Options Greeks

When traders talk about "the Greeks," they are referring to the different ways risk can be measured as it relates to a particular option or position. A different Greek letter (e.g., delta, gamma, vega) corresponds with each unique measurement. Together, these analytical tools enable traders to manage risk. Of these, the delta is one of the most common. Often, you'll see strategies or positions referred to as delta neutral. According to the theoretical pricing models, a delta neutral strategy has essentially no risk from market movement. In other words, the market could go up or down and the position will continue to perform as expected provided that certain predictable adjustments are made along the way.

The risk measures presented here are:

• Delta
• Gamma
• Theta
• Rho

The delta, derived from a theoretical pricing model like Black-Scholes, is a number between 0 and +/-100 that has a variety of different uses and interpretations including:

• A hedge ratio
• Change in price of an option given a $1 change in the underlying stock
• The probability that an option will finish in-the-money

The table below shows call and put deltas over a range of strike prices. Note that the at-the-money 105 strike has 48 and -52 deltas for calls and puts respectively. Deep in-the-money deltas approach +/-100 while far out-of-the-money deltas approach 0.

Stock Price: $104
37 days to expiration

Option	Delta	Option	Delta
70 Call	100	70 Put	-1
75 Call	97	75 Put	-3
80 Call	91	80 Put	-9
85 Call	85	85 Put	-15
90 Call	78	90 Put	-21
95 Call	68	95 Put	-32
100 Call	59	100 Put	-40
105 Call	48	105 Put	-52
110 Call	37	110 Put	-63
115 Call	27	115 Put	-73
120 Call	18	120 Put	-82
125 Call	10	125 Put	-90
130 Call	6	130 Put	-93
135 Call	2	135 Put	-99

Hedge Ratio

Traders who use the delta as a hedge ratio do so to know how many shares of stock to buy or sell in order to establish a theoretically riskless hedge. By doing so, they are able to establish a position that should make money regardless of market direction.

Before establishing a hedge, it's important to remember the following:

• The delta of a stock position is always in a 1:1 ratio with the number of shares (in other words, 100 shares of stock have a 100 delta)
• Calls have a positive delta
• Puts have a negative delta

Position	Delta
Long 100 shares of stock	+ 100
Short 100 shares of stock	- 100
Long 1 call (45 delta)	+ 45
Short 1 call (55 delta)	- 55
Long 1 put (70 delta)	- 70
Short 1 put (60 delta)	+ 60

Like other arbitrage strategies, delta neutral strategies are often used to capitalize on price discrepancies in the market. If the delta of a call option is 45 and the trader wants to stay delta neutral, it will be necessary to sell 45 shares of the underlying stock for every call contract purchased. Similarly, if the calls are sold, stock will have to be purchased to create a neutral hedge. As the stock price moves, the option delta also changes. For this reason, it may be necessary to adjust the position by buying or selling stock to remain delta neutral.

Initial Position	Delta
Stock Price: $35
Long 1 40 call	38
Sell 38 shares @ $35	-38
Position Deltas	0

With a stock trading at $35, let's imagine that your theoretical pricing model shows 40 calls offered $0.75 below their theoretical value. Since each contract represents 100 shares, this adds up to a theoretically riskless profit of $75 per contract. To lock in this profit, you would buy the 40 calls (with a 38 delta) and sell 38 shares of stock for every 40 call you purchased. In this way, you establish a delta neutral position.

Stock Price: $39

Position	Delta
Long 1 40 call	49
Short 38 shares @ $35	-38
Position Deltas	+11
Adjustment
Sell 11 shares @ $39	-11
Adjusted Delta	0

Later, if the stock jumps to $39 and the 40 call delta increases to 49, it will be necessary to sell an additional 11 shares of stock to remain delta neutral.

Stock Price: $30

Position	Delta
Long 1 40 call	+ 25
Short 38 shares @ $35	- 38
Short 11 shares @ $39 (from adjustment)	- 11
Delta	- 24
Adjustment
Buy 24 shares @ $30	+ 24
Adjusted Delta	0

If the stock plummets from $39 to $30 and the delta of the 40 call drops to 25, it will be necessary to buy 24 shares (49 - 25) to remain delta neutral.

