Option Greeks

The “Greeks” refer to a group of parameters that measure risk in an options position. The Greeks are typically used to help investors and traders risk-manage individual options positions, as well as the overall portfolio.

The Greeks are referred to as such because each dimension of risk is represented by a Greek letter. The primary Greeks are delta, gamma, theta, vega and rho.

These five parameters provide investors and traders with important insight into how a given position will respond under a range of possible scenarios in an underlying stock or ETF. And much like any worthy endeavor, it can take some time to master these concepts.

In options trading, the “minor” Greeks refer to second-order parameters that provide additional insight into the pricing behavior of an option.

Some of the minor Greeks include lambda, vomma, speed, zomma, color, ultima and charm. These secondary Greeks aren’t frequently cited, except by financial engineers and highly sophisticated derivatives institutions.

Options positions are slightly more complicated than stock or ETF positions because options can fluctuate in value as a result of several different factors.

For example, the value of an option can fluctuate based on changes in the value of the associated underlying. But an option can also fluctuate in value based on the passage of time, due to changes in volatility, or due to changes in prevailing interest rates.

The option Greeks provide insight into these changes, and how they can affect an option’s value, which is why they are so essential to traders and investors active in options markets.

The five mostly commonly referenced Greeks are:

1. Delta
2. Gamma
3. Theta
4. Vega
5. Rho

Delta is arguably the best-known Greek because it reports on the sensitivity of an option's value to changes in the associated underlying stock or ETF. This is critical because ultimately all market participants want to understand why they are making or losing money in an options position as it relates to movement in the underlying.

Along those lines, the delta of an option reports how much an option will change in value for every $1 move in the underlying security.

For example, if a call option is worth $3.00 and has a delta of 0.50, and the underlying increases from $50 to $51 per share, then the value of that option theoretically increases by $0.50—from $3.00 to $3.50. That might be good or bad for the position, depending on if it is a long option, or a short option.

Call options have deltas between 0 and 1, because as the underlying asset increases in price, call options increase in price—and vice versa. Alternatively, put options have deltas between -1 and 0, because as the underlying increases in value, put options decrease in value—and vice versa.

Gamma reports by how much the delta of a given option will theoretically increase or decrease for each dollar move in the underlying. The value for gamma ranges between 0 and 1.

Looking at an example, imagine that a long call option worth $1.00 has a delta of 0.40 and a gamma of 0.10. Now imagine that the underlying increases in value by $1. In this example, that means that the gamma (0.10) must be added to the old delta (0.40) to get the new delta of the option (0.10 + 0.40 = 0.50).

Theta is the Greek that measures the rate of change in an option's theoretical value relative to the passage of time. This concept is often referred to as "time decay," because all else being equal, options lose value as they get closer to expiration.

For example, if an option is worth $1 with five days until expiration, the theta of that option might be equal to $0.20. That means for each day that passes, the option will theoretically lose $0.20 in value per day.

Vega is the Greek that measures an option's sensitivity to the changes in the volatility of the underlying. Vega is one of the “Greeks,” but it is the only one of the five that is not actually represented by a Greek letter.

Vega is typically expressed as the amount of money that an option's value will gain or lose when volatility rises or falls by 1%. Theoretically, all options (both calls and puts) gain in value when volatility rises, and vice versa.

Options with longer dated maturities generally have a higher percentage of vega than closer dated maturities. Additionally, vega is typically higher for at-the-money (ATM) options as compared to out-of-the-money (OTM) options.

When it comes to changes in the interest rate environment, rho is the greek of choice—rho measures the sensitivity of an option's value to changes in interest rates.

Rho can be overlooked at times, because interest rates often remain stagnant for many months (or many years) at a time. For this reason, rho doesn't generate the same type of buzz as delta, gamma, theta and vega.

But it's still important that market participants understand how rho works, and how changes in interest rates can affect the options market.

Mathematically, rho represents the amount that an option will gain or lose in value for every 1% move in interest rates.

For example, imagine a given option is worth $2.00 and has a positive rho of $0.50. If interest rates increase by 1%, that option would theoretically increase in value by $0.50, and be worth $2.50. Alternatively, if interest rates were to drop by 1%, then the option would be worth $1.50.

It's the same with negative rho options but reversed. For example, an option worth $2.00 with a negative rho of $0.50 would see its value decline to $1.50 if interest rates increased by 1%. Alternatively, that same option would rise in value to $2.50 if interest rates dropped by 1%.

Rho is positive for long calls and short puts, and rho is negative for short calls and long puts. In other words, an increase in interest rate is generally good news for long calls and short puts, whereas a decrease in rates tends to benefit short calls and long puts.

In general, rho tends to play a bigger role in the value of longer-term options, as opposed to near-term options (much like vega). Rho also tends to be larger for at-the-money (ATM) options, as compared to out-of-the-money (OTM) options.

Vega measures an option's sensitivity to changes in the implied volatility of the underlying asset. While vega is included in the group of "Greeks" used in option analysis, it is the only one not represented by an actual Greek letter.

