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Option Greeks

Stock Option prices are affected by:

  • The price of the stock.
  • The Implied Volatility or "IV".
  • The time to expiration of the option.
  • The current risk-free interest rate.

"The Greeks" are names given to measurements of how each of these various factors will affect the price of an option, assuming the other factors remain steady. They are called The Greeks because the names are Greek letters (or the pseudo-Greek "Vega").

These three Greeks are important for most option traders:

  • Delta measures the change of an option price per $1 change of the underlying.
  • Vega measures the change of an option price per 1% change of volatility.
  • Theta measures the change of an option price per 1 day less time to expiration.

Two other Greeks also affect option prices, but their day-to-day importance is not usually as great as the first three:

  • Rho measures the change of an option price per 1% change in interest rates.
  • Gamma measures the change of the Delta per $1 change of the underlying.

If reading about The Greeks puts you to sleep, remember that option trading is probably 80% about formulating an opinion about what a stock is or isn't going to do. If you can do that, you can be a successful option trader. If you trade options for awhile and then start to have questions like "why did my option go up .50 last week, but only .40 this week, on the same stock movement?", or "why do some of my options go crazy at earnings time and expiration time?", or "why is my call decreasing in value even though the stock is going up?", then return to this page and do a little reading. It might all start to make sense.

DELTA

From your high school math or chemistry classes, you may remember that the term "delta" is used to show the change in some value. Delta in option trading has the same meaning. The Delta of a stock option is a measure of how much the option can be expected to gain or lose per $1 gain in the stock price.

For example, if an option will theoretically gain .50 when the stock gains $1.00, then the Delta is half the stock movement. This can be expressed as a delta of .5, or 50%. If an option will theoretically gain .80 when the stock gains $1.00, then the delta is .8 or 80%.

Calls have either a zero Delta or a positive Delta up to 1.

Puts have either a zero Delta or a negative Delta to -1.

A zero Delta on an option means that the option (put or call) is so far out of the money that stock price movement is not having any effect on the value of the option. A zero Delta on an option can also exist on an option that is anywhere out of the money with very little time remaining to expiration.

All other calls can be expected to increase in value if the stock price increases, therefore the Delta will be between 0 and 1. A call Delta of 1 might exist on a very deep in the money call, and means the call value is increasing at the same rate as the stock.

All other puts can be expected to decrease in value if the stock price increases, therefore the Delta will be between 0 and -1. A put Delta of -1 might exist on a very deep in the money put, and means the put value is increasing at the same rate as the stock is decreasing.

If you watch option prices long enough, you will notice times when option prices gain or lose faster than the stock is gaining or losing. That is not Delta however, it is a volatility effect caused by quickly changing expectations of those trading the options.

The Delta of an option is only valid over a small range of stock movement. The more the option is in-the-money, the closer the delta will get to 1 for calls or -1 for puts. The more the option is out-of-the-money, the closer the delta will get to 0. An ATM call will typically have a Delta near .5, and an ATM put will have a Delta near -.5.

The practical application of Delta is this: the more in-the-money an option is, the more the option will move in response to movement in the underlying. The more out-of-the-money an option is, the less the option will move in response to movement in the underlying.

The table below shows the Delta value for a range of strike prices, as a stock rises from $50 to $51. The deep In-the-Money call rises dollar for dollar with the stock, so it has a Delta of 1. The far Out-of-the-Money call increases .04 with a $1 move in the stock, giving it a Delta of .04.

Stock currently $50, 30 days to expiration, IV 33.33%
Strike Price
Long Call value with stock at $50
Long Call value if stock rises to $51

Delta

40 (deep ITM)
$10.08
$11.08
1
45 (ITM)
$5.39
$6.29
.90
50 (ATM)
$1.98
$2.54
.56
55 (OTM)
$.45
$.66
.21
60 (far OTM)
$.07
$.11
.04

Knowing the current Delta of an option allows you to estimate what the option will be worth if the stock rises or falls by $1. For instance, using the information in the table, we could estimate that the 50 strike call would be worth about $3.10 if the stock rose from $51 to $52 today: $2.54 + .56 = $3.10. The estimate is slightly low, since the 50 strike option is moving more ITM when the stock rises to 52, which causes the Delta to increase. An options calculator shows the Delta increases to .65 and the option would theoretically be worth $3.19.

