 # Option Greeks: The Ultimate Guide

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## What are the option greeks?

Option greeks represent the various risks that option contracts are exposed to. Each risk is represented by a letter of the Greek alphabet. There are a total of 5 option greeks: delta, gamma, theta, vega, and rho. Let’s begin by introducing our first option greek.

## Delta

If you’re trading options and want to figure out how an option’s value might appreciate as its underlying asset moves up or down in price, then you’re going to want to take a look at that option’s delta value.

### What is Delta in options?

Delta is the option greek that is most sensitive to price changes in the underlying asset.  It estimates the expected rate of change in an option’s price based on a 1 point increase in the underlying asset’s price.

### How to use Delta when trading stock options

Picture a stock and how its prices move up and down. Every movement that a stock makes, will have an effect on its options prices. Delta tries to predict how much the price of that option will change if the underlying stock price were to increase by one dollar.

### Example of Delta

Imagine you are looking at a call option that costs \$2.50 and its underlying stock is trading at \$50 per share.  A delta of 0.5 means that the option price will be expected to gain \$0.50 if the stock price increases by one dollar.

Note: Keep in mind, delta also assumes other factors like time decay and implied volatility remains equal.

As the underlying’s price increases from \$50 to \$51, the value of the call option is expected to change by the delta value of 0.5, increasing the option’s value from \$2.50 to \$3.00. Factoring in the multiplier of 100, the option’s estimated value would be \$300, resulting in a gain of \$50.

### Using Delta in an Option Chain

Delta values can often be shown in an option chain. You can use the delta value by adding it to the current price of the option to get an idea of how much that option would appreciate or depreciate based on a one-dollar change in the underlying stock price.

### Positive and Negative Delta

Another thing you should know is that call options have positive delta values, while put options have negative delta values. Let’s take a look at an example of a negative delta using a put option.

Let’s say a Put option that costs \$2.00 has a delta value -0.7. Its underlying stock is trading at \$50 per share and you want to know what would happen to the option price if the underlying stock price were to increase by \$1 per share. A delta of -0.7 means that the option price will be expected to lose \$0.70 if the stock price increases by one dollar, assuming all else is equal. As the underlying’s price increases, from \$50 to \$51, the put option is expected to change by the delta value of -0.7. This will decrease the option’s value from \$2.00 to \$1.30. Factoring in the multiplier of 100, the option’s estimated value would be \$130, resulting in a loss of \$70.

### Delta is used to express the probability that an option expires in the money

Aside from being used to estimate changes in an option’s price, delta is also commonly used as a way to express the probability of an option expiring in the money.

For example, an at-the-money call option with a 0.5 delta value would assume the option has a 50% chance of expiring in the money. This makes sense because at-the-money options have strike prices that are around the same price where the stock is currently trading. So, between the time you purchase the option and the time it expires, it has a 50/50 chance that it will either be in or out of the money.

As call options move in the money, their delta values increase until they reach a value of 1, which would be a near 100% probability that the option expires in the money. This makes sense because the stock price would have to drop considerably for an in the money option to be out of the money by the time it expires. As call options move out of the money, their delta values decrease until they reach a value of 0, which would be a near 0% probability that the option expires in the money. This makes sense because the stock price would have to increase considerably for an out of the money option to be in of the money by the time it expires.

This concept remains the same for puts, we would simply omit the negative sign and just look at the value. So, an at the money put option with a -0.5 delta value would assume the option has a 50% chance of expiring in the money.  Just the at the money call example, an at the money put has a 50/50 chance that it will either be in or out of the money. Deep in the money puts near a value of -1, would have a near 100% probability that the option expires in the money. Deep out of the money puts have delta values near 0, which would have a near 0% probability that the option expires in the money.

1. Delta measures the expected rate of change in an option contract’s price, for a 1-point increase in the underlying asset’s price.
2. Delta values can be positive or negative. Call options have positive delta values and range from 0 to 1 while Put options have negative delta values and range from 0 to -1.
3. Delta values can be used to express the probability that an option expires in the money.

