The options Greeks explained: delta, gamma, theta, vega and rho
The options Greeks are a set of measures that describe how an option's price changes when one input changes and the others stay fixed. Delta covers the underlying price, gamma covers how delta itself changes, theta covers the passage of time, vega covers implied volatility and rho covers interest rates. Together they break an option's risk into pieces that can be measured, compared and hedged separately.
Delta and gamma: the price Greeks
Delta is how much an option's price changes for a one-dollar move in the underlying. A call with a delta of 0.40 gains about 40 cents if the stock rises a dollar. Calls have deltas between 0 and 1, puts between minus 1 and 0. Because each contract covers 100 shares, a 0.40 delta call behaves like about 40 shares.
Gamma is how much delta changes for a one-dollar move. A call with a delta of 0.40 and a gamma of 0.05 will have a delta of about 0.45 after a one-dollar rise. Gamma is largest for at-the-money options and grows as expiry approaches, which is why short-dated options near the money are so sensitive.
Delta tells you your exposure now; gamma tells you how quickly that exposure will change. Long options always have positive gamma, so their delta moves in your favour as price moves: calls gain delta as the stock rises, puts gain negative delta as it falls.
Theta: the cost of time
Theta is how much an option's price falls each day with everything else unchanged. It is quoted as a negative number for long options. An option with a theta of minus 0.05 loses about five cents a day from time decay alone.
Theta is largest in dollar terms for at-the-money options and accelerates as expiry approaches. It is also the mirror image of gamma: options with high gamma tend to have high theta. A trader who owns gamma pays for it in theta, and a trader who sells options collects theta in exchange for being short gamma.
Theta is not constant through an option's life. For an at-the-money option it rises steadily as expiry approaches, with the steepest decay in the final days. For far out-of-the-money options it can rise and then fall away, because once the option is nearly worthless there is little value left to decay.
Vega and rho: volatility and rates
Vega is how much an option's price changes for a one-point change in implied volatility. An option with a vega of 0.12 gains about 12 cents if implied volatility rises from 25 to 26 percent. Vega is largest for at-the-money options and for longer-dated options, which have more time for volatility to matter.
Rho is how much an option's price changes for a one-percentage-point change in interest rates. Higher rates raise call values and lower put values, because owning a call instead of the stock frees up cash that earns interest. Rho is usually small for short-dated options and matters more for long-dated contracts, such as options a year or more out.
Vega is the Greek that most often surprises new option buyers. A position can be right on direction and still lose if implied volatility falls, as happens after earnings.
The second-order Greeks
Beyond the main five are second-order Greeks that describe how the first-order ones change. Gamma is itself one of these. Vanna measures how delta changes when implied volatility changes. Charm measures how delta changes as time passes. Vomma, or volga, measures how vega changes with volatility.
These matter most to market makers and others who hedge large books, because they describe how a hedge will drift. Vanna and charm in particular explain why dealer hedging can change even when the stock price does not move, which is why they appear so often in discussions of expiration flows.
Reading the Greeks together
Every option is a bundle of these exposures. A short-dated at-the-money option is mostly a gamma and theta trade: large sensitivity to price movement, paid for with fast time decay. A long-dated option is more of a vega trade: its value depends heavily on what implied volatility does over the coming months.
Position Greeks are found by adding up the Greeks of every leg, each multiplied by the number of contracts and the contract multiplier. A spread that buys one call and sells another can have a small net delta, little vega and modest theta, which is exactly why spreads are used to isolate one exposure and remove others.
The Greeks are model outputs, and they hold only for small changes. A large move or a jump in volatility changes all of them at once, so they describe the shape of the risk near current conditions rather than the outcome of a big event.
| Greek | Measures sensitivity to | Largest for |
|---|---|---|
| Delta | Underlying price | Deep in-the-money options (near 1 or -1) |
| Gamma | Change in delta as price moves | At-the-money, short-dated options |
| Theta | Passage of time | At-the-money, short-dated options |
| Vega | Implied volatility | At-the-money, long-dated options |
| Rho | Interest rates | Long-dated options |
Common questions
Which Greek is most important?
It depends on the trade. For short-dated options, delta, gamma and theta dominate. For long-dated options, vega usually matters most.
Why is theta negative?
Long options lose value as time passes, all else equal, because there is less time for a favourable move. Sellers of options have positive theta.
Do the Greeks change over time?
Yes, constantly. They depend on price, time, volatility and rates, so every one of them shifts as those inputs move.
Delta is exposure to price, gamma is how fast that exposure changes, theta is the daily cost of time, vega is exposure to implied volatility and rho to rates. Every option is a mix of them, and spreads exist to keep some and remove others.
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