Some option traders like to buy long options with small deltas: OTM options. Their thinking is that if the stock moves in their favor, the delta will also increase, giving their position an extra percentage "boost". From the table, you can see that someone long the 60 call when the stock moved from $50 to $51 only gained 4 cents. However, that 4 cents was a 57% increase from 7 cents.

Other option traders like to buy long options with high deltas: ITM options. Their thinking is that if the stock moves in the expected direction, the high delta means that the option will capture most of the stock's dollar movement. Someone long the 40 call when the stock moved from $50 to $51 gained $1, but that was "only" a 9.9% increase from $10.08.

Buying long options based on their deltas is really just another way of making the decision to use ITM, ATM, or OTM options. See the page "Choosing a long option Strike Price" for more information.

The idea of Delta is also used in a trading strategy known as "Delta Neutral Trading". An entire position can have a Delta value made up of the sum of the individual option Deltas. For instance, a position that has a Delta of 100 could be expected to gain $100 if the stock rises by $1. See the Delta Neutral Trading page for more information.

And finally, what could be called the "Delta" of long stock is 1, because the stock will gain $1 when the stock gains $1 (!). And the Delta of short stock is -1, because short stock will lose $1 when the stock gains $1.

This is pointed out because if you are absolutely positively convinced a stock is going to go up, your best investment as far as dollar gains is the stock itself. Most call options will have a Delta less than 1, which means they will not rise as fast as the stock.

But with a knowledge of Delta, you know that the ATM calls will have a Delta of about .5. So if you buy two ATM calls, your dollar gains will be (2 x .5 = 1), or about equal to the gains on 100 shares of stock. And since you will spend less to buy two calls than buy 100 shares of stock, your leverage is increased.

Furthermore, if the stock does move as you expect, the Delta on the bought calls will move higher than .5, because they will be more In-the-Money. So over the first dollar of stock movement, you might make (200 x .5 = $100). Over the next dollar of stock movement, with the Delta increasing, you might make (200 x .55) = $110. The higher the stock price goes, the more dollar gains your two calls will make compared to the stock itself.

THETA

Theta is the Greek used to describe the change in an option price caused by the passage of time. All options have an expiration date, therefore the amount of time left to expiration of an option has a very strong influence on the value of the option.

The effect of Theta can easily be seen on the options graphs used on this site. As shown below, this 30 strike long call position entered for $1.19 when the stock was at $30, is showing no gain or loss at that price today (the blue time line). If the stock stays at $30 after half the time to expiration passes, the green time line shows that the option will lose value.

Theta Effect Option Graph

An options calculator shows that the option will be worth .85. The option lost .34 out of $1.19 or 28.6% of its value in 15 days, just from the passage of time. If the stock is still at $30 at expiration, it will be worthless. The red time line shows a loss of the entire $1.19 at the 30 strike at expiration. The option lost an additional 71.4% of its value in the second 15 day period. As you can see from the graph and the example, a long option will lose value just from the passage of time, and at an increasing rate as the option gets closer to expiration.

Even though Theta for a long option is usually given as a negative value per day, you could say that the average Theta for the first 15 days was (-.34/15) = -.023. The average Theta for the second 15 days was (-.85/15) = -.057.

Options with a longer time to expiration are less affected by the passage of time.

Options with a shorter time to expiration are more affected by the passage of time.

The practical application of this is that if you are a long option investor, you certainly want to buy options that have more than 2 or 3 weeks of life left, possibly even a month or two of life left. If you buy options with less time left than that, the passage of time will work against you dramatically. You might get the stock movement you expected, but find that the value of the option is hardly budging.