## Gamma

### What is Gamma in options?

Gamma is defined as the expected rate of change in an option’s delta value for every 1-point increase in the underlying asset’s price.

### Can Delta change?

Yes, It can! The further an option moves in the money, the more delta begins moving towards its maximum value. For calls, the maximum value is positive 1 and for puts, the maximum value is negative 1. Therefore, an option’s delta is not a fixed value. How much delta changes, or how much it is expected to change by, is what gamma tries to tell us.

### Gamma is always positive

Unlike delta, calls and puts only have positive gamma values. This is because gamma is an absolute value that does not consider the direction of delta. It only shows the value that delta is expected to change by.

### Which options are most sensitive to Gamma?

Gamma has the most effect on an option’s delta value when an option is near or at the money. The further the underlying asset travels away from the option’s strike price, the more gamma will decrease until it reaches zero. This is because deep out of the money and deep in the money options have very low odds that an options delta will change.

### Using Gamma to estimate the new delta value

To help determine what the new delta value might be, you can simply add the gamma value to the current delta value. The sum of these two values will give you an idea of what the new delta value would be if the underlying asset were to increase by 1 point.

### Example of Gamma

Let’s use a 50-strike call option as an example. Let’s say the price of the underlying stock is \$49 and the value of the 50-strike call option is worth \$2.10. The option has a delta of 0.40. and a gamma of 0.10. As we just learned, the delta value means the value of the call option is expected to increase by \$0.40 cents per dollar as the stock price increase from \$49 to \$50. However, once the stock price reaches \$50, the delta value will no longer be 0.40, but 0.50. The change in delta by 0.10 points is what a gamma value of 0.10 represents.

### Gamma measures the expected change in the probability that an option will expire In The Money

Since delta is commonly used to express the probability that an option expires in the money, gamma will provide an insight as to how much those probabilities would change for a 1 point move in the underlying. For example, an at the money call with a delta value of 0.5 estimates a 50% probability that the call will expire in the money. If it has a gamma value of 0.15, those odds are expected to change by 15% if the stock increases by one dollar. So, if the underlying increases by one dollar, the delta value is expected to be 0.65, which estimates a 65% chance that the call will expire in the money.

### Gamma Risk

As an option approaches expiration, the effect that gamma has on an option’s delta increases. This is known as “Gamma Risk.” The larger an option’s delta, the more money the option will gain or lose for a 1 point move in the underlying.

Since options near expiration have lesser time value, option price movements begin to track more closely to movements in the underlying price. The increase in delta values can lead to volatile swings in an option’s value. For this reason, many option traders prefer not to carry any open option positions in the final days of an option’s life cycle.

1. The acceleration of the option position is relative to the underlying stock price.
2. Gamma has the most effect on an option’s delta value when an option is near or at the money.
3. Gamma is the change in probability of the position expiring in the money.

## Theta

### What is Theta in Options?

Theta is the option greek that measures the expected decline in an option’s price for a 1-day passage of time. To better understand this, let’s look at how time plays a role in the pricing of an option’s contract.

### Time Value

Options are wasting assets. They can only be traded for a certain period of time and then they expire. When an option first becomes available to trade they are packed with a time value, which we’ll define later on. For now, try to accept the concept that time value is highest when an option first becomes available to trade and is gone by the time an option expires.

### Example of Time Value and its effect on option pricing

If we compare two options with the same underlying and same strike price but with different expiry dates, the one with more time to expiry will be more expensive. This makes sense because when you trade options, you are speculating the likelihood that the underlying price will be above or below the strike price before the option expires.

### Try it yourself!

The next time you are comparing options, open up the option chain in your platform and compare the prices of all the options with the same strike price against those with different expiration dates. You will notice that the options with more time until expiration will be more expensive.

### Why options with more time are the most expensive

Options with more time are the most expensive because they bring on more risk to an option seller. This is because option sellers are obligated to deliver the underlying shares to (or receive shares from) an option buyer at the strike price should the option expire in the money.

This “time risk” is built into an options pricing which brings us to our next point.

