The Other Greeks

To create a delta neutral trade, you need to select a calculated ratio of short and long positions that combine to create an overall position delta of zero. To accomplish this goal, it is helpful to review a variety of risk exposure measurements. The option Greeks are a set of measurements that can be used to explore the risk exposures of specific trades. Since options and other trading instruments have a variety of risk exposures that can vary dramatically over time or as markets move, it is essential to understand the various risks associated with each trade you place.


Options traders have a multitude of different ways to make money by trading options. Traders can profit when a stock price moves substantially or trades in a range. They can also make or lose money when implied volatility increases or decreases. To assess the advantage that one spread might have over another, it is vital to consider the risks involved in each spread. When making these kinds of assessments, options traders typically refer to the following risk measurements: delta, gamma, theta, and vega. These four elements of options risk are referred to as the option “Greeks.” 

Let’s take a deeper look at the most commonly used Greeks and how they can be used in options trading. First, I would like to go over a couple of technical issues in regard to the Greeks.  These numbers are calculated using higher-level mathematics and the Black-Scholes option pricing model. My objective is not to explain those computations, but to shed some light on the practical uses of these concepts. Additionally, I would suggest using an options software program to calculate these numbers so that you are not wasting precious time on tedious mathematics. 

Lastly, it is important to realize that these numbers are strictly theoretical, meaning that model values may not be the same as those calculated in real-world situations. Each risk measurement (except vega) is named after a different letter in the Greek alphabet—delta, gamma, and theta. In the beginning, it is important to be aware of all of the Greeks, although understanding the delta is the most crucial to your success. Comprehending the definition of each of the Greeks will give you the tools to decipher option pricing as well as risk. Each of the terms has its own specific use in day-to-day trading by most professional traders as well as in my own trading approach.

• Delta. Change in the price (premium) of an option relative to the price change of the underlying security.
• Gamma. Change in the delta of an option with respect to the change in price of its underlying security.
• Theta. Change in the price of an option with respect to a change in its time to expiration.
• Vega. Change in the price of an option with respect to its change in volatility.

Each of these risk measurements contains specific important trading information. As you become more acquainted with the various aspects of options trading, you will find more and more uses for each of them. For example, they each make a unique contribution to an option’s premium. The two most important components of an option’s premium are intrinsic value and time value (extrinsic value). In an effort to understand the elements that influence the value of an option, various option pricing models were created, including Black-Scholes and Cox-Rubinstein.

To comprehend the Greeks, we must understand that they are derived from these types of theoretical pricing models. The values that are needed as inputs into the option pricing models are related to the Greeks. However, the inputs for the models are not the Greeks themselves. A common mistake among options traders is to refer to vega as implied volatility. When we refer to the Greeks, we are talking about risk that will ultimately affect the option’s price. Therefore, a more accurate description of vega would be the option’s price sensitivity to implied volatility changes.


The concept of the option’s delta seems to be the first Greek that everyone learns. It’s basically a measure of how much the value of an option will change given a change in the underlying stock. When the strike price of the option is close or at-the-money, the delta of the option will be around 50 for long call options and –50 for long put options. In the case of a call option, the option’s delta could be higher if the value of the stock has exceeded the option’s strike price significantly. If our call option’s strike price were much higher than the price of the shares, the value of the delta would be smaller. For example, if XYZ stock is trading for $50 per share and I own the $60 strike price call, my delta may be around 30. 

Recall that delta is computed using an option pricing model. It will vary based on the difference between the stock price and the strike price of the option as well as the time left until expiration. In this case, let’s assume the delta of this option is 30, or .30.  Therefore, my position will theoretically make $30 for every $1 increase in XYZ stock based on the option’s delta. There are many ways that traders can use the delta, or hedge ratio, in their options analysis. A very basic way to use delta is in hedging a shares position. Let’s suppose that I have 500 shares of XYZ and that I want to purchase some puts to protect my position. 

Most traders would purchase five at-the-money puts. This creates a synthetic call position. The idea is that the trader can exercise the puts if the market moves against him. In this respect the purchased options become like insurance for the stock trader. However, there is another way to look at this scenario. If I have 500 shares of XYZ stock, I can hedge the delta of the stock by purchasing 10 of the XYZ at-the-money puts. Since the delta of each share of stock is 1 and the delta of each at-the-money put is –50, I would need 10 puts to hedge the deltas of the long stock position. The results are similar to a straddle.


Long 500 shares of XYZ   Delta = +500
Long 10 ATM puts             Delta = –500
   Net delta                                                 0


Gamma tells us how fast the delta of the option changes for every 1 point move in the underlying stock. For this reason, some traders refer to gamma as the delta of the delta. However, gamma is different from delta in that it is always expressed as a positive number regardless of whether it relates to a put or a call. If the price of the stock increases $1 and the delta increases or decreases by a value of 15 then the gamma is 15. Remember, we are using our option pricing model to make this determination. An other interesting characteristic of gamma is that it is largest for the at-the-money options. This means that the deltas for the at-the-money options are more sensitive to a change in the price of the underlying stock.

