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If it is predicted that the price of a stock will increase from P1 to between P2 and P3 in time T (assume the distribution of the price will be evenly distributed between the range of [P2, P3] at time T), how to optimize to find the best call option to purchase to maximize the profit?

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    $\begingroup$ What have you tried so far? $\endgroup$ Nov 7 '21 at 4:50
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Assuming the options available to you are priced using the Black-Scholes model and because your predicted prices of the stock at time $T$ are evenly distributed between $P_2$ and $P_3$ where $P_3 \ge P_2$, you should simply take the option with the strike price $K = P_2$ (or any lowest available).

This is because the fee (price) for a call option decreases more slowly than its strike price increases. Any call option with a strike price higher than the lowest available will cost you more in gross profit than it saves on the fee under these assumptions.

You can see how the price of a call option typically changes (decreases) as the strike price increases in the middle left chart below where $S_0$ is the stock's current price, $r$ is the risk-free interest rate, $\sigma$ is the stock's volatility, and $T$ is the time from today in years. The figure is from page 258 of the ninth edition (2018) of the "Options, Futures, and Other Derivatives" by John C. Hull.

Effect of changes in stock price, strike price, and expiration date on option prices

I guess a better proof of this recommendation would be to show, using calculus, that the first differential of a call option's price with respect to its strike price, under normal conditions, is always greater than $-1$ with the Black-Scholes model but that's beyond me.

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  • $\begingroup$ The Black-Scholes formula is for European options. What should be used for US options? $\endgroup$ Nov 7 '21 at 15:02
  • $\begingroup$ LOL you mean American Options $\endgroup$
    – noob2
    Nov 7 '21 at 17:51
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    $\begingroup$ IMO, same argument holds for American options. $\endgroup$ Nov 7 '21 at 18:44

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