# Option Greeks

Greeks are statistics which measure sensitivity of option prices to various factors, such as underlying price (delta, gamma), time to expiration (theta), volatility (vega), and interest rate (rho). They help a trader understand risk exposures of option positions: for example, when an option portfolio has positive delta, its value will increase if underlying price goes up, or similarly, if it has negative vega, its value will decrease if implied volatility rises. Understanding Greeks is therefore very useful for understanding and managing risk in options trading.

This tutorial introduces the most important Greeks – delta, gamma, theta, vega, and rho.

## Don’t Fear the Greeks

Beginning option traders sometimes avoid the Greeks, fearing they are too math-heavy. While the formulas for calculating Greeks may seem complicated, knowing them is by no means necessary to use Greeks in trading. Nowadays, software like your broker’s trading platform or our calculators can do the calculations for you. Your part is to understand what the Greek numbers mean, how they are related and how they can change under different circumstances.

Therefore, you won’t find any formulas in these tutorials. Focus is on the logic and practical use.

Those interested in the formulas can find them in Black-Scholes Greeks Formulas and Option Greeks Excel Formulas.

Below you can find a summary of main things to know about each Greek. Follow the links to individual tutorials for more details.

## Delta

• Delta measures how option price will change if underlying price increases by \$1.
• Call option delta is from 0 to +1. Put option delta is from 0 to -1.
• Out of the money options have delta near zero. In the money options near +1 (calls) or -1 (puts).
• Delta itself changes with underlying price (this is measured by gamma). Therefore, delta is only accurate for small underlying price changes.
• Like all other Greeks, delta is additive. Total delta of a position with multiple options is the sum of all options’ deltas.
• Delta hedging makes delta zero – makes a position immune to small underlying price changes. It requires ongoing monitoring and rebalancing.
• Delta also changes with volatility and passing time. Lower volatility or lower time to expiration push delta closer to the extremes (0 or +1 or -1).

See more in the Option Delta Tutorial.

## Gamma

• Gamma measures how much delta will change if underlying price increases by \$1.
• All options have positive gamma. All short option positions have negative gamma.
• Gamma is highest at the money. At the money gamma increases with passing time or decreasing volatility.
• Positive gamma means your profits accelerate in big moves.
• Negative gamma means your losses accelerate and can be very dangerous.

See more in the Option Gamma Tutorial.

## Theta

• Theta measures how much an option’s price will change in one day.
• All options (with some rare exceptions) have negative theta – lose value with passing time.
• Theta is greatest at the money. At the money theta is greatest just before expiration.
• An increase in volatility increases time value and thereby theta.
• Short option positions have positive theta and profit from passing time.
• Positive theta goes hand in hand with negative gamma. There is no free lunch.

See more in the Option Theta Tutorial.

## Vega

• Vega measures how option price will change if implied volatility rises by one percentage point.
• All options have positive vega – gain value with rising volatility.
• Vega is greatest at the money (but out of the money in percentage terms).
• The more time to expiration, the higher vega.

See more in the Option Vega Tutorial.

## Rho

• Rho measures how option premium will change if the risk-free interest rate increases by one percentage point.
• Call options on most underlyings have positive rho; put options have negative rho.
• Rho is generally greater (in absolute terms) with more time to expiration.
• For many underlyings like currencies or bonds, interest rates may also affect underlying price, and thereby option prices. This indirect effect, though often greater than the direct effect, is not measured by rho.

See more in the Option Rho Tutorial.

An important property of all the Greeks is that they are additive across different options on the same underlying security. Therefore, it is easy to calculate Greeks for option strategies like straddles or condors by simply adding them up for all the individual options (the short options with opposite sign).

For example, a bull call spread which is long a call option with 0.80 delta and short a call option with 0.30 delta has total delta of 0.80 – 0.30 = 0.50, which means the total value of the position will increase by approximately \$0.50 if the underlying price increases by \$1.

When you hold multiple contracts, simply multiply the Greeks by the number of contracts.

For example, a position of 10 contracts of the call spread above would have delta of 10 times 0.50 = 5.00, which means its value would grow by \$5 if underlying price increased by \$1.

You can also use Greeks for positions which also include the underlying. A long position in the underlying security always has delta of +1; short position has delta of -1. Because the underlying has no optionality and no time value, the other Greeks (gamma, theta, vega, rho) are zero. Knowing this, you can also calculate and use Greeks for positions like covered calls or collars.

For example, a covered call strategy holding 100 shares of a stock and short one call option contract (which represents 100 shares) with 0.20 delta, 0.03 gamma, -0.04 theta, and 0.08 vega has total delta of 0.80 (1 – 0.20), gamma of -0.03, theta of +0.04, and vega of -0.08 (always the opposite sign, as the stock has gamma/theta/vega of zero and you subtract the short option’s Greeks). This means the position’s total value rises if underlying stock goes up (positive delta), or as time passes (positive theta), but decreases if the short option’s implied volatility goes up (negative vega).

The Greeks tell you which kinds of risks you are exposed to and what will happen to your profit or loss under different scenarios. They help you make decisions such as which strikes and/or expirations to use, how many contracts to trade (to keep the risk within your limits), or how to adjust your position when circumstances change.

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