Choosing a Call Strike for a Uniform Terminal Price Forecast
Summary
The document considers choosing a call option when a stock’s terminal price is forecast to be uniformly distributed over a specified range. Assuming the available calls are priced with the Black–Scholes model, the answer recommends the lowest available strike at or below the bottom of that range. The reasoning is that raising the strike reduces the option premium by less than it reduces the payoff across the forecast range, so a higher strike is said to lower gross profit net of the fee under these assumptions.
The response refers to the typical way call prices vary with strike and suggests that a calculus argument could formalize the claim-price sensitivity behind the recommendation, but it does not provide that proof. The advice is narrow: it relies on the stated terminal-price distribution and pricing assumptions and does not compare alternatives using a specified utility, risk measure, volatility forecast, or transaction costs. The document gives a rationale, not empirical performance evidence.
Key ideas
- The recommendation assumes a uniform distribution for the stock’s terminal price over the forecast range.
- Under Black–Scholes pricing, the response favors the lowest available strike at or below the range’s lower bound.
- The stated rationale compares the change in call premium with the change in strike-related payoff.
- The response offers no formal derivation or empirical evidence and depends on its pricing assumptions.
Tags
Full text
# Optimize call option purchase # 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? ## Answer by Alper (score 4) https://quant.stackexchange.com/a/68675 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. 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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.