Choosing a Call Strike by Expected Payoff and Historical Data

Summary

The document proposes choosing a European call strike by comparing its premium with the payoff at expiration, then asks whether historical price data can help select the strike that maximizes profit. It attempts to replace the uncertain call payoff with a probability calculation using the standard normal cumulative distribution function.

The central idea—compare premium income with expected exercise losses—is relevant to option selling, but the derivation is incorrect. Expected call payoff requires integrating the payoff amount, (S−K) when S exceeds K, over the distribution of the future asset price; it is not simply the probability of finishing above the strike multiplied by the strike. The document supplies no data, tested strategy, or answer to its question. It also leaves unspecified the price distribution, estimation window, and risk measure. A sound strike-selection method would need a justified distributional model and would assess expected return alongside tail risk, transaction costs, and the seller’s exposure.

Key ideas

• A call seller’s expiration profit is the premium received minus the call’s payoff.
• The expected payoff depends on both the probability and the size of outcomes above the strike.
• The proposed cumulative-probability substitution does not correctly calculate expected call payoff.
• Historical data alone does not specify a strike-selection rule or account for tail risk and trading costs.

Tags

# A naive approach to choose a strike


# A naive approach to choose a strike












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.