## THE GREEKS OF OPTION

Purchasing a stock has an obvious risk/reward profile. If the stock goes up, you make money: If it goes down, you lose money. The reverse is true if you sell a stock short. We refer to this loss exposure as directional risk (refer to Graph ) . Furthermore, the amount of the profit or loss is easy to anticipate. If you purchase 100 shares of XYZ, for each $1 increase in price the position increases in value by $100, and for each $1 reduction in price, the position loses $100 in value. If you sell 100 shares of stock short, for each $1 decrease in price you will make $100, and you will lose $100 for each $1 increase in price. By contrast, determining the risk/reward profile of an option position is much more complicated. As we have seen, an option's value can be affected by a change in anyone of these five variables :

* Stock price

* Time until expiration

* Volatility

* Interest rate

* Amount and timing of dividends

When two or more of these inputs changes, the changes can either act to offset each other in whole or in part or can work in concert to magnify either the increase or decrease in price. The situation is further complicated when options are used in combination.

Fortunately; there are analytical tools available to simplify the analysis of option positions. These tools are commonly referred to collectively as the Greeks and individually as delta, theta, gamma, rho, vega, and omega. Individually; they each measure some aspect of an option position's market risk/reward profile. This statement is true whether the position is a simple one involving one or perhaps a few different options or an extremely complicated position (such as a professional floor trader who has scores of different option contracts-some of which are long and some of which are short-and who may have a long or short position in the underlying asset).

Collectively; the Greeks provide the practiced trader with a comprehensive assessment of a position's risk/reward profile. Not only will the trader have an accurate picture of which market conditions will enhance the value of the position and which will subject it to a loss of value, but the trader will be in a position to determine what adjustments are appropriate in order to reflect the trader's current expectations concerning the stock. The trader can also reduce exposure to one or more aspects of market risk.

### The Greeks

The various Greeks, the variables with which they are associated, and a short introductory definition of each are summarized in . This chapter not only explores how the Greeks affect option pricing individually but also describes how they affect option pricing in combination. With the availability of option-analytical software, it is neithernecessary nor particularly useful for you to learn the mathematical formulas involved. What is important, however, is that you grasp conceptually the insight that these tools provide. This knowledge will help you identify your risks and respond to them appropriately: Let's start our inquiry by examining the impact of price movement of the underlying asset on an option's theoretical value. We will isolate the impact of change in price of the underlying asset by keeping the other inputs constant while varying the price of the stock. Consider Stock ABC

that has a volatility of 50. Using an interest rate of 5 percent, assume that the company does not issue a dividend. A table of the theoretical values of the 60-level calls and puts with 30 days to go until their expiration is shown in This table highlights two important aspects of the relationship between the price of an option and the price of the underlying stock:

1. As the price of ABC increases, the price of the 60-level call also increases-while the price of the corresponding put decreases. With the stock at $50, the theoretical values of the call and put are $.41 and $10.25, respectively: By the time the price of the stock reaches $55, for example, the theoretical value of the call has risen to $1.46. The put's theoretical value has declined to $6.23.

2. As the price of ABC increases, the price of the call increases while the price of the put decreases. With the stock at $50, a $1 increase in the stock theoretically produces a $.14 increase in the price of the 60- level call and a $.87 decrease in the price of the corresponding put. The call increases in price by $.29 when the stock increases from $54 to $55 and by $.80 when the stock increases from $66 to $67. Correspondingly, the put declines by $.72 when the stock goes from $54 ton $55 but only by $.21 as the stock increases from $66 to $67.

The relationship between the change in the price of a stock and the corresponding change in the price of an option is referred to as the option's delta.

### Delta

The delta is the most widely known of the Greeks and is an extremely important gauge of any option strategy. Delta measures how sensitive an option's price is to change in the value of the underlying stock. There are two particularly useful ways to look at delta: the measure of how much the option's price will change compared to a change in the price of the underlying asset, and the approximate probability that the option will finish ITM. We will explore both of these perspectives in some detail.

#### Related Change

An option's delta is the ratio of the change in its the or etical value to a small change in the price of the underlying stock. More commonly (but marginally less precise), delta is defined as how much an option's price changes for every $1 change in the price of the stock. A positive delta means that an option's price moves in the same direction as price movement in the underlying asset. An option that has a positive delta will increase in value as the underlying asset increases in value and will decrease in value as the price of the underlying asset decreases.

Conversely; a negative delta reflects the fact that an option's price moves in the opposite direction from the price movement in the underlying asset.