Throughout the life of the position, ongoing adjustments may be necessary to maintain the risk neutral position. At expiration, any out-of-the-money options expire worthless, any in-the-money options are sold (or exercised), and any long or short stock position is liquidated. At that point, the net result of all the trades should approximate the $75 profit per contract predicted by the model. This is the essence of how delta neutral trading works.

Now, let's examine how traders use delta to measure the change in price of an option as the underlying moves.

The Delta as a Measure of Changing Option Prices

One of the other common uses of the delta is as a measure of the change in an option's value given a $1 change in the underlying. For example, imagine that a stock trading at $75 has at the money options with the following prices and deltas.

Stock Price: $75

Option	Price	Delta
75 Call	$ 5	+ 52
75 Put	$ 4.75	- 48

If volatility and all other factors remain the same and the stock price rises to $76, the price of the options will change by the amount of the deltas. More specifically, the call price will increase by $0.52 and the put price will decrease (because of the negative delta) by $0.48. Thus, the new option prices will be $5.52 and $4.27 respectively.

It is important to note that the delta of the 75 call and 75 put changes as the stock price moves. If you think in the case of an large price move, $10 for example, this makes sense. If the 75 put delta didn't change, a $10 price increase would imply that the put value would drop by $4.80 ($10 per share x -.48) bringing the value to -$0.05. That, however, is impossible because options never have negative prices. With the stock at $85, the 75 put will be worth significantly less than $4.75, but it will still have a positive value.

Deep In- and Out-of-the-Money Options

While the deltas of at-the-money options tend to hover near 50, the deltas of deep in- and out-of-money options tend to approach +/-100 and 0 respectively.

Using the example above, a 50 call might be considered deep in-the-money with the stock at $75. As such, it's value would consist primarily of its $25 of intrinsic value. For this reason, deep in-the-money options tend to move in tandem with the underlying stock. For example, a 100 delta option implies a $1 move in the price of the option for every $1 move in the underlying stock. Therefore, if the stock price dropped from $75 to $73, the 50 call would drop from $25 to $23.

Deep out-of-the-money options, the 50 put for example, have deltas that approach 0. In other words, since the option has no intrinsic value and as little as 1/16 of time value, it would take more than a $1 move in the stock to have an impact on the value of the put. In this case, it might take a $5 drop in the stock price to get the 50 put as high as 1/8.

With these examples in mind, it will be easy to understand the third interpretation that views the delta as a probability.

The Delta as Probability

Although purists might argue that the delta was not intended as a probability, there are many who view the delta as the likelihood that an option will finish in-the-money.

Consider the following option chain where deltas have been substituted for prices:

Stock Price: $104
37 days to expiration

Option	Delta	Option	Delta
70 Call	100	70 Put	-1
75 Call	97	75 Put	-3
80 Call	91	80 Put	-9
85 Call	85	85 Put	-15
90 Call	78	90 Put	-21
95 Call	68	95 Put	-32
100 Call	59	100 Put	-40
105 Call	48	105 Put	-52
110 Call	37	110 Put	-63
115 Call	27	115 Put	-73
120 Call	18	120 Put	-82
125 Call	10	125 Put	-90
130 Call	6	130 Put	-93
135 Call	2	135 Put	-99

First, let's look at the at-the-money options. In this case, the closest strike to $104 is the 105 strike. Here, we see that the 105 calls have a + 48 delta while the 100 puts have a -52 delta. When viewing delta as a probability, it doesn't matter whether the value is positive or negative. Only the number is important. Thus, the 52 delta of the put can be interpreted as a 52% probability the option will finish in-the-money. Considering the option is already $1 in-the-money with the stock at $104, it makes sense that the option would have a slightly better than even chance of finishing in-the-money. Similarly, the 105 call has a slightly less than even chance of finishing in-the-money. More precisely, the probability is 48%.

At every strike, the sum of the call and put deltas--all taken as a positive number--add up to approximately 100.