Vega basically measures an option's sensitivity to volatility fluctuations. A higher vega value implies that the option price will be more sensitive to changes in volatility, while a lower vega indicates that the option price will be less sensitive to changes in volatility.

Vega is typically expressed as the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls by 1%. Owned options (both calls and puts) typically increase in value when volatility increases, and decrease in value when volatility decreases.

A trader that believes implied volatility will increase might therefore gravitate toward owning options with higher vega, to benefit from that potential scenario.

Conversely, traders who anticipate a decrease in volatility might prefer to own options with lower vega to mitigate potential losses from declining prices. Alternatively, a trader expecting a decline in implied volatility might elect to sell an option, to capitalize on a potential drop in volatility.

Options with longer dated maturities will have a higher percentage of vega than closer dated maturities. Additionally, vega is higher for at-the-money (ATM) options as compared to out-of-the-money (OTM) options.

Now imagine a trader owns a hypothetical call option with a vega of 0.05. This means that for every 1% increase in implied volatility, the option's price is expected to increase by $0.05, assuming all other factors remain constant.

Therefore, with a 1% increase in implied volatility, the option's price would be expected to increase by $0.05. On the other hand, if mplied volatility were to decrease by 1%, the option's price would be expected to decrease by the same amount.

Now assume the option is currently priced at $2.00 with an implied volatility of 20%. If the implied volatility increases by 1% to 21%, the value of the option would therefore increase to $2.05.

Theta is the "Greek" that measures the rate of change in an option's theoretical value relative to the passage of time. This concept is often referred to as "time decay," because all else being equal, options lose value as they get closer to expiration.

Theta represents the amount by which an option's price is expected to decrease per day, assuming all other factors remain constant. It quantifies the impact of time on an option's value.

For example, if an option is worth $1 with five days until expiration, the theta of that option might be equal to $0.20. That means for each day that passes, the option will theoretically lose $0.20 in value per day.

Theta is negative for most options because, as time progresses, the option's remaining lifespan decreases, leading to a decrease in its value. That’s because options have an expiration date, and as time passes, there is less time for the option to move in the desired direction, reducing its probability of being profitable.

The rate of time decay accelerates as the option approaches its expiration date. This means that options with shorter time to expiration will generally have higher theta values than those with longer time to expiration.

Theta is an essential consideration for options traders, and traders should keep in mind that holding owned options for long periods of time may result in significant losses attributable to time decay.

Now imagine a trader owns a hypothetical call option with a theta of -0.03. This means that, all else being equal, the option's price is expected to decrease by $0.03 per day as time passes.

Let's assume the option is currently priced at $2.00 with 30 days remaining until expiration. If one day passes, the option theoretcally decreases in value by $0.03, and is now worth $1.97.

It’s important to keep in mind that like most of the Greeks, theta is dynamic and can change from day to day. Generally speaking, theta tends to increase as an option gets closer to expiration. Changes in the value of the underlying, or to the other Greeks, can also impact the theta of an option.

Delta is arguably the best-known Greek because it reports on the sensitivity of an option's value to changes in the associated underlying stock or ETF. This is critical because ultimately all market participants want to understand why they are making or losing money in an options position as it relates to movement in the underlying.

Along those lines, the delta of an option reports how much an option will change in value for every $1 move in the underlying security.

Delta is expressed as a number between 0 and 1 for call options and between -1 and 0 for put options. It represents the change in the option's price for a $1 change in the price of the underlying asset.

At-the-money (ATM) options generally have deltas around 0.50 for calls, and -0.50 for puts. As expiration draws closer, the deltas for in-the-money (ITM) options rise toward 1 (for calls) or decline toward -1 (for puts). For out-of-the-money (OTM) options, delta moves toward zero as expiration draws closer.

For call options:

• If a call option has a Delta of 0.50, it means that for every $1 increase in the price of the underlying asset, the option's price is expected to increase by $0.50.

• If a call option has a Delta of 0.80, it means that for every $1 increase in the price of the underlying asset, the option's price is expected to increase by $0.80.

For put options:

• If a put option has a Delta of -0.50, it means that for every $1 decrease in the price of the underlying asset, the option's price is expected to increase by $0.50.

• If a put option has a Delta of -0.80, it means that for every $1 decrease in the price of the underlying asset, the option's price is expected to increase by $0.80.

Looking at another example, imagine a trader owns a hypothetical call option worth $3.00 that has a delta of 0.50. If the underlying increases by $1 per share then the value of that option theoretically increases by $0.50—from $3.00 to $3.50. 

Using a put in an example, imagine a put option that’s worth $5.00 with a delta of -0.30. If the underlying increases in value by $1.00, then the put option will decline in value from $5.00 to $4.70. But if the underlying declines in value by $1.00, the put option will increase in value from $5.00 to $5.30. 

Like the other Greeks, delta is dynamic and will change as the underlying price fluctuates.

In the options world, gamma is the “Greek” that measures the rate of change in an option's delta for every $1 move in the underlying asset's price.

