# Option Greeks: Delta, Gamma, Theta, Vega, Rho

Trading options without an understanding of the Greeks is like flying a plane without knowing how to read the instruments. You may not have a problem when everything is going smoothly, but you’ll likely end up crashing and burning when any problems arise.

Unfortunately, many options traders are flying blind without a basic understanding of the Greeks – delta, gamma, theta, vega, and rho – or the concepts underlying them. The complex names and mathematical formulas can be off-putting, but in reality, it’s more important to understand what these numbers mean rather than how they are calculated.

These factors influence option buyers and sellers in different ways:

The influence of these factors on option pricing is summarized using the Greeks, which are fancy words for numerical values that help quantify the risk and reward profile of a given option or option strategy. Options traders use Greeks to quickly and easily understand the complex interplay of these factors in a way that is mathematically exact.

Those familiar with option pricing models – such as the Black-Scholes Model – may note that interest rates play a role in prices. While this tutorial will discuss how interest rates impact calculations, they don’t usually play a role in typical strategy design or outcomes, so they will be left out of the discussion for most parts of this tutorial.

### How Greeks Can Help

Many options tutorials use calendar spreads as an introductory strategy. By simultaneously purchasing two options of the same type and strike price with different expirations, options traders with a neutral outlook can profit from a sideways market.

Some of these tutorials will highlight how Theta – a measure of the impact of time decay – impacts the position, but the hidden cost of this strategy actually stems from Vega – a measure of the impact of implied volatility. If you sell an at-the-money front month option and an at-the-money back month option, the Vega value of these options are net long volatility.

This means that a fall in implied volatility will create a loss on the position, assuming everything else remains the same. In fact, small changes in Vega values can have a bigger impact than Theta values. A failure to understand Vega could be extremely costly in these cases, since you might not understand why the option position is losing value.

## Delta

Delta is the most commonly used and easiest to understand of the Greeks, which measures the rate of change of an option price given a \$1.00 increase in the price of the underlying asset. The value of a Delta is influenced by the time remaining until expiration and the strike price of the option relative to the underlying price of the asset. ### Probability of Being In-the-Money

The Delta is commonly used when determining the likelihood of an option being in-the-money at expiration. For example, an out-of-the-money call option with a 0.20 Delta has roughly a 20% chance of being in-the-money at expiration where as a deep in-the-money call option with a 0.95 Delta has a roughly 95% chance of being in-the-money at expiration. The assumption is that the prices follow a log normal distribution (e.g. like a coin flip).

On a high level, this means that traders can use Delta to measure how risky a given option or strategy is. Higher Deltas may be suitable for high-risk, high-reward strategies with low win rates, while lower Deltas may be ideally suited for low-risk strategies with high win rates.

### Directional Risk Assessment

The Delta is also used when determining directional risk. Positive Deltas are long market assumptions, negative Deltas are short market assumptions, and neutral Deltas are neutral market assumptions. When you buy a call option, you want a positive Delta since the price will increase along with the underlying asset price. When you buy a put option, you want a negative Delta where the price will decrease if the underlying asset price increases.

## Gamma

Gamma is one of the more obscure Greeks, but it has important implications in analyzing option strategies. It measures the rate of change of Delta, which is how much an option price changes given a one-point movement in the underlying asset. Delta increases or decreases along with the underlying asset price, whereas Gamma is a constant that measures the rate of change of Delta (see table below for an example of an in-the-money call option).

 Stock Price Option Price Delta Gamma \$51.00 \$2.95 0.55 0.05 \$50.00 \$2.45 0.50 0.05 \$49.00 \$2.00 0.45 0.05

For example, suppose that two options have the same Delta value, but one option has a high Gamma and one has a low Gamma. The option with the higher Gamma will have a higher risk since an unfavorable move in the underlying stock will have an oversized impact. High Gamma values mean that the option tends to experience volatile swings, which is a bad thing for most traders looking for predictable opportunities.