Deep In- and Out-of-the-Money Deltas

Looking at the 135 strike, we see the call and put deltas at 2 and 99 respectively. With the stock at $104, this can be interpreted to mean there is a 99% probability the 135 put will finish in the money. At the same time, there remains an outside probability (roughly 2%) the stock will rally above 135 so the 135 calls finish in the money. Not great odds no matter how you look at it.

How Deltas Behave Closer to Expiration

The closer the options get to expiration, the more the deltas tend to approach 0 and +/-100. Using the example above, if we fast forward from 37 until expiration to just 9 days, the deltas for each strike are markedly different. For the sake of comparison, we'll assume the stock price didn't move during the 28 days.

Stock Price: $104

	Days to Expiration			Days to Expiration
	37	9		37	9
Option	Delta	Delta	Option	Delta	Delta
70 Call	100	100	70 Put	-1	0
75 Call	97	100	75 Put	-3	0
80 Call	91	100	80 Put	-9	0
85 Call	85	99	85 Put	-15	-1
90 Call	78	95	90 Put	-21	-5
95 Call	68	87	95 Put	-32	-13
100 Call	59	71	100 Put	-40	-29
105 Call	48	48	105 Put	-52	-52
110 Call	37	26	110 Put	-63	-74
115 Call	27	10	115 Put	-73	-89
120 Call	18	3	120 Put	-82	-97
125 Call	10	1	125 Put	-90	-99
130 Call	6	0	130 Put	-93	-100
135 Call	2	0	135 Put	-99	-100

As you can see, the further the option is out-of-the-money, the more its delta approaches 0 or +/- 100. Looking at the at-the-money 105 strike, we see that the deltas remain exactly the same. However, just one strike away, the 100 calls gain 12 deltas, while the 100 puts lose 11 deltas. Similarly, the out-of-the-money 110 calls lose 11 deltas. In other words, with only 9 days remaining until expiration, the probability that the 110 calls would finish in-the-money is only 26%. Just 28 days earlier, the same option had a 37% probability of finishing in-the-money.

A few strikes away, the difference is even more pronounced. The 125 calls which once had a 10% probability of finishing in-the-money now have only a 1% probability of doing so. Conversely, the 125 put now has a 99% probability of finishing in the money whereas before the probability was only 90%.

Pin Risk

It sometimes happens that the stock price at expiration is exactly the same as one of the strike prices. In the example above, if the stock closed at $105 on expiration, the 105 calls and puts would technically have a 50 delta up until the moment of expiration because the stock's next move, theoretically, has an equal probability of being up or down.

If it becomes apparent the stock will settle on a particular strike price, traders generally get out of the position if they are short options at the strike because they have no way to know how many contracts on which they will be assigned. This uncertainty is known as pin risk because they may find themselves unexpectedly short or long if they receive an assignment notice and the stock moves sharply against them.

Although gamma, as a risk measurement, is more useful to professionals who manage large positions, an understanding of the concept can certainly enhance every investor's knowledge and appreciation of option behavior.

The gamma measures the change in delta of an option as the underlying price changes. Perhaps the best way to understand this is to look again at the delta across a range of strike prices.

Stock Price: $104
37 days to expiration

Option	Delta	Option	Delta
65 Call	100	65 Put	0
70 Call	100	70 Put	-1
75 Call	97	75 Put	-3
80 Call	91	80 Put	-9
85 Call	85	85 Put	-15
90 Call	78	90 Put	-21
95 Call	68	95 Put	-32
100 Call	59	100 Put	-40
105 Call	48	105 Put	-52
110 Call	37	110 Put	-63
115 Call	27	115 Put	-73
120 Call	18	120 Put	-82
125 Call	10	125 Put	-90
130 Call	6	130 Put	-93
135 Call	2	135 Put	-99
140 Call	0	140 Put	-100

As you can see in the table above, the deltas range from 0 to +/-100. As the price of the underlying stock changes, the option deltas also change. The amount these deltas change is referred to as the gamma. More specifically, gamma is the change in delta for every point change in the underlying.

