Beyond the Expected Value

[WITH CODE] The Full Probability Distribution of a European Call Option at Expiry: Derivation, synthetic stress test, and what it tells you about your position.

Hello and welcome back to another paid post!

Today we will take a look at the full expiry-payoff distribution of a European call option: the probability of zero payoff, the range of positive payoffs, and what changes when the assumed price process changes.

Let’s dive right in.

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What Black-Scholes Is and Is Not

The Black-Scholes formula is one of the most used equations in all of finance. But it is worth being precise about what it actually gives you, because there is a common confusion about this.

Black–Scholes gives the discounted risk-neutral expected payoff of a European option under its stated assumptions. That is one number: a price today, not a physical forecast of every possible expiry outcome.

For pricing under those assumptions, that is the object you need. For risk analysis, the rest of the assumed distribution can still matter.

But for risk management, one number is not enough. If you hold a call option on a stock that is currently trading just below the strike with two months to expiry, what you actually want to know is:

• What is the probability this expires worthless?

• If it does pay off, what range of outcomes should I plan for?

• How does that probability change as the stock moves day by day?

The familiar Black–Scholes price does not display those answers by itself, although the model assumptions imply a distribution. What follows is a derivation of that distribution, a controlled synthetic stress test, and a narrower account of what the result can and cannot support.

The payoff distribution depends on the probability measure and process you choose. A physical-measure distribution describes assumed real-world outcomes; a risk-neutral distribution is the object used for no-arbitrage pricing.

Those expectations coincide only in a special parameterization. An undiscounted physical expected payoff should not be labelled a Black–Scholes price.