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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 $\delta(i) \times \text {probability}$

Expected Value

$\sum\limits _{n}^{y}=1$

$\cfrac y{c\times n}(p_\text{maxprice}-p_\text{strike})\times p(y) =e^{rt}+1$

$c$ 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?

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the (p_max-p_strike)/c goes out leaving just: sum y/n*f(y) this is a riemann sum the solution is integral from p_strike to p_max y*p(y)dy where p(y) is a truncated normal distribution –  Mort Goldman Oct 29 '12 at 2:37
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