An option's price can be influenced by a number of factors that can either help or hurt traders depending on the type of positions they have taken. Successful traders understand the factors that influence options pricing, which include the socalled "Greeks"—a set of risk measures so named after the Greek letters that denote them, which indicate how sensitive an option is to timevalue decay, changes in implied volatility, and movements in the price of its underlying security.
These four primary Greek risk measures are known as an option's theta, vega, delta, and gamma. Below, we examine each in greater detail.
Key Takeaways
 An option's "Greeks" describes its various risk parameters.
 For instance, delta is a measure of the change in an option's price or premium resulting from a change in the underlying asset, while theta measures its price decay as time passes.
 Gamma measures delta's rate of change over time, as well as the rate of change in the underlying asset. Gamma helps forecast price moves in the underlying asset.
 Vega measures the risk of changes in implied volatility or the forwardlooking expected volatility of the underlying asset price.
Understanding Options Contracts
Options contracts are used for hedging a portfolio. That is, the goal is to offset potential unfavorable moves in other investments. Options contracts are also used for speculating on whether an asset's price might rise or fall.
In short, a call option gives the holder of the option the right to buy the underlying asset while a put option allows the holder to sell the underlying asset.
Options can be exercised, meaning they can be converted to shares of the underlying asset at a specified price called the strike price. Every option has an end date called an expiration date, and a cost or value associated with it called the premium. The premium or price of an option is usually based on an option pricing model, like BlackScholes, which leads to fluctuations in price. Greeks are usually viewed in conjunction with an option price model to help understand and gauge associated risks.
Volatility
How much an option's premium, or market value, fluctuates leading up to its expiration is called volatility. Price fluctuations can be caused by any number of factors, including the financial conditions of the company, economic conditions, geopolitical risks, and moves in the overall markets.
Implied volatility represents the market's view of the likelihood that an asset's price will change. Investors use implied volatility, called implied vol, to forecast or anticipate future moves in the security or stock and in the option's price. If volatility is expected to increase, meaning implied volatility is rising, the premium for an option will likely increase as well.
Profitability
There are a few terms that describe whether an option is profitable or unprofitable. When comparing the strike price to the price of the underlying stock or asset, if the difference results in a profit, that amount is called the intrinsic value.
An atthemoney option means that the option's strike price and the underlying asset's price are equal. An inthemoney option means that a profit exists due to the option's strike price being more favorable to the underlying's price.
Conversely, an outofthemoney (OTM) option means that no profit exists when comparing the option's strike price to the underlying's price. For example, a call option that's outofthemoney means the underlying price is less than the strike price. On the other hand, a put option is OTM when the underlying's price is higher than the strike price.
Influences on an Option's Price
Table 1 below lists the major influences on both a call and put option's price. The plus or minus sign indicates an option's price direction resulting from a change in one of the listed variables.
For example, when there is a rise in implied volatility, there is an increase in the price of an option as long as other variables remain static.
Table 1: Major influences on an option's price  

Options  Increase in Volatility  Decrease in Volatility  Increase in Time to Expiration  Decrease in Time to Expiration  Increase in the Underlying  Decrease in the Underlying 
Calls  +    +    +   
Puts  +    +      + 
Bear in mind that results will differ depending on whether a trader is long or short. If a trader is long on a call option, a rise in implied volatility will be favorable because higher volatility is typically priced into the option premium. On the other hand, if a trader has established a short call option position, a rise in implied volatility will have an inverse (or negative) effect.
The writer of a naked option, whether a put or a call, would not benefit from a rise in volatility because writers want the price of the option to decline. Writers are sellers of options. When a writer sells a call option, the writer doesn't want the stock price to rise above the strike because the buyer would exercise the option if it does. In other words, if the stock's price rose high enough, the seller would have to sell shares to the option holder at the strike price when the market price was higher.
Sellers of options get paid a premium to help compensate for the risk of having their options exercised against them. Selling options is also called shorting.
Tables 2 and 3 present the same variables in terms of long and short call options (Table 2) and long and short put options (Table 3). Note that a decrease in implied volatility, reduced time to expiration, and a fall in the price of the underlying security will benefit the short call holder.
At the same time, an increase in volatility, a greater time remaining on the option, and a rise in the underlying will benefit the long call holder.
A short put holder benefits from a decrease in implied volatility, a reduced time remaining until expiration, and a rise in the price of the underlying security, while a long put holder benefits from an increase in implied volatility, a greater time remaining until expiration, and a decrease in the price of the underlying security.
Table 2: Major influences on a short and long call option's price  

Call
Options

Increase in Volatility  Decrease in Volatility  Increase in Time to Expiration  Decrease in Time to Expiration  Increase in the Underlying  Decrease in the Underlying 
Long  +    +    +   
Short    +    +    + 
Table 3: Major influences on a short and long put option's price  