While I have been talking about delta and gamma in relation to the underlying stock price, it is important to note that they are also influenced by time and volatility. Statistical (or historical) volatility is a measure of the fluctuation of the underlying stock. As I have already noted, delta is a measure of how the options price will change when the underlying stock changes. Therefore, the delta of the options will be generally higher for a higher-volatility stock versus a lower-volatility stock. This is due to the fact that the stock’s volatility and the option’s delta are related to the movement of the stock. Also, ITM and ATM option deltas fall faster than OTM options as they approach expiration.


The theta of an option is a measure of the time decay of an option. Theta can also be defined as the amount by which the price of an option exceeds its intrinsic value. Generally speaking, theta decreases as an option approaches expiration. Theta is one of the most important concepts for a beginning option trader to understand for it basically explains the effect of time on the premium of the options that have been purchased or sold. The less time that an option has until expiration, the faster that option is going to lose its value. Theta is a way of measuring the rate at which this value is lost. The further out in time you go, the smaller the time decay will be for an option.

Therefore, if you want to buy an option, it is advantageous to purchase longer-term contracts. If you are using a strategy that profits from time decay, then you will want to be short the shorter-term options so that the loss in value due to time decay happens quickly. Since an option loses value as time passes, theta is expressed as a negative number. For example, an option (put or call) with a theta of –.15 will lose 15 cents per day. As noted earlier, time decay is not linear. For that reason, options with less time until expiration will have a higher (negative) theta than those with only a few days of life remaining.


Vega tells us how much the price of the options will change for every 1 percent change in implied volatility. So, if we purchased the XYZ option for $100 and its vega is 20, we can expect the cost of the option to increase by $20 when implied volatility moves up by 1 percent. Vega tends to be highest for options that are at-the-money and decreases as the option reaches its expiration date. It is interesting to note that vega does not share the correlation to the stock’s fluctuation that delta and gamma do. This is because vega is dependent on the measure of implied volatility rather than statistical volatility. 

This is an important distinction for traders who like to trade options straddles. We all know that there is time value associated with the value of an option. The rate at which the option’s time premium depreciates on a daily basis is called theta. It is typically highest for at-the-money options and is expressed as a negative value. So, if I have an option that has a theta of –.50, I can expect the value of my option to decrease 50 cents per day until the option’s expiration. This characteristic of the option’s time premium has particular interest to the trader of credit spreads.


As options traders become more experienced with creating spreads, they should become more aware of the types of risks involved with each spread. To reach this level of trading competence, options traders should combine the values of the Greeks used to create the optimal options spread. The result will allow the trader to more accurately assess the risks of any given options spread. Understanding the relative impact of the Greeks on positions you hold is indispensable. Here are six of the more salient mathematical relationships of these Greek variables:

1. The delta of an at-the-money option is about 50. Out-of-the-money options have smaller deltas and they decrease the farther out-of-the-money you go. In-the-money options have greater deltas and they increase the farther in-the-money you go. Call deltas are positive and
put deltas are negative.

2. When you sell options, theta is positive and gamma is negative. This means you make money through time decay, but price movement is undesirable. So profits you’re trying to earn through option time decay when you sell puts and calls may never be realized if the stock moves quickly in price. Also, rallies in price of the underlying asset will cause your overall position to become increasingly delta-short and to lose money. Conversely, declines in the underlying asset price will cause your position to become increasingly delta-long and to lose money.

3. When you buy options, theta is negative and gamma is positive. This means you lose money through time decay but price movement is desirable. So profits you’re attempting to earn through volatile moves of the underlying stock may never be realized if time decay causes losses. Also, rallies in price result in your position becoming increasingly delta-long and declines result in your position becoming increasingly delta-short.

4. Theta and gamma increase as you get close to expiration, and they’re greatest for at-the-money options. This means the stakes grow if you’re short at-the-money because either the put or the call can easily become in-the-money and move point-for-point with the equity. You can’t adjust quickly enough to accommodate such a situation.

5. When you sell options, vega is negative. This means if implied volatility increases, your position will lose money, and if it decreases, your position will make money. When you buy options, vega is positive, so increases in implied volatility are profitable and decreases are unprofitable.

6. Vega is greatest for options far from expiration. Vega becomes less of a factor while theta and gamma become more significant as options approach expiration.


Since the rate of time decay varies from one options contract to the next, the strategist generally wants to know how time is impacting the overall position. For instance, a straddle, which involves the purchase of both a put and a call, can lose significant value due to time decay. In addition, given that the rate of time decay is not linear, straddles using short-term options are generally not advised. Instead, longer-term options are more suitable for straddles and the trade should be closed well before (30 days or more before) expiration. Some strategies, on the other hand, can use time decay to the option trader’s advantage. 

Have you ever heard that 85 percent of options expire worthless? While that is probably not entirely true, a large number of options do expire worthless each month. As a result, some traders prefer to sell, rather than buy, options because, unlike the option buyers, the option seller benefits from forces of time decay. The simplest strategy that attempts to profit from time decay is the covered call—or buying shares of XYZ Corp. and selling XYZ calls. Perhaps a better alternative, however, is the calendar spread. This type of spread involves purchasing a longer-term call option on a stock and selling a shorter-term call on the same stock. 