An option that has a negative delta will decrease in value as the underlying asset increases and will increase in value as the underlying asset decreases. Because a long call and short put increase in value as the underlying asset increases (in other words, their value rises and falls along with the underlying asset), they have a positive delta. Conversely, both a long put and a short call have negative deltas, because their values decrease as the underlying asset increases in price and increase when the

stock price declines. Going back to the table presented earlier in this chapter, with ABC stock trading at $50, the fact that the 60-level call's price would theoretically increase by $.13 when the stock price increased $1 to reach $51 indicates a delta of .13. In comparison, the delta of the 60- level put with the stock trading at $50 would be - .87. Because one equity option contract generally represents options on 100 shares of the underlying asset, the delta is most commonly expressed as the aggregate change in price of the option contract for a $1 change in the underlying asset. In other words, the delta for one option is multiplied by 100 (the number of shares represented by one contract). Under this approach, the 60-level call with ABC trading at $50 would have a delta of 13, not .13. From now on, we will refer to option deltas by using this aggregate designation.

Many option strategies and positions include holdings in the underlying asset, which can be long or short. When you are determining the delta for a complicated position that includes a combination of options and/or positions in the underlying asset, the aggregate delta of the entire position is most relevant. Using the same ratio definition of the delta as for options, the delta of one share of long stock is + 1, while the delta of one share of short stock is -1. Therefore, if a position included 1,000 shares of stock, this stock would contribute 1,000 deltas to the aggregate position delta. On the other hand, if the position included a short of 1,000 shares, this short position would contribute -1,000 deltas to the aggregate position delta.

#### If the holding is: It will contribute the following:

Long underlying security Positive delta

Short underlying security Negative delta

Long call Positive delta

Short call Negative delta

Long put Negative delta

Short put Positive delta

Although you can calculate an option's delta precisely by using an option-pricing model, many experienced option traders approximate option deltas by using the following rules of thumb:

* An ATM option typically has a delta of about +50 for calls and -50 for puts.

* With the stock trading at or near a strike price, give the nearest ITM options a delta of 75 and the next-closest ITM options a delta of 90. Then, estimate the deltas of all other ITM options to be 100.

* With the stock trading at or near a strike price, give the nearest OTM options a delta of 25 and the next-closest OTM options a delta of 10. All other OTM options should receive a zero delta.

#### Determining Options Deltas

Option Strike Price Call Delta Put Delta

45 +100 0

50 +90 -10

55 +75 -25

60 +50 -50

65 +25 -75

70 +10 -90

75 0 -100

For example, if XYZ were trading at $60, we would approximate option deltas as shown in Table.

As you learned in the previous chapter, being long a call and short a put with the same strike and expiration is the equivalent of being long 100 shares of the underlying asset (synthetic long stock). Because long 100 shares always represent an aggregate 100 deltas, combining the deltas of the long call and the short put with the same strike price and expiration date must also always equal 100. Similarly; the aggregate delta of a short call, long put position with the same strike price and expiration date will be -100, the equivalent of being short 100 shares of the underlying asset. This example highlights an important relationship between the deltas of puts and calls with the same strike price and expiration date. If you know one, you can easily determine the other. For example, if the March 60 call has a delta of +38, the delta of the March 60 put is -62.

#### Probability of an ITM Finish upon Expiration

#### An easy way to think of delta is as the probability that the option will finish ITM upon expiration. The ATM calls and puts each have a 50-50 (or 50 percent) chance of finishing ITM upon expiration and both carry an approximate delta of 50. The ITM calls and puts have a much greater chance of finishing ITM upon expiration than their ATM counterparts with a deeper ITM call or put having a higher likelihood of finishing ITM than one that is less ITM. Their deltas reflect those respective probabilities. The OTM options have the least percentage chance of finishing ITM and not surprisingly; their small deltas reflect this decreased probability.

#### Delta and Time

The time until expiration impacts an option's delta. Considering delta as the probability of finishing ITM makes it easier to understand this effect. This thought leads to the following relationship:* The delta for an ITM option will move towards 100 as time expiration decreases. The likelihood of the option staying ITM increases as

* the time until expiration decreases. The delta of an ATM option will remain at 50, because it still has a 50-50 chance of finishing ITM.

* The delta of an OTM option will move towards zero as expiration approaches, because the likelihood of the option finishing ITM decreases as the time until expiration decreases.

examples of delta and time until expiration. As we can see from these examples, strike prices at differing expiration dates have different deltas. Hence, as the time until expiration increases, the probability that the underlying asset will move towards.

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