For example, if you bought the 30 strike option with 15 days left for .85, even if the stock got to $30.85 in the next 15 days, the option would be worth .85 and you would just break even. If instead you went out to the next expiration to reduce the negative effect of theta, and bought the 30 strike option expiring in 43 days for $1.42, the option would be worth $1.62 with the stock at $30.85 in 15 days, for a gain of 14% in two weeks. You got the same stock movement over the same time period, but only made money on the option that had more time before expiration.

Another practical application of theta is in exiting a long option position. Many buyers of long options use two weeks to expiration as a cutoff date to either be profitable, or exit with a manageable loss. They know that if they continue to hold the option and the stock does not move in the last two weeks, their loss will get quite a bit bigger.

Viewed from the perspective of the option seller, you would want to sell options with only two weeks left to expiration, because time will be working for you and at an increasing rate as expiration gets closer. For instance, someone using the covered call strategy might wait for two weeks to expiration and then write an OTM call against his stock. To cause the trader to lose his stock or need to buy the option back, not only would the stock need to move enough to get ITM, it would need to do it in just two weeks. Every day that passes with the stock going nowhere is money in the bank for this trader. But see the Covered Call Strategies Disadvantages page for the downside concerns with the strategy.

VEGA

Vega is the pseudo-Greek used to describe the change in an option price caused by volatility.

The practical application of Vega is to know that if you are already long an option, an increase in volatility helps you, but a decrease hurts you. You want to buy long options when volatility is lower than normal so that a return to normal volatility helps your position.

If you are already short an option, an increase in volatility hurts you, but a decrease helps you. You want to sell options when the volatility is above normal, so that a return to normal volatility helps your position.

For example, the table below shows that if you bought the long call with the volatility at 31, and the volatility increased to 35, even if the stock price did not move your position would be .23 better off.

If you sold the call with the volatility at 35, and the volatility decreased to 31, your position would be .23 better off even if the stock price does not change.

50 Strike Long Call with stock at 50, 30 days left, various volatilities
Volatility

Call Value

Vega
31
$1.84
.06
32
$1.90
.06
33
$1.96
.06
34
$2.02
.06
35
$2.07
.05

The graph below shows a 30 strike option position with 30 days left, bought when the stock was at $30 and the Implied Volatility was 33.33%. On the check date, represented by the green time line 15 days from the entry, we have raised the IV to 56%. Not only has the option position with this increase in IV completely overcome Theta or time decay, it has increased beyond that to where the option position shows a gain even with no movement in the stock price.

Such increases in implied volatility are possible when dramatic changes might take place in a stock price. The most common event that might change a stock price dramatically is an earnings announcement. Others might be an expected analyst upgrade or downgrade, a expected buyout offer, a possible earnings preannouncement, an expected FDA approval or disapproval, etc.

Some traders are always on the lookout for opportunities where an increase or decrease in volatility might help a position. Using the same example as above, say you found a stock trading with an IV of 33% with 15 days to go until an earnings announcement. Checking past records, you find that on the day of the earnings announcement, the IV typically shoots up to 56%. Buying such an option now, and then just waiting for the expected increase in volatility, could be profitable all by itself. If the stock also rose in anticipation of good earnings, that would be an extra boost.

Traders utilizing such a strategy usually try to sell all or part of their position while the IV is high, before earnings are announced. They know that they could be hurt both by the stock falling after an earnings disappointment, and by the IV returning to normal levels. The IV quickly returning to normal levels after an earnings announcement or other event is called a "volatility crush" by option traders.

The "volatility crush" is a trading strategy in itself. A trader may look for options that have had tremendous increases in volatility on the earnings date, and use strategies such as a deep ITM buy-write to sell the high IV options. Then when the volatility is crushed after earnings are announced, the trade can be exited for a gain, as long as the stock did not move too much.

Neither strategy is a sure thing. Do some paper trading at earnings time to be sure you understand the risks and rewards before committing real money.