What is important to notice in the table below in the way gamma increases near the at-the-money strike and decreases as you get further from the current stock price in either direction. For demonstration purposes only, we'll assume that the gamma is, on average, 1/5 the difference between the deltas of the 2 strikes.

Stock Price: $104
37 days to expiration

Option	Delta	Gamma	Option	Delta	Gamma
65 Call	100	0	65 Put	0	0.2
70 Call	100	0.6	70 Put	-1	0.4
75 Call	97	1.2	75 Put	-3	1.2
80 Call	91	1.2	80 Put	-9	1.2
85 Call	85	1.4	85 Put	-15	1.2
90 Call	78	2.0	90 Put	-21	2.2
95 Call	68	1.8	95 Put	-32	2.4
100 Call	59	2.2	100 Put	-40	2.4
105 Call	48	2.2	105 Put	-52	2.2
110 Call	37	2.0	110 Put	-63	2.0
115 Call	27	1.8	115 Put	-73	1.8
120 Call	18	1.6	120 Put	-82	1.6
125 Call	10	0.8	125 Put	-90	1.4
130 Call	6	0.8	130 Put	-93	1.2
135 Call	2	0.4	135 Put	-99	0.2
140 Call	0	0	140 Put	-100	0

Looking at the at-the-money 105 strike, the calls have delta of 48 while the puts have a -52 delta. A gamma of 2.2 for these options suggests that the delta is going to change by +2.2 for every point increase in the underlying. For simplicity, we'll round the gamma to 2.0. If the stock moves from $104 to $105, the 105 call and put will have 50 (48 + 2) and -50 (-52 + 2) delta respectively.

Although it might seem confusing that gamma is positive for both calls and puts, it begins to make sense when you look at the big picture. As the stock price increases, the call deltas increase and approach 100. Meanwhile, the deltas of the puts also become more positive as the stock price increases. Only this time, because puts have a negative delta, the more positive the delta, the closer it will be to zero.

How Gamma Behaves Closer to Expiration

Using the example above, if we fast forward from 37 to 9 days before expiration, the deltas and gammas for each strike are markedly different. For the sake of comparison, we'll assume that the stock price didn't move during the 28 days

Stock Price: $104

	Days to Expiration					Days to Expiration
	37	37	9	9		37	37	9	9
Option	Delta	Gamma	Delta	Gamma	Option	Delta	Gamma	Delta	Gamma
65 Call	100	0	100	0	65 Put	0	0.2	0	0
70 Call	100	0.6	100	0	70 Put	-1	0.4	0	0
75 Call	97	1.2	100	0	75 Put	-3	1.2	0	0
80 Call	91	1.2	100	0.2	80 Put	-9	1.2	0	0.2
85 Call	85	1.4	99	0.8	85 Put	-15	1.2	-1	0.8
90 Call	78	2.0	95	1.6	90 Put	-21	2.2	-5	1.6
95 Call	68	1.8	87	3.2	95 Put	-32	2.4	-13	3.2
100 Call	59	2.2	71	4.6	100 Put	-40	2.4	-29	4.6
105 Call	48	2.2	48	4.4	105 Put	-52	2.2	-52	4.4
110 Call	37	2.0	26	3.2	110 Put	-63	2.0	-74	3.0
115 Call	27	1.8	10	1.4	115 Put	-73	1.8	-89	1.6
120 Call	18	1.6	3	0.4	120 Put	-82	1.6	-97	0.4
125 Call	10	0.8	1	0.2	125 Put	-90	1.4	-99	0.2
130 Call	6	0.8	0	0	130 Put	-93	1.2	-100	0
135 Call	2	0.4	0	0	135 Put	-99	0.2	-100	0
140 Call	0	0	0	0	140 Put	-100	0	-100	0

The closer the options get to expiration, the more the deltas tend to approach 0 and +/-100. With 37 days until expiration, the call deltas ranged from 100 at the 65 strike to 0 at the 140 strike. With 9 days to go, the range is more concentrated.

It's also worth noting that the real action is happening near the at-the-money strikes. As you can see, the gamma increases dramatically as the difference between the strike deltas becomes more pronounced.

