# Optimize call option purchase

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?

• What have you tried so far? Nov 7 '21 at 4:50

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).
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.
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.