### How Gamma Impacts Strategies

A good way to think of Gamma is the measure of the stability of an option’s probability. If Delta represents the probability of being in-the-money at expiration, Gamma represents the stability of that probability over time. An option with a high Gamma and a 0.75 Delta may have less of a chance of expiring in-the-money than a low Gamma option with the same Delta.

## Theta

Theta is the decay of an option’s value over time. Often times, traders refer to it as the “silent killer” of option buyers since it occurs slowly over time. Traders may look great on paper, but Theta can turn out to be a death from a thousand cuts. Theta values are always negative for long options and will always have a zero-time value at expiration, since time only moves in one direction and time runs out when an option expires.

For example, the table below shows AAPL options that expire in three months assuming a \$157.44 current price. The in-the-money call option with a \$139.00 strike price has a Theta value of -0.025, which means that the option price will decline about \$0.03 per day. The at-the-money call option with a \$157.50 strike price has a lower Theta of -0.054, which means that it will decay at an accelerated rate of \$0.05 per day.

Theta is the decay of an option’s value over time. Often times, traders refer to it as the “silent killer” of option buyers since it occurs slowly over time. Traders may look great on paper, but Theta can turn out to be a death from a thousand cuts. Theta values are always negative for long options and will always have a zero-time value at expiration, since time only moves in one direction and time runs out when an option expires.

For example, the table below shows AAPL options that expire in three months assuming a \$157.44 current price. The in-the-money call option with a \$139.00 strike price has a Theta value of -0.025, which means that the option price will decline about \$0.03 per day. The at-the-money call option with a \$157.50 strike price has a lower Theta of -0.054, which means that it will decay at an accelerated rate of \$0.05 per day.

Theta values appear smooth and linear over the long-term, but the slopes become much steeper for at-the-money options as the expiration date grows near. The reason is that the extrinsic value of the in- and out-of-the-money options is very low near expiration since the likelihood of the price reaching the strike price is low. At-the-money options may be more likely to reach these prices, but if they don’t, the extrinsic value must be discounted over a short period.

### How Theta Impacts Strategies

Theta is good for sellers and bad for buyers. A good way to conceptualize the idea is to imagine an hourglass where one side is the buyer and one side is the seller. The buyer must decide whether to exercise the option before their time runs out, but in the meantime, the value is flowing from the buyer’s side to the seller’s side of the hourglass. The movement may not be extremely rapid, but it’s a continuous loss of value for the buyer.

## Vega

Vega measures the rate of change in the implied volatility of an option or position, which is similar to the way that Delta measures the change in an underlying asset price. Implied volatility is the expected volatility of the underlying asset over the life of the option – not the current or historical volatility of the asset. In particular, Vega shows traders how much an option price will change for each 1% move in implied volatility.

For example, the table below shows AAPL options that expire in three months assuming a \$157.44 current price. The in-the-money call option with a \$139.00 strike price has a Vega of 0.085, which means that the option’s price will increase about \$0.09 for each 1% increase in implied volatility. Similarly, the out-of-the-money call option with a \$160.00 strike price has a Vega of 0.212, which means that it will increase about \$0.21 for each 1% increase in IV.

### How Vega Impacts Strategies

Vega values are positive for buying options and negative for selling options. For example, credit spreads or naked options have negative values and debit spreads or call options have positive values. At-the-money options with the most days until expiration tend to have the highest Vega values since they benefit the most from an increase in volatility, while the opposite is true for out-of-the-money options with few days until expiration.

## Rho

Rho is the rate at which the price of a derivative changes relative to a change in the risk-free rate of interest. Rho measures the sensitivity of an option or options portfolio to a change in interest rate. Rho may also refer to the aggregated risk exposure to interest rate changes that exist for a book of several options positions.

For example, if an option or options portfolio has a rho of 1.0, then for every 1 percentage-point increase in interest rates, the value of the option (or portfolio) increases 1 percent. Options that are most sensitive to changes in interest rates are those that are at-the-money and with the longest time to expiration.

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