If you are trying to price an option if the stock surges you can reap a very large return, but most of the time the return is $-p_1$ where $p_1$ is the amount you invested
The problem i'm running into is how do you get an expected value for your call option if you have numerous weighted probabilities? There is a tiny probability of a big payoff, if the stock goes to $\$1000$ or whatever. If the price ends at the strike or lower, you lose your premium so the return is always negative $-p_1$. This is easy to calculate because it's just the probability of the stock ending below the strike price to theoretically $0$. But what about between a theoretical maximum price and the strike price? Then you have many weighted sums where the weight is equal to the $\delta(i) \times \text {probability}
\sum\limits _{n}^{y}=1
\cfrac y{c\times n}(p_\text{maxprice}-p_\text{strike})\times p(y) =e^{rt}+1
c is the call option price that you're trying to solve for
r,\space t is risk-free interest and time
where p(y)$ is the probability of the price being between $\cfrac{y}n (p_\text{maxprice}-p_\text{strike})$ and $\cfrac{y-1}n (p_\text{maxprice}-p_\text{strike})$
I can get the probability using Truncated Normal Distribution
but how do I find a closed way of doing this taking into account the weighting?