http://www.optionsplaybook.com/options-introduction/option-greeks/
Delta isn’t a probability, but apparently people treat it like one (or approximately). Seems like the whole risk-neutral “probability” stuff. Also, this wouldn’t be the first time that people in finance (and tons of other fields) have been sloppy (or outright misused) statistics/probability.
I just read both of those.
It’s sad what people will believe.
And while we’re on the subject of stupid option stuff, where did the idea arise that an ATM call option will have a delta of 0.5? That’s just plain silly.
I guess only when the time-premium of money is zero, or the time horizon is infinite 
Or a binary option, or both. I don’t have MCS to test though.
My guess is the same. When the time to maturity is zero, delta of an ATM would be 0.5ish.
I think some people assume that it’s equally likely to rise or fall over the next step in time. Similar to what they are saying in the link XK posted, if its the very last instant before expiration, and the option is ATM, it could make sense under the assumption of equiprobable up or down moves.
Other than that, it seems like a reallllly bad and loose approximation (based on your monte carlo simulations). I will say that I have heard people use it before as a probability, but I have more frequently heard it as a partial derivative of the option value with respect to the underlying’s price (you know, because that’s what it is…).
It is certainly true (in the B-S-M model) that as T approaches zero, the delta of an ATM call approaches 0.5.
It’s also true that if rf = 0 and σ = 0, the delta of an ATM call is 0.5. Not that anybody cares at that point.
In all other cases, the delta of an ATM call is greater than 0.5.
They’re saying just what I was saying.
It’s easy to see if you look as the formula for _d_1 in B-S-M: unless rf and σ are both zero, _d_1 > 0 for an ATM call, so N(_d_1) > 0.5
Honestly, though, what percentage of people in finance are well-versed in multi-variate calculus?
Or versed at all?
We could consider that percentage to be the probability of a randomly selected person in financing being well-versed in multi-variate calculus. Furthermore, we could approximate that probability by using the delta of that person . . . .
Sorry . . . got carried away there.
this thread is a car crash… according to some random online calculator… A stock price is $20.67 will give a $20.00 strike with 0.5 delta and the other inputs…
any non genius will tell you a stock price of $20.67 and strike of $20 you have a slightly > 50% chance of expiring in the money.
I’m not sure that I follow you.
To begin with, a delta of 0.5 and a strike price of $20 corresponds to a spot price of $19.36, not $20.67.
If I may be so bold, where’d you get that number?
We were all thinking it (at least I was)…this also comes back to the notion that mass acceptance of an idea or term doesn’t make it accurate. However, if they had some credible research to show this idea holds, I’d be willing to buy it. So far, we’ve found things that indicate it isn’t a very good approximation…
Cool.
What’s your point?
I believe it was you that was making the point that a 0.5 deta option has a 2% chance of expiring in the money. my point is that its actually over 50%
Got it.
Your point’s wrong.
By a mile.
He has a point.
The probability should be much higher that the stock will be >$20 than 2%, even though it’s not 50% either. It’s actually 41.6% for 180 days at 50-delta implied stock price using a norrmal return distribution, as per BSM.
With all due respect, I’m quite certain he hasn’t.
I’d love to know how you arrived at that number.
What is the spot price for the stock with a 50 delta?
What is the probability distribution of returns on that stock?
With 30% volatility in the continuously compounded returns and a 2% effective annual risk-free rate I got a spot price of $19.36 from the B-S-M model. I’m pretty confident in my model for the price at expiration.
I’d love to see the details of your methodology.
$19.36, like you’ve said earlier.
What’s the probability that the price will be above $20 in 180 days time with a volatility of returns equal to 30%? Much higher than 2%. I used a calculator to get to the actual figure.