So ive been meaning to write this for a couple weeks now in response to a thread about delta of options being thought of as the probability the option expires in the money.
The Black Scholes equation is as follows:
c = SN(d1) - PV(K)N(d2)
delta is the sensitivity of the option price to changes in the underlying and by simply differentiation:
dc/dS = N(d1)
Mathematically speaking, the risk neutral probability that an option expires in the money is actually N(d2), not N(d1) as is often believed.
N(d2) = pr ( ln(S) > ln(x) ) at expiry
If we ignore discounting, and take the idea that N(d1) is the risk neutral probability as often believed, therefore we get N(d1) = N(d2)
c = SN(d1) - KN(d1) = N(d1)*(S - X)
but as you can see this can make the call option price negative if S < X. Therefore N(d1) will be strictly greater than N(d2).
The problem is that there are two uncertainties of expiration:
1. If we get anything at all: represented by N(d2)
2. How much we get: represented by N(d1)
The key to take away is that when you are starting out learning about options, it is really not helpful to think of theas the probability it expires in the money. It is much more useful and practical to think about the sensitivity to underlying.