Options Greeks

The options Greeks are sensitivity measures that estimate how an option's price, or premium, responds to a change in one of the factors that set it, with the other factors held constant. FINRA describes them as "a number of key factors that influence the price of options contracts." The five commonly shown are delta (the underlying price), gamma (the change in delta), theta (time), vega (implied volatility) and rho (interest rates). They come from option pricing models, so they are estimates of how a price should move, not guarantees of how it will. The Options Industry Council's definitions are hedged accordingly: delta is "a theoretical estimate," and theta describes decay "in theory."

What each one measures. Every Greek answers one question, per share, with the other inputs held fixed. Delta is the expected change in the option's price for a $1 move in the underlying. Gamma is the expected change in delta for that same $1 move. Theta is the expected change in price as one day passes, the daily rate at which an option's extrinsic value erodes; the word is also used loosely for the time value itself. Vega is the expected change for a one-percentage-point change in implied volatility, and rho the expected change for a one-percentage-point change in interest rates.

What they respond to. The Options Industry Council lists the pricing inputs behind the Greeks as the stock price, the strike price, the time to expiration, implied volatility, the interest rate and anticipated dividends. The stock price and implied volatility change constantly during market hours, while the strike, the interest rate and the dividend assumption may not change at all for the life of the contract. Delta and gamma follow the stock price, theta time, vega implied volatility and rho the interest rate.

Reading them together, and their limits. On an option chain the Greeks are shown per share, so turning one into dollars for a standard equity option means multiplying by the contract's multiplier, usually 100. Each Greek holds every other input fixed, which never happens in practice: a stock move usually comes with a change in implied volatility and a day's decay. The Greeks are best read as a snapshot of where the risks in a position lie today, recomputed as the position changes.

Each Greek is a per-share estimate of price change for a one-unit change in one input: $1 in the stock price for delta, $1 in the stock price for the change in delta (gamma), one day for theta, one percentage point of implied volatility for vega, and one percentage point of interest rates for rho. Multiplying by the number of shares the contract covers converts it into dollars for a position.

For example, with illustrative figures: a stock trades at $50 and a $50 call with 35 days to expiration is quoted at $2.07, with a delta of 0.54, a theta of −0.03 and a vega of 0.06. One contract covers 100 shares, so it costs $207.

• If the stock rises $1, the call should be worth about $2.61 (an increase of 0.54 × $1), or $261 a contract.
• If a day passes and nothing else changes, the call should lose about $0.03, or $3 a contract.
• If implied volatility rises 2 percentage points, the call should gain about $0.12 (0.06 × 2), or $12 a contract.
• If all three happen on the same day, adding the estimates gives about $2.07 + $0.54 − $0.03 + $0.12 = $2.70 a share, or $270 a contract.

That combined figure is only a first approximation: the Greeks themselves shift as the inputs move, so the actual price change will differ from the sum of the separate estimates.

Pros

• Turn an option's price behavior into a few numbers that can be compared across contracts.
• Delta gives a quick measure of how much stock exposure an option position carries.
• Theta and vega show what a position is exposed to besides the stock's direction, which is where many beginners are surprised.

Cons

• They are estimates from a pricing model and depend on its assumptions.
• Each assumes all other inputs stay fixed, which they rarely do.
• They change continuously, so a reading is only current for a moment.
• Being per share, they understate dollar exposure by a factor of 100 for a standard contract if the multiplier is forgotten.