Options Greeks

The partial derivatives of an option's model price, one per input, used as the trader's control surface: how much the position moves when the stock moves, when volatility moves, when a day passes. option-pricing introduces them as the layer this wiki's predict → decide → execute stack does not otherwise have, because a Greek is not a forecast — it is a statement about the position you are already holding.

What this corpus can define, and what it still cannot

Two of them are defined by sources here.

Delta — Δ(t,S) = ∂V/∂S, the change in option value per unit change in the underlying. It arrives in black-scholes as w₁, and not as a risk statistic: it is the hedge ratio, the number of options to sell against one share so the combined position stops caring about the stock. The whole no-arbitrage derivation rests on holding that ratio continuously.

Gamma — Γ(t,S) = ∂Δ/∂S = ∂²V/∂S², the change in delta per unit change in the underlying, defined in exactly those terms by pervasive-impact-of-option-trading. Purchased options have positive gamma, written options negative. Gamma is why a delta hedge is work rather than a one-time trade, and its sign decides which direction that work pushes the stock — the mechanism on dealer-gamma-exposure.

Vega, theta and rho — sensitivity to volatility, to time, and to the interest rate — are still not defined by anything in this spoke. They appear as a feature bullet in staskh-trading-skills, which computes Greeks from Yahoo Finance data, and nowhere else. Naming them is not sourcing them, so the gap stays recorded rather than filled from memory.

Why the derivative is a control surface and not a prediction

Every Greek is a local statement: it holds for a small move, at today's inputs, under the model that produced it. Delta from black-scholes is delta under Black-Scholes assumptions, including the constant volatility that option-pricing shows the market does not actually quote. Two consequences run through the sources here.

The first is that Greeks decay as data ages. staskh-trading-skills screens on Greeks computed from a feed roughly 15 minutes delayed, and delay hurts a derivative more than it hurts a price, because the quantity being reported is the slope at a point the market has already left.

The second is that hedging a Greek is itself trading. Keeping delta at zero means buying and selling the underlying as gamma moves the ratio, and that flow is large enough to be measured in the stock's own returns — about 12% of an optioned stock's daily absolute return in pervasive-impact-of-option-trading. The control surface is not free to steer.

Cross-spoke

The calculus — partial derivatives of a solution to the heat equation, and the stochastic-process setup underneath — belongs to ../mathematics-wiki. What stays here is the trading use: the hedge ratio, the rebalancing it forces, and what that rebalancing does to the price the ratio was computed from.

Related

option-pricing · black-scholes · dealer-gamma-exposure · unusual-options-activity · pervasive-impact-of-option-trading · gamma-fragility · staskh-trading-skills · synthesis