Option pricing

The layer this wiki’s stack does not have. algorithmic-trading organizes the corpus as predict → decide → execute: forecast something about a price, choose a position, get it filled. An option adds a step that sits before all three — the instrument has a theoretical value derived from the underlying, so there is a right answer to what it costs, and a trade can be wrong on the price without being wrong about the direction.

What is being priced

A call option is the right, not the obligation, to buy an asset at a fixed exercise price (strike) within a fixed period; a European option can be exercised only at maturity, an American one at any time up to it. Value comes from asymmetry: unlimited participation in a rise, a floor at zero on a fall, paid for up front.

black-scholes derives the price of the European call from a no-arbitrage condition and lands on five inputs: the stock price, the strike, the time to maturity, the short-term interest rate, and the variance rate of the stock’s return. The expected return on the stock is not among them.

That last omission is what makes the layer tractable and is also where the practitioner’s problem starts. Four of the five inputs are observable. Volatility is not, so in practice the formula is run backwards: take the market price of the option as given and solve for the variance that would produce it. That number is the implied volatility, and it is what an options market is really quoting.

The Greeks

The partial derivatives of the pricing formula, used as the trader’s control surface — how the position responds to each input moving. black-scholes supplies the first one explicitly: w₁, the sensitivity of option value to stock price, later named delta, and the paper uses it as the hedge ratio rather than as a risk statistic. Selling 1/w₁ options against one share is what makes the hedged position insensitive to the stock, and continuously re-adjusting that ratio is the argument the whole derivation rests on.

The corpus holds the rest only as a feature list. staskh-trading-skills computes Greeks and screens for poor-man’s-covered-call setups from Yahoo Finance data — which is roughly 15 minutes delayed, and delay is most damaging to exactly these quantities, since every one of them is a derivative with respect to a price that has already moved. No source here defines gamma, vega, theta or rho, or measures a hedging practice. That is a real gap and it is recorded rather than filled.

Why it changes the spoke’s evidence problem

Everywhere else in this corpus, “does the strategy work” is settled by a backtest the author ran (backtesting). Options are the one instrument here where a model price is available independently of any simulation, so a claim can be checked against theory rather than against history. Black and Scholes did exactly that against 1970s call data and found systematic deviations that transaction costs made untradeable.

That is a different kind of answer from the ones the rest of the wiki produces, and a warning about them: the deviation was real, replicated, and still not a strategy. A retail platform selling automated options trading (trading-bot-platform) is selling into the market where the theoretical value has been public since 1973 and the edge has been arbitraged for fifty years.

Cross-spoke

The mathematics — stochastic calculus, the heat equation, the lognormal random walk — belongs to ../mathematics-wiki. What stays here is the trading use: the hedge ratio, the volatility input, and what having a model price does to the evidence question.