# Trying to price options using infinite series

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$\sum\limits _{n}^{y}=1\cfrac y{c\times n}(p_\text{maxprice}-p_\text{strike})\times p(y) =e^{rt}+1c$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/nf(y) this is a riemann sum the solution is integral from p_strike to p_max yp(y)dy where p(y) is a truncated normal distribution –  Mort Goldman Oct 29 '12 at 2:37