At the same time at-the-money options rapidly gain gamma, the out-of-the-money options lose it. Just two strikes away from the at-the-money 105 strike we see the 115 calls and puts losing gamma as expiration nears.

Gamma and the Professional Trader

Seeing how rapidly the delta changes as expiration approaches makes it easier to appreciate just how important it is for professional traders to carefully monitor their positions. Using gamma to anticipate the change in delta is what makes it such a valuable measure of risk. Looking at the overall gamma, traders can see at a glance how much longer or shorter they will be given a move in the underlying stock.

In the discussion on theta or time decay, we make the point that an option position either benefits from the passage of time or from market movement, but not both. In this sense, gamma is considered the flip side of theta because if time hurts a position (i.e., negative theta), price movement (i.e., positive gamma) will help it and vice versa. For example, the short straddle is a position that is hurt by market movement but helped by the passage of time. The straddle writer wants the market to remain steady because the more the underlying moves, the more likely it is that the position will lose money. In contrast, a person holding a long straddle is in a race against time hoping to see the market move before the options expire.

Theta is the Greek letter used to represent the impact of time on an option's value. All options lose value as they get closer to expiration. However, the rate at which an individual option loses value is primarily a function of how much time remains until expiration. Options tend to lose the most value in the final 30 days. At that point, the price decay accelerates.

Only the extrinsic portion of an option's value is subject to time decay. An in-the-money option will retain at least its intrinsic value until expiration. In other words, if an underlying stock is trading at $42, the 40 call will always have at least $2 of intrinsic value whether there are three or 300 days remaining until expiration. Any value above $2 will be extrinsic value and therefore subject to time decay.

Theta, or time decay, is usually expressed as a negative number to represent the loss of value as time passes. Since the time remaining on an option can never increase, time decay is a one-way street. Thus, if the theta is given as -.37, they option will lose $0.37 per day in value.

However, it is important to note that theta changes over time. Assuming the price of the stock doesn't change, an out-of-the-money $3.50 option with a theta of -.20 will be worth $3.30 tomorrow. At that point, the theta may only be -.18. If so, the option will only be worth $3.12 the following day if prices remain constant. Gradually, the value of the option will approach zero as long as it remains out-of-the-money.

In the adjacent table, the theta of the AT&T Aug 35 call is -.10. If the stock price remains unchanged, the Aug 35 calls will only be worth $1.65 on the following day.

The Relationship Between Theta and Strike Price

When we looked at the extrinsic value of an option in the section on pricing options, we saw that at-the-money options have the highest extrinsic value. For this reason, these options also have the highest thetas.

Deep in- and out-of-the-money options have lower thetas because they have less extrinsic value than at-the-money options. The less value they have, they less they can lose through decay. Hence, the lower thetas.

When Time Works For You

The only way to have a positive theta position is to be short options. This makes sense when you consider that short option positions (e.g., the short straddle) tend to do best in stable markets. Wide swings up or down will hurt these positions. Only the passage of time will help. Neutral strategies like the long butterfly also benefit from the passage of time. The less time to expiration, the less chance the underlying stock has to move up or down into unprofitable territory.

Every option position represents a trade off between time and market movement. You can't benefit from both. If the passage of time helps a position, price movement will hurt it and vice versa. In Greek terms, price movement, the flip side of theta, is known as gamma. Any position that has a positive theta (i.e., a position that benefits from the passage of time) will by definition have a negative gamma. Similarly, a negative theta position (i.e., one that is hurt by the passage of time), will have a positive gamma.

Rho is the Greek letter used to represent the impact of the prevailing interest rate on an option's value. More specifically, the rho measures the change in an option's value given a change in interest rates.

An increase in interest rates raises the carrying costs associated with holding an option position. As such, it decreases the value of the options. Conversely, a decrease in interest rates increases the value of options. However, the impact of interest rates on price is so small, relatively speaking, that it makes very little difference overall. Familiarity with delta, vega, gamma, and theta is much more important because each has a significant measurable impact on option prices.