The price that one pays to purchase an option is called the option premium. An option premium can be further broken down into two parts: intrinsic value and extrinsic value. Intrinsic value is the portion of an option’s premium which is the difference between the strike price and the underlying price. Intrinsic value is not affected by the passing of time and thus, the intrinsic value of an option’s premium is not affected by theta. Extrinsic value, on the other hand, is the portion of an option’s premium which is affected by risk factors such as implied volatility, Vega, interest rates, rho, and time or theta. When an option first becomes available for trading, it is loaded with extrinsic value, with time being the biggest factor in the option’s extrinsic value. Theta is the option greek that measures the expected decline in an option’s price for a 1-day passage of time.

### Theta and Time Decay in Options

Both calls and puts have negative theta values. This is because theta is attempting to measure the daily rate at which the option’s extrinsic value is expected to decay by. Have you ever noticed how theta tends to peak around the strike price? This is because theta has the most effect on an option’s value when an option is near or at the money.

### How to use Theta when trading Stock options

To determine how much the passage of one day’s time will affect your option, take the current value of the option and add the theta value to it. The result from this simple calculation will give you an idea of what the option would be worth the following day, assuming all of the other risk factors such as the stock price and implied volatility remain equal.

### Example of using Theta Decay

Let’s say we are looking at a 50-Strike call option and the price of the underlying stock is also at \$50. Let’s also assume there are only 7 days left before the option expires. The option is currently worth \$0.15, or \$15 considering the option multiplier, and has a theta value of -\$0.02, or -\$2.

If the following day the price of the underlying stock and implied volatility remain unchanged, then according to theta, we can assume the 50-strike call option would have decayed by \$2 cents, or \$2 dollars, after factoring in the option multiplier. Therefore, theta decay has decreased the value of the option from \$15 to \$13.

### Is Theta a Constant?

No. As we learned, theta measures the amount an option’s price is expected to decline by the passage of 1 calendar day. Each day that passes, the time value of an option’s price is reduced by the theta amount, this is known as “Theta Decay” or “Time Decay.”

However, theta’s value is not linear; in other words, it does not decrease an equal amount each day. It is exponential. Theta decay increases and accelerates as time approaches an option’s expiration date. The closer to expiration an option becomes, the faster the time value decays. At expiration, the option has no time value, just intrinsic value, which is the amount that the option is in the money. If the option is out of the money at expiration, then the option is worthless.

Therefore, when you buy options, theta is harmful to your long option positions. If you sell options, theta is helpful to your short option positions because time decay is reducing part of the option’s value.

1. Theta measures the expected decline in an option’s price for a 1-day passage of time, which is known as “Time Decay.”
2. Theta increases exponentially as an option approaches expiration.
3. Theta is harmful to long option positions but helpful to short option positions.

## Vega

### What is Vega in options?

Vega is the option greek that measures the expected change in an option’s price based on a 1% change in the underlying asset’s implied volatility, which is expressed as an annualized percentage on an option chain.

### What is implied volatility?

For now, let’s just say implied volatility is the perceived risk of the expected change in an option’s price over the next 12 months. An option with higher implied volatility will tend to cost more than one with lower implied volatility.

For example, if we compare two call options that belong to two different stocks currently trading at \$50 per share, we will be able to see which stock is expected to move more based on the option’s implied volatility value. In addition to the underlying price being the same, both of these options expire on the same day, have the same strike price, and delta values.

Yet, the option on the right is more expensive. This is because this option has a higher implied volatility, which adds to an option’s extrinsic value. Remember, extrinsic value is the portion of an option’s premium that’s affected by risk factors such as time, interest rates, and of course, implied volatility.

### Example of Vega

Now that we addressed implied volatility, let’s see how vega plays a role in all of this. Just like the other option greeks, vega is plotted as a value on an option chain, which has its own column. Vega tries to tell us how much an option’s price is expected to change if implied volatility for the underlying stock were to increase by 1%. To illustrate this, let’s say XYZ Company is currently trading at \$50 per share and the at the money call option costs \$2.50. Let’s assume the option has a vega value of .08. Let’s also assume that the implied volatility is currently 30%.

