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# Greeks

## What Are the Greeks?

The variables that are used to assess risk in the options market are commonly referred to as "the Greeks." A Greek symbol is used to designate each of these risks.

Each Greek variable is a result of an imperfect assumption or relationship of the option with another underlying variable. Traders use different Greek values, such as delta, theta, and others, to assess options risk and manage option portfolios.

### Key Takeaways

• The Greeks are symbols assigned to the various risk characteristics that an options position entails.
• The most common Greeks used include the delta, gamma, theta, and vega, which are the first partial derivatives of the options pricing model.
• Greeks are used by options traders and portfolio managers to understand how their options investments will behave as prices move, and to hedge their positions accordingly.

## Understanding the Greeks

Greeks encompass many variables. These include delta, theta, gamma, vega, and rho, among others. Each one of these Greeks has a number associated with it, and that number tells traders something about how the option moves or the risk associated with that option. The primary Greeks (delta, vega, theta, gamma, and rho) are calculated each as a first partial derivative of the options pricing model (for instance, the Black-Scholes model).

The number or value associated with a Greek changes over time. Therefore, sophisticated options traders may calculate these values daily to assess any changes that may affect their positions or outlook, or simply to check if their portfolio needs to be rebalanced. Below are several of the main Greeks traders look at.

## Delta

Delta (Δ) represents the rate of change between the option's price and a \$1 change in the underlying asset's price. In other words, the price sensitivity of the option is relative to the underlying asset. The delta of a call option has a range between 0 and 1, while the delta of a put option has a range between 0 and -1. For example, assume an investor is long a call option with a delta of 0.50. Therefore, if the underlying stock increases by \$1, the option's price would theoretically increase by 50 cents.

For options traders, delta also represents the hedge ratio for creating a delta-neutral position. For example, if you purchase a standard American call option with a 0.40 delta, you will need to sell 40 shares of stock to be fully hedged. Net delta for a portfolio of options can also be used to obtain the portfolio's hedge ratio.

A less common usage of an option's delta is the current probability that the option will expire in-the-money. For instance, a 0.40 delta call option today has an implied 40% probability of finishing in-the-money.

## Theta

Theta (Θ) represents the rate of change between the option price and time, or time sensitivity—sometimes known as an option's time decay. Theta indicates the amount an option's price would decrease as the time to expiration decreases, all else equal. For example, assume an investor is long an option with a theta of -0.50. The option's price would decrease by 50 cents every day that passes, all else being equal.

Theta increases when options are at-the-money, and decreases when options are in- and out-of-the money. Options closer to expiration also have accelerating time decay. Long calls and long puts will usually have negative theta; short calls and short puts will have positive theta. By comparison, an instrument whose value is not eroded by time, such as a stock, would have zero theta.

## Gamma

Gamma (Γ) represents the rate of change between an option's delta and the underlying asset's price. This is called second-order (second-derivative) price sensitivity. Gamma indicates the amount the delta would change given a \$1 move in the underlying security. For example, assume an investor is long on a call option on hypothetical stock XYZ. The call option has a delta of 0.50 and a gamma of 0.10. Therefore, if stock XYZ increases or decreases by \$1, the call option's delta would increase or decrease by 0.10.

Options traders may opt to not only hedge delta but also gamma in order to be delta-gamma neutral, meaning that as the underlying price moves, the delta will remain close to zero.

## Vega

Vega (ν) represents the rate of change between an option's value and the underlying asset's implied volatility. This is the option's sensitivity to volatility. Vega indicates the amount an option's price changes given a 1% change in implied volatility. For example, an option with a vega of 0.10 indicates the option's value is expected to change by 10 cents if the implied volatility changes by 1%.

Because increased volatility implies that the underlying instrument is more likely to experience extreme values, a rise in volatility will correspondingly increase the value of an option. Conversely, a decrease in volatility will negatively affect the value of the option. Vega is at its maximum for at-the-money options that have longer times until expiration.

Gamma is used to determine how stable an option's delta is: Higher gamma values indicate that delta could change dramatically in response to even small movements in the underlying's price. Gamma is higher for options that are at-the-money and lower for options that are in- and out-of-the-money and accelerates in magnitude as expiration approaches. Gamma values are generally smaller the further away from the date of expiration; options with longer expirations are less sensitive to delta changes. As expiration approaches, gamma values are typically larger, as price changes have more impact on gamma.

Greek-language buffs will point out that there is no actual Greek letter vega. There are various theories about how this symbol, which represents the Greek letter nu, found its way into stock-trading lingo.

## Rho

Rho (ρ) represents the rate of change between an option's value and a 1% change in the interest rate. This measures sensitivity to the interest rate. For example, assume a call option has a rho of 0.05 and a price of \$1.25. If interest rates rise by 1%, the value of the call option would increase to \$1.30, all else being equal. The opposite is true for put options. Rho is greatest for at-the-money options with long times until expiration.

## Minor Greeks

Some other Greeks, which aren't discussed as often, are lambda, epsilon, vomma, vera, zomma, and ultima. These Greeks are second- or third-derivatives of the pricing model and affect things such as the change in delta with a change in volatility and so on. They are increasingly used in options trading strategies, as computer software can quickly compute and account for these complex and sometimes esoteric risk factors.

## What Is Delta?

Delta (Δ) is the rate of change between an option's price and a \$1 change in the underlying asset's price. The delta indicates how sensitive the price of the option is to the price of the underlying asset. The delta of a call option has a range between zero and one, while the delta of a put option has a range between zero and -1.

## What Is Gamma?

Gamma (Γ) indicates the amount that an option's option's delta would change in response to a \$1 move in the underlying security. Gamma determines how stable the option's delta is. A higher gamma indicates that the delta could change dramatically in response to even small movements in the underlying's price. A lower gamma points to less volatility.

## What Is Theta?

Theta (Θ) measures the rate of decline in the value of an option over time. Theta is generally expressed as a negative number and can be read as the amount by which an option's value declines every day as it moves closer to its maturity.

## What Is Vega?

Vega (ν) indicates an option's price sensitivity to changes in the volatility of the underlying asset. Vega represents the amount that an option contract's price changes in reaction to a 1% change in the implied volatility of the underlying asset. For example, an option with a Vega of 0.10 indicates the option's value is expected to change by 10 cents if the implied volatility changes by 1%.

Article Sources
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1. Corporate Finance Institute. "Vega (ν)." Accessed Feb. 2, 2022.

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