The various Greek letters are used in the options market to describe parameters of risk when taking an options position. This has led to the term “Greeks” being frequently used in discussions of options trading. Each letter is assigned to a situation where risk is present due to the relationship between the option and some other variable. Traders use these different Greek values to determine the risk parameters of their positions and to manage these positions properly.

## Option Markets Greeks

There are a host of variables that are described by the various Greeks. Some of the major Greeks in the options markets are delta, gamma, rho, theta, and vega. In the options markets each of these has a number associated with it and that number gives traders some idea of the risk associated with the option, or how it moves. The primary Greeks, which are the ones mentioned above, are each calculated as a first partial derivative of the option pricing model.

This number is not fixed however, and it does change over time. Thus, a trader might calculate these values each day in order to determine if market changes have affected their portfolio to the degree it needs to be rebalanced.

Below we have a closer look at a number of the primary Greeks in use in the options markets.

## Delta

Delta is the first of the Greek letters used in the options market and it represents the rate of change between the price of the option and a change of \$1 in the price of the underlying asset. This describes the price sensitivity of the option relative to its underlying asset. For a call option the delta ranges from zero to one, while a put option has a delta which ranges from zero to negative one. Delta works like this: suppose a call option has a delta of 0.6. That means a \$1 change in the price of the underlying asset would cause an increase of \$0.60 in the option.

There is also a strategy in which traders create a delta-neutral position, and as you might expect delta in this position represents the hedge ratio of the position. For example, if you purchase the call option with a delta of 0.6 you would need to sell 60 shares of the stock at the same time to be fully hedged in that position. It is also possible to use the net delta for an entire portfolio of options to determine the hedge ratio for the entire portfolio.

Delta can also be used to forecast the probability of an option finishing in-the-money. For example, our call option with the 0.6 delta is said to have a 60% probability of finishing in-the-money.

## Gamma

Gamma is the rate of change between the delta of an option and the price of the underlying asset. Where delta is a first order measure of price sensitivity, gamma is considered a second order price sensitivity measure. Gamma measures the expected change in delta for each \$1 change in the price of the underlying asset. For example, if a trader is long one call option with a delta of 0.6 and a gamma of 0.1 it is presumed that each \$1 change in the value of the underlying asset will cause the delta of the call to increase or decrease by 0.1.

Gamma is useful in determining the stability of delta for an option. The higher gamma is the more likely it is that delta will change dramatically in response to even small changes in the price of the underlying asset. Gamma rises as options get closer to being at-the-money. It also accelerates as expiration nears. So, gamma values are typically quite low for options that are far from expiration, but as that expiration approaches gamma increases.

Options traders can opt to hedge for gamma as well as delta, thus creating a position that is delta-gamma-neutral. In such a position the delta of the option will remain close to zero even as the price of the underlying asset moves.

## Rho

Rho is a measure of the rate of change in an options price and a 1% change in the interest rate. It measures the sensitivity of the option to interest rates. For example, assume a call option has a rho of 0.1 and a current price of \$1.40. If interest rates increase 1% then the price of the option would increase to \$1.50, with everything else remaining equal. Put options increase in price when interest rates drop.  is highest in options that are at-the-money with a long period of time until expiration.

## Theta

Theta is a representation of the rate of change between the option price and time. This is also known as time sensitivity, or time decay. The theta is a representation of how much an option’s price will decrease as the expiration of the option nears, with all else considered equal. For example, if a trader has a long position in an option with a theta of 0.6 it is presumed the price of the option will decrease by \$0.60 each day, with all else being assumed equal.

The closer an option is to being in the money, the closer theta will be to zero. Long calls and puts have a negative theta, while short calls and puts have a positive theta. Options also have accelerated time decay as they get closer to their expiration. As a comparison, an asset that does not have price eroded by time will always have a theta of zero.

## Vega

Vega is a measure of the rate of change in the implied volatility of the underlying asset and the option value. In other words, it is the sensitivity the option has to volatility of the underlying. Vega indicates how much the price of an option will change given a 1% change in the volatility of the underlying asset. For example, if an option has a vega of 0.2 it implies the price of the option will change by \$0.20 for every 1% change in volatility for the underlying.

Increased volatility of the underlying also increases the probability of the asset reaching extreme values. This increases the value of an option on that underlying asset. Conversely a decrease in volatility negatively impacts the price of an option. Vega is highest in options that are at-the-money with a long period of time until expiration.

An interesting point is that there actually is no Greek letter named vega. There are a number of theories explaining how this term was included in the group of Greek letters expressing option risks.

## Minor Greeks

There are several other minor Greeks that aren’t frequently used or discussed. These include colour, epsilon, lambda, speed, ultima, vera, vomma, and zomma. All of these are second or even third derivatives of the pricing model and they impact such parameters as the change in delta in relation to the change in volatility or interest rate and such. While human option traders rarely use these variables, they are increasingly used by computer software algorithms for trading, since such software can quickly calculate and account for these complex risk factors in options trading.

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