Option Greeks

The Greeks

Table 4 describes the four primary risk measures—the Greeks—that a trader should consider before opening an option position.


Measures Impact of a Change in Time Remaining

Measures Impact of a Change in the Price of Underlying

Measures the Rate of Change of Delta

Measures Impact of a Change in Volatility


Delta is a measure of the change in an option's price (that is, the premium of an option) resulting from a change in the underlying security. The value of the delta ranges from -100 to 0 for puts and 0 to 100 for calls (-1.00 and 1.00 without the decimal shift, respectively). Puts generate a negative delta because they have a negative relationship with the underlying security—that is, put premiums fall when the underlying security rises, and vice versa.

Conversely, call options have a positive relationship with the price of the underlying asset. If the underlying asset's price rises, so do the call premium, provided there are no changes in other variables such as implied volatility or time remaining until expiration. If the price of the underlying asset falls, the call premium will also decline, provided all other things remain constant.

A good way to visualize delta is to think of a race track. The tires represent the delta, and the gas pedal represents the underlying price. Low delta options are like race cars with economy tires. They won't get a lot of traction when you rapidly accelerate. On the other hand, high delta options are like drag racing tires. They provide a lot of traction when you step on the gas. Delta values closer to 1.00 or -1.00 provide the highest levels of traction.

Example of Delta

For example, suppose that one out-of-the-money option has a delta of 0.25, and another in-the-money option has a delta of 0.80. A $1 increase in the price of the underlying asset will lead to a $0.25 increase in the first option and a $0.80 increase in the second option. Traders looking for the greatest traction may want to consider high deltas, although these options tend to be more expensive in terms of their cost basis since they're likely to expire in the money.

An at-the-money option, meaning the option's strike price and the underlying asset's price are equal, has a delta value of approximately 50 (0.5 without the decimal shift).34 That means the premium will rise or fall by half a point with a one-point move up or down in the underlying security.

In another example, if an at-the-money wheat call option has a delta of 0.5 and wheat rises by 10 cents, the premium on the option will increase by approximately 5 cents (0.5 x 10 = 5) or $250 (each cent in a premium is worth $50).

Delta changes as the options become more profitable or in-the-money. In-the-money means that a profit exists due to the option's strike price being more favorable to the underlying's price. As the option gets further in the money, delta approaches 1.00 on a call and -1.00 on a put with the extremes eliciting a one-for-one relationship between changes in the option price and changes in the price of the underlying.

In effect, at delta values of -1.00 and 1.00, the option behaves like the underlying security in terms of price changes. This behavior occurs with little or no time value as most of the value of the option is intrinsic.

Probability of Being Profitable

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, whereas 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-northe mal distribution, like a coin flip.

Generally speaking, this means traders can use delta to measure the directional risk of a given option or options strategy. Higher deltas may be suitable for higher-risk, higher-reward strategies that are more speculative, while lower deltas may be ideally suited for lower-risk strategies with high win rates.

Delta and Directional Risk

Delta is also used when determining directional risk. Positive deltas are long (buy) market assumptions, negative deltas are short (sell) 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.

Three things to keep in mind with delta:

  1. Delta tends to increase closer to expiration for near or at-the-money options.

  2. Delta is further evaluated by gamma, which is a measure of the delta's rate of change.

  3. Delta can also change in reaction to implied volatility changes.


Gamma measures the rate of changes in delta over time. Since delta values are constantly changing with the underlying asset's price, gamma is used to measure the rate of change and provide traders with an idea of what to expect in the future. Gamma values are highest for at-the-money options and lowest for those deep in or out-of-the-money.

While delta changes based on the underlying asset price, gamma is a constant that represents the rate of change of delta. This makes gamma useful for determining the stability of delta, which can be used to determine the likelihood of an option reaching the strike price at expiration.

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 asset 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.

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.

Example of Gamma

Table 5 shows how much delta changes following a one-point move in the price of the underlying. When call options are deep out-of-the-money, they generally have a small delta because changes in the underlying generate tiny changes in pricing. However, the delta becomes larger as the call option gets closer to the money.