As a reminder, the delta of an option tells us how much an option's price will move for every $1 move in the underlying. For example, a call option with a delta of 0.30 will move up or down by $0.30 for every dollar move up or down in the underlying.

However, when an underlying moves, the delta of the option also changes. Gamma therefore is the amount that an option's delta will change for every $1 move in the underlying.

Gamma will always be higher for at-the-money (ATM) and options in-the-money (ITM) options because they are more sensitive to movement in the underlying than out-of-the-money (OTM) options.

Options with higher gamma are therefore more responsive to price changes in the underlying, which in turns means the deltas of these options can change more quickly.

Like the other Greeks, gamma is dynamic and will change as the underlying price fluctuates. Traders often monitor gamma at the position and portfolio level to help manage risk.

When it comes to changes in the interest rate environment, rho is the greek of choice—rho measures the sensitivity of an option's value to changes in interest rates.

Rho can be overlooked at times, because interest rates often remain stagnant for many months (or many years) at a time. For this reason, rho doesn't generate the same type of buzz as delta, theta, gamma and vega.

However, t's still important that market participants understand how rho works, and how changes in interest rates can affect the options market.

Mathematically, rho represents the amount that an option will gain or lose in value for every 1% move in interest rates.

For example, imagine a hypothetical option is worth $2.00 and has a positive rho of 0.50. If interest rates increase by 1%, that option would theoretically increase in value by $0.50, and be worth $2.50. Alternatively, if interest rates were to drop by 1%, then the option would be worth $1.50.

The math works similarly for options with negative rho. For example, an option worth $2.00 with a negative rho of $0.50 would see its value decline to $1.50 if interest rates increased by 1%. Alternatively, that same option would rise in value to $2.50 if interest rates dropped by 1%.

Rho is positive for long calls and short puts, and rho is negative for short calls and long puts. In other words, an increase in interest rate is generally good news for long calls and short puts, whereas a decrease in rates tends to benefit short calls and long puts.

In general, rho tends to play to bigger role in the value of longer-term options, as opposed to near-term options (much like vega). Rho also tends to be larger for at-the-money (ATM) options, as compared to out-of-the-money (OTM) options

In options trading, the “minor” Greeks refer to second-order parameters that provide additional insight into the pricing behavior of an option.

Some of the minor Greeks include lambda, vomma, speed, zomma, color, ultima and charm. These secondary Greeks aren’t frequently cited, except by financial engineers and highly sophisticated derivatives institutions.

Lambda measures the percentage change in the price of an option for a 1% change in the price of the underlying asset. It is similar to the concept of delta but expressed in percentage terms rather than absolute terms. Lambda essentially measures the leverage or sensitivity of an option's price to changes in the underlying asset's price.

For example, if a call option has a lambda of 0.50, it means that for every 1% increase in the price of the underlying asset, the price of the call option is expected to increase by 0.50% (or 50 basis points. Similarly, if the underlying asset's price decreases by 1%, the price of the call option would be expected to decrease by 0.50%.

Vomma measures the rate at which an option's vega changes with respect to changes in implied volatility. It quantifies the sensitivity of an option's vega to fluctuations in implied volatility.

Vomma is essentially the second derivative of an option's price with respect to changes in implied volatility.

A higher vomma value implies that the option's vega will be more sensitive to changes in implied volatility, leading to potentially larger price changes in response to volatility fluctuations.

Speed measures the rate of change of gamma with respect to changes in the underlying asset's price. 

Speed therefore provides insight into how gamma itself changes as the underlying price moves. Speed can help traders understand how gamma will evolve as the underlying asset's price changes.

Zomma quantifies the rate of change of gamma with respect to changes in implied volatility.

Zomma therefore represents the sensitivity of gamma to changes in implied volatility. Zomma helps traders assess the impact of volatility changes on the stability of an option's Gamma

Color refers to the sensitivity of the gamma to changes in the time until expiration. Color therefore measures how the Gamma changes as time passes, or as the option approaches its expiration date.

Color can be positive or negative, indicating the direction of the change in gamma with respect to time. 

A positive color means that as time passes, the gamma of the option increases, indicating that the option's delta becomes more sensitive to changes in the underlying asset's price. This implies that the option becomes more responsive to price movements as it approaches expiration.

Conversely, a negative color means that as time passes, the gamma of the option decreases, indicating that the option's delta becomes less sensitive to changes in the underlying asset's price. This suggests that the option becomes less responsive to price movements as it approaches expiration.

Ultima is basically a secondary measure of an options sensitivity to changes in implied volatility. It quantifies the rate of change of vega with respect to changes in implied volatility.

Ultima helps traders understand how changes in volatility impact an option's vega and subsequently the price of the option. A higher ultima value implies that the option's vega will exhibit greater sensitivity to changes in implied volatility, resulting in larger price swings in response to volatility shifts.

Charm measures the rate at which delta changes with respect to time. It quantifies the impact of time decay on delta. It helps traders assess how delta will change as the option approaches expiration. A positive charm value indicates that delta increases as time passes for both calls and puts, while a negative value indicates the opposite.