Put
Options

Increase in Volatility  Decrease in Volatility  Increase in Time to Expiration  Decrease in Time to Expiration  Increase in the Underlying  Decrease in the Underlying 
Long  +    +      + 
Short    +    +  +   
Interest rates play a negligible role in a position during the life of most option trades. However, a lesserknown Greek, rho, measures the impact of changes in interest rates on an option's price. Typically, higher interest rates make call options more expensive and put options less expensive, all other things being equal.
All of the above provides context for an examination of the risk categories used to gauge the relative impact of these variables.
Keep in mind that the Greeks help traders to project changes in an option's price.
The Greeks
Table 4 describes the four primary risk measures—the Greeks—that a trader should consider before opening an option position.
Table 4: The main Greeks  

Vega  Theta  Delta  Gamma 
Measures Impact of a Change in Volatility  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 
Delta
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 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 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 does 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 outofthemoney option has a delta of 0.25, and another inthemoney 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 inthemoney.
An atthemoney 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). That means the premium will rise or fall by half a point with a onepoint move up or down in the underlying security.
In another example, if an atthemoney 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 inthemoney. Inthemoney 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 oneforone 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 inthemoney at expiration. For example, an outofthemoney call option with a 0.20 delta has roughly a 20% chance of being inthemoney at expiration, whereas a deepinthemoney call option with a 0.95 delta has a roughly 95% chance of being inthemoney at expiration.
The assumption is that the prices follow a lognormal 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 higherrisk, higherreward strategies that are more speculative, while lower deltas may be ideally suited for lowerrisk 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:
 Delta tends to increase closer to expiration for near or atthemoney options.
 Delta is further evaluated by gamma, which is a measure of delta's rate of change.
 Delta can also change in reaction to implied volatility changes.
Gamma
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 atthemoney options and lowest for those deep in or outofthemoney.
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 inthemoney 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 inthemoney than a low gamma option with the same delta.
Example of Gamma
Table 5 shows how much delta changes following a onepoint move in the price of the underlying. When call options are deep outofthemoney, 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.
Table 5: Example of Delta after a onepoint move in the price of the underlying  

Strike Price  925  926  927  928  929  930  931  932  933  934 
P/L  425  300  175  50  75  200  325  475  600  750 
Delta  48.36  49.16  49.96  50.76  51.55  52.34  53.13  53.92  54.70  55.49 
Gamma  0.80  0.80  0.80  0.80  0.79  0.79  0.79  0.79  0.78  0.78 
Theta  45.01  45.11  45.20  45.28  45.35  45.40  45.44  45.47  45.48  45.48 
Vega  96.30  96.49  96.65  96.78  96.87  96.94  96.98  96.99  96.96  96.91 
In Table 5, 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 atthemoney strike of 930, and each column represents a onepoint change in the underlying.
Atthemoney gamma is 0.79, which means that for every onepoint move of the underlying, delta will increase by exactly 0.79. (For both delta and gamma, the decimal has been shifted two digits by multiplying by 100.)
If you move right to the next column, which represents a onepoint move higher to 931 from 930, you can see that 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 onepoint rise in the underlying.
There are some additional points to keep in mind about gamma:
 Gamma is the smallest for deep outofthemoney and deepinthemoney options.
 Gamma is highest when the option gets near the money.
 Gamma is positive for long options and negative for short options.
Theta
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 inthemoney 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.
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 atthemoney call option. The strike price is 930.
As you can see, theta increases as the expiration date gets 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.
Table 6: Example of Theta values for short S&P Dec 930 call option  

  T+0  T+6  T+13  T+19 
Theta  45.4  51.85  65.2  93.3 
Theta values appear smooth and linear over the longterm, but the slopes become much steeper for atthemoney options as the expiration date grows near. The extrinsic value or time value of the in and outofthemoney 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. Atthemoney 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:
 Theta can be high for outofthemoney options if they carry a lot of implied volatility.
 Theta is typically highest for atthemoney options since less time is needed to earn a profit with a price move in the underlying.
 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
Vega measures the risk of changes in implied volatility or the forwardlooking 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. 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:
 Vega can increase or decrease without price changes of the underlying asset, due to changes in implied volatility.
 Vega can increase in reaction to quick moves in the underlying asset.
 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 atthemoney 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 thirdderivatives 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.
The Bottom Line
The Greeks help to provide important measurements of an option position's risks and potential rewards. Once you have a clear understanding of the basics, you can begin to apply this to your current strategies. It is not enough to just know the total capital at risk in an options position. To understand the probability of a trade making money, it is essential to be able to determine a variety of riskexposure measurements.
Since conditions are constantly changing, the Greeks provide traders with a means of determining how sensitive a specific trade is to price fluctuations, volatility fluctuations, and the passage of time. Combining an understanding of the Greeks with the powerful insights the risk graphs provide can take your options trading to another level.