The goal is to hold the long-term option while the short-term contract loses value at a faster rate due to the nonlinear nature of time decay. There are a number of different ways to construct calendar spreads. Some diagonal spreads, butterflies, and condors are examples of other strategies that can benefit from the loss of time value. In each instance, the strategist is generally not interested in seeing the underlying asset make a dramatic move higher or lower, but rather seeing the underlying stock trade within a range while time decay eats away at the value of its options.


Volatility can be defined as a measurement of the amount by which an underlying asset is expected to fluctuate in a given period of time. It is one of the most important variables in options trading, significantly impacting the price of an option’s premium as well as contributing heavily to an option’s time value. As previously mentioned, there are two basic kinds of volatility: implied and historical (statistical). Implied volatility is computed using the actual market prices of an option and one of a number of pricing models (Black-Scholes for shares and indexes; Black for futures). 

For example, if the market price of an option increases without a change in the price of the underlying instrument, the option’s implied volatility will have risen. Historical volatility is calculated by using the standard deviation of underlying asset price changes from close-to-close of trading going back 21 to 23 days or some other predetermined period. In more basic terms, historical volatility gauges price movement in terms of past performance. Implied volatility approximates how much the marketplace thinks prices will move. Understanding volatility can help you to choose and implement the appropriate option strategy. 

It holds the key to improving your market timing as well as helping you to avoid the purchase of overpriced options or the sale of underpriced options. In basic terms, volatility is the speed of change in the market. Some people refer to it as confusion in the market. I prefer to think of it as insurance. If you were to sell an insurance policy to a 35-year-old who drives a basic Honda, the stable driver and stable car would equal a low insurance premium. Now, let’s sell an insurance policy to an 18-year-old, fresh out of high school with no driving record. Furthermore, let’s say he’s driving a brand-new red Corvette. 

His policy will cost more than the policy for the Honda. The 18-year-old lives in a state of high volatility! The term vega represents the measurement of the change in the price of the option in relation to the change in the volatility of the underlying asset. As the option moves quicker within time, we have a change in volatility: Volatility moves up. If the S&P’s volatility was sitting just below 17, perhaps now it’s at 17.50. You can equate that .50 rise to an approximate 3 percent increase in options. Can options increase even if the price of the underlying asset moves nowhere? Yes. 

This frequently happens in the bonds market just before the government issues the employment report on the first Friday of the month. Before the Friday report is released, demand causes option volatility to increase. After the report is issued, volatility usually reverts to its normal levels. In general, it is profitable to buy options in low volatility and sell them during periods of high volatility. When trading options, you can use a computer to look at various indicators to assess whether an option’s price is abnormal when compared to the movement of the underlying asset. 

This abnormality in price is caused mostly by an option’s implied volatility, or perception of the future movement of the asset. Implied volatility is a computed value calculated by using an option pricing model for volume, as well as strike price, expiration date, and the price of the underlying asset. It matches the theoretical option price with the current market price of the option. Many times, option prices reflect higher or lower option volatility than the asset itself. The best thing about implied volatility is that it is very cyclical; that is, it tends to move back and forth within a given range. 

Sometimes it may remain high or low for a while, and at other times it might reach a new high or low. The key to utilizing implied volatility is in knowing that when it changes direction, it often moves quickly in the new direction. Buying options when the implied volatility is high causes some trades to end up losing even when the price of the underlying asset moves in your direction. You can take advantage of this situation by selling options and receiving their premium as a credit to your account instead of buying options. For example, if you buy an option on IBM when the implied volatility is at a high you may pay $6.50 for the option. 

If the market stays where it is, the implied volatility will drop and the option may then be priced at only $4.75 with this drop in volatility. I generally search the computer for price discrepancies that indicate that an option is very cheap or expensive compared to its underlying asset. When an option’s actual price differs from the theoretical price by any significant amount, I take advantage of the situation by purchasing options with low volatility and selling options with high volatility, expecting the prices to get back in line as the expiration date approaches.

To place a long volatility trade, I want the volatility to increase. I look for a market where the implied volatility for the ATM options has dropped down toward its historic lows. Next, I wait for the implied volatility to turn around and start going back up. In its most basic form, volatility means change. It can be summed up just like that. Markets that move erratically—such as the energy markets in times of crisis, or grains in short supply—command higher option premiums than markets that lag. I look at volatilities on a daily basis and many times find options to be priced higher than they should be. 

This is known as volatility skew. Most option pricing models give the trader an edge in estimating an option’s worth and thereby identifying skew. Computers are an invaluable resource in searching for these kinds of opportunities. For example, deeply OTM options tend to have higher implied volatility levels than ATM options. This leads to the overpricing of OTM options based on a volatility scale. Increased volatility of OTM options occurs for a variety of reasons. Many traders prefer to buy two