Strike Price925926927928929930931932933934
























































In the above table, the delta is rising as we read the figures from left to right, and it is shown with values for gamma at different levels of the underlying. The column showing profit/loss (P/L) of -200 represents the at-the-money strike of 930, and each column represents a one-point change in the underlying.

At-the-money gamma is -0.79, which means that for every one-point move of the underlying, delta will increase by exactly 0.79. (For both delta and gamma, the decimal has been shifted by two digits by multiplying by 100.)

If you move right to the next column, which represents a one-point move higher to 931 from 930, you can see that the delta is -53.13, an increase of .79 from -52.34. Delta rises as this short call option moves into the money, and the negative sign means that the position is losing because it is a short position. (In other words, the position delta is negative.) Therefore, with a negative delta of -51.34, the position will lose 0.51 (rounded) points in premium with the next one-point rise in the underlying.

There are some additional points to keep in mind about gamma:

  1. Gamma is the smallest for deep out-of-the-money and deep-in-the-money options.

  2. Gamma is highest when the option gets near the money.

  3. Gamma is positive for long options and negative for short options.


Theta measures the rate of time decay in the value of an option or its premium. Time decay represents the erosion of an option's value or price due to the passage of time. As time passes, the chance of an option being profitable or in the money lessens. Time decay tends to accelerate as the expiration date of an option draws closer because there's less time left to earn a profit from the trade.6

Theta is always negative for a single option since time moves in the same direction. As soon as an option is purchased by a trader, the clock starts ticking, and the value of the option immediately begins to diminish until it expires, worthless, at the predefined expiration date.

Theta is good for sellers and bad for buyers. A good way to visualize it is to imagine an hourglass in which one side is the buyer, and the other is the seller. The buyer must decide whether to exercise the option before 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.

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.

Example of Theta

An option premium that has no intrinsic value will decline at an increasing rate as expiration nears.

Table 6 shows theta values at different time intervals for an S&P 500 Dec at-the-money call option. The strike price is 930.

As you can see, theta increases as the expiration date get closer (T+25 is expiration). At T+19 or six days before expiration, theta has reached 93.3, which in this case tells us that the option is now losing $93.30 per day, up from $45.40 per day at T+0 when the hypothetical trader opened the position.







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 extrinsic value or time value of the in- and out-of-the-money options is very low near expiration because the likelihood of the price reaching the strike price is low.

In other words, there's a lower likelihood of earning a profit near expiration as time runs out. At-the-money options may be more likely to reach these prices and earn a profit, but if they don't, the extrinsic value must be discounted over a short period.

Some additional points about theta to consider when trading:

  1. Theta can be high for out-of-the-money options if they carry a lot of implied volatility.

  2. Theta is typically highest for at-the-money options since less time is needed to earn a profit with a price move in the underlying.

  3. Theta will increase sharply as time decay accelerates in the last few weeks before expiration and can severely undermine a long option holder's position, especially if implied volatility declines at the same time.


Vega measures the risk of changes in implied volatility or the forward-looking expected volatility of the underlying asset price. While delta measures actual price changes, vega is focused on changes in expectations for future volatility.

Higher volatility makes options more expensive since there’s a greater likelihood of hitting the strike price at some point.

Vega tells us approximately how much an option price will increase or decrease given an increase or decrease in the level of implied volatility.7 Option sellers benefit from a fall in implied volatility, but it is just the reverse for option buyers.

It’s important to remember that implied volatility reflects price action in the options market. When option prices are bid up because there are more buyers, implied volatility will increase.

Long option traders benefit from pricing being bid up, and short option traders benefit from prices being bid down. This is why long options have a positive vega, and short options have a negative vega.

Additional points to keep in mind regarding vega:

  1. Vega can increase or decrease without price changes of the underlying asset, due to changes in implied volatility.

  2. Vega can increase in reaction to quick moves in the underlying asset.

  3. Vega falls as the option gets closer to expiration.

Minor Greeks

In addition to the main Greek risk factors described above, options traders may also look to other, more nuanced risk factors. One example is rho (p), which represents the rate of change between an option's value and a 1% change in the interest rates. This measures sensitivity to interest rates.

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 will 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.

Some other minor Greeks that aren't discussed as often include lambda, epsilon, vomma, vera, speed, zomma, color, and ultima.

These minor 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.

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