RHO

Rho is the Greek used to describe the change in an option price caused by changes in the risk-free interest rate.

The risk-free interest rate is assumed to be the return on an investment in which you have a near certainty of getting all of your investment back. The US Government issued T-Bill is assumed to be such an investment.

The risk-free interest rate has a very small effect on the price of an option in the first place. And since the risk-free interest rate is unlikely to change dramatically while you hold an option position, Rho is usually unimportant as a practical matter.

For example, buying a 30 strike long call with 30 days left and 33% IV with the current risk-free interest rate at 2% would cost $1.17. If the risk-free interest rate suddenly changed to 5% on the day of entry, the same option would be theoretically worth $1.21.

The reasoning for option values changing with interest rates is that since an option is an investment, it must compete with the return of other investments. If those other investments are paying higher interest, the value of an option should also be higher.

All other factors affecting an option price, including the stock price, the time to expiration, and the implied volatility, are much more important than changes in the interest rate. Even if the interest rate changed dramatically from today to tomorrow, the prices of stocks would likely change to reflect the new interest rate environment, and the effect of the change in option prices from interest rates alone would be lost in the shuffle.

GAMMA

Gamma is the Greek used to describe the change in Delta as the stock price changes. If the Delta was to change by .09 when the stock price changed by $1, then the Gamma is .09.

Using the same information as with the discussion on Delta, but now adding the option value and Delta for a second $1 move higher, we get the information below. The Gamma is how much the Delta changed as the stock rose from $51 to $52.

Stock currently $50, 30 days to expiration, IV 33.33%
Strike Price
Long Call value with stock at $50
Long Call value if stock rises to $51

Delta from $50 to $51

Long Call value if stock rises to $52

Delta from $51 to $52

Gamma
40 (deep ITM)
$10.08
$11.08
1
$12.08
1
0
45 (ITM)
$5.39
$6.29
.90
$7.21
.92
.02
50 (ATM)
$1.98
$2.54
.56
$3.19
.65
.09
55 (OTM)
$.45
$.66
.21
$.93
.27
.06
60 (far OTM)
$.07
$.11
.04
$.18
.07
.03

Gamma is highest on options that have strike prices near the current stock price. This means that the Delta increases the most (the option has a higher Gamma) when a stock starts to go In-the-Money, and the Delta increases at a slower rate (the option has a lower Gamma) the more In-the-Money the option goes.

Most option traders do not need to be too concerned about Gamma in their everyday trading. It is enough to know that Delta increases faster when an option is just going in-the-money. For the buyer of long options based on their Delta, this means that the best "bang for the buck" is obtained right when the stock goes from at-the-money to in-the-money. They can still benefit if the stock moves higher of course, but at a decreasing rate.

One combination Delta-Gamma effect is very noticeable at expiration time. The Gamma gets very high for an option trading At-the-Money, because very small movements in the stock price can change the Delta dramatically.

Say you have a position that includes short 30 strike calls. On expiration day you find that the stock is bouncing around $29.80 to $30.00 with just an hour until closing. You would like to take off the risk of the stock going over $30, but you find that the option is trading in the market at .25, which is about a quarter of what you received a month ago for selling the option. In other words, the action in the final hour of trading could gain or lose 25% of your entire premium. How is that possible?

It is possible because every trader knows that if this stock rises by just .20 from $29.80, the option will be In-the-Money, and as soon as it goes In-the-Money, it will have a Delta of 1 - the option will start moving 1:1 with the stock. So if the stock happens to end up at $30.50, the 30 strike option will have an intrinsic worth of .50. Nobody is going to let you get out of your short options for free if those options might be worth .50 a share or more in the next hour.

This expiration-day effect is the basis of a strategy for some traders. On expiration day, they look for stocks trading near strike prices, that look like they might move beyond the strike price. If they can buy an option for .25 that then becomes worth .50, they can double their money in an hour. But remember, they can also lose the entire investment in an hour.

 

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