option.
The Delta of the option is the primary driver of an options behavior and provides a predictable model of what the option will do relative the underlying product. Many traders refer to the delta as the probability of the option expiring In-The-Money. The even more detailed math proves that the probability of a touch is approximately twice the value or 2xDelta of the call.
It's close enough anyways...
e-qt * N(d1) x 2
So the probability of a touch is approximately twice the delta of the option.
The Delta of the option is the primary driver of an options behavior and provides a predictable model of what the option will do relative the underlying product. Many traders refer to the delta as the probability of the option expiring In-The-Money. The even more detailed math proves that the probability of a touch is approximately twice the value or 2xDelta of the call.
It's close enough anyways...
I would say this is fairly correct...
Black Scholes is more correct than you think. You just need to adjust for skew... and toss in a dividend.
In any way, you have to incorporate changes in IV... otherwise you're missing a major component.
Thanks. Can I approximate dividends with a dividend rate, assuming the ex-dividend date is far away?
I would think it would have to be qualified as pertaining to OTM options only since a call wtih a delta of .6 does not have a 120% chance of being touched as 1) it is already ITM and 2) 100% is the cap.
Basically, any option with a delta greater than .5 can expect to be tested at least once. A one standard deviation option with a delta around .25 has a 50% chance of a touch. Plan for it.
That works for the probability of the underlying price touching a given barrier before expiry, but I don't it works for the probability of an option price touching a barrier set in terms of that option's price. I think the problem is in the "multiply by two" part. The logic with the underlying price is that once it touches the barrier exactly there is a roughly 50/50 chance it will be at or above that price by expiry; ergo, multiply by two looking backwards. For an option price (except for deep in the money options) on the other hand, when it touches the barrier, the chance it will be at or above by expiry is less than 50%. How much less? I don't know, it appears to be a non-linear function of iv and time to expiry, so no simple formula will work.
I may be wrong, as trying to work options behavior out analytically or from first principles has seldom worked for me. So I will run a simulation (using gb2 distro to account for skew and kutosis) when I get a chance.