A vega of .08 means that the option price will be expected to gain \$8 (factoring in the option multiplier) if implied volatility increases by one percent assuming other factors like time decay, interest rates, and the stock price remain equal.

So, as implied volatility increases from 30% to 31% the option price would be expected to change from \$2.50 to \$2.58. Factoring in the multiplier of 100, the option’s estimated value would be \$258, resulting in a gain of \$8.

Likewise, if implied volatility decreases from 30% to 29% the Call Option would be expected to decrease by its vega value. If implied volatility reaches 29%, then the option price would be expected to change by the delta value of 0.08, decreasing the option’s value from \$2.50 to \$2.42. Factoring in the multiplier of 100, the option’s estimated value would be \$242, resulting in a loss of \$8.

In general, options tend to increase in value when the underlying stock’s volatility increases. So, volatility helps the owner of an option and hurts the writer of an option.

### Using Vega in an Option Chain

Another characteristic of vega is that it tends to be highest with at the money options. This makes sense because the chance the underlying price will change from what it is currently will be pretty high.

Also, options that have the most time until expiry will tend to have the highest vega exposure. This is because more time gives the option more opportunity to experience changes in its implied volatility. A stock with a few days left to expire will have less opportunity to experience changes in implied volatility compared to an option with 6 months left to expire. Despite having higher vega exposure, a common misconception is longer-dated options are more sensitive to changes in implied volatility. This isn’t necessarily true. Longer-dated options are loaded with other extrinsic factors such as time value, making them more expensive and less popular than shorter-dated options. As a result, implied volatility values tend to be more stable in longer-dated options.

1. Vega is the option greek that measures the expected change in an options price based on a one percent change in the underlying asset’s implied volatility value.
2. Vega tends to be highest with at the money options and Options that have the most time until expiry will tend to have the highest vega.
3. Vega is generally helpful to long option positions and is harmful to short option positions.

## Rho

### What is Rho in options?

Rho, is the option greek that measures an option’s sensitivity to interest rates. Rho represents the expected rate of change in an option’s price based on a one-percentage-point change in the U.S. Treasury bill’s risk-free interest rate. The options most sensitive to interest-rate changes are those that have the longest time before they expire, as well as those that are at the money.

### Using Rho in an Option Chain

Options that are most sensitive to rho are those that are in the money. So, if we look at the rho values for these options, we should find that they have the largest rho values compared to the other options expiring on the same date.

Also, if we compare the same type of options with the same strike price, the options with the most time left to expire are expected to have higher rho values. This is because rho also rises as the time to expiration increases.

For example, options with expiration dates at least two years away, also known as LEAPS, are far more sensitive to changes in interest rates than options that expire in 90 days.

### Rho can be Positive or Negative

In general, calls, have positive rho, and puts have negative rho. Positive rho means a rise in interest rates increases the price of a call by the amount of rho. Negative rho means that the rise in interest rates hurts the price of a put, causing it to fall by the amount of rho.

### Example of Rho

For example, let’s look at a call priced at \$2.50 with a rho valuation of 0.25. If interest rates rise by one percentage point, from 1% to 2%, then the value of the call would be expected to increase by \$0.25, to \$2.75, all other risk factors being equal.

In puts, negative rho means the option price moves in the opposite direction of interest rates.

In this example, the put has a rho of -0.25. If the option is priced at \$2.50 and interest rates rise by one percentage point, the put would see its price fall by \$0.25 to \$2.25, all else being equal. Stock options have a multiplier of 100, therefore an option quoted at \$2.50 actually costs \$250, and a rho of 0.25 would mean a \$25 change in the value of the option.

The call would see a profit of \$25 as the price moves to \$275. The put would see a \$25 loss as its value moves to \$225. If interest rates instead fall one percentage point from 2% to 1%, the put with a negative rho of 0.25 would see its price rise by \$0.25 to \$2.75, while a call with the same price and a positive rho of 0.25 would see its value decline to \$2.25.

### Federal Reserve and Interest Rates

Rho becomes important when interest rates are expected to change, such as in the days before the Federal Reserve Bank’s policy board, the Federal Open Market Committee (the FOMC), holds a policy meeting.