I am trying to approximate the price of a European put with strike K and expiration T. I have to assume that the spot price $S_0$ is a lot less than $K$ or a lot more than $ K$.
I understand how this would work with digital options, where the $\frac{S_0}{K}$ would either be very large or very small. Thus, the Black-Scholes formula would yield something closer to 1 or 0.
How would I would this work with a European call or put?
My idea of a solution would be that the put price would be:
$$e^{-rT}(-S_0N(-d_1))$$
Where the $d_2$ is either ignored (if closer to 0) or just one (if closer to 1). But this seems too simplified.
What would be the correct way to approach this problem?