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The idea is to choose a strike base on the premium and historical data to have maximum profit.

For example a selling a (European) call.

$$Profit = Premium_K - (S(t) -K)^+$$

Replacing $(S(t) -K)^+$ for the Cdf

$$Profit = Premium_K - (1- \Phi(K)) K$$

I know that it is a naive approach, but is there any merit on it?

Second question: How can I improve it?

Thank you,

Edit 1

Change in the above text: Payoff to Profit (as point out by nbbo2) and Pdf to cdf (since I realize I made a typo)

From the first equation to the second one:

Graphically $(S(t) -K)^+$ is the area bellow the pdf or

We do not know $S(t)$ since it is a future event, but we know the probability of a certain $S$ occur: $$(S(t) -K)^+ = \int_K^\infty\phi(S) dS = (1- \Phi(K)) K$$ where: $$\phi(x) = pdf$$ $$\Phi(x) = cdf$$

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  • $\begingroup$ If an option is fairly priced, it's expected profit should be 0 (all of the terms in your equation should cancel out). It's only if you think it's mispriced because one of the inputs to the pricing model is wrong that you should expect a profit. $\endgroup$
    – D Stanley
    Oct 31, 2023 at 13:30
  • $\begingroup$ I agree with you, "IF". There are a lot of research to find what is the option fair price :) $\endgroup$
    – aoliv
    Nov 2, 2023 at 19:55
  • $\begingroup$ My point is that the "premium" is calculated with the same parameters that you would use in your CDF, and thus would cancel out. The only way they would be different is if you believed the parameters for the CDF should be different than what the market uses to calculate the premium. $\endgroup$
    – D Stanley
    Nov 2, 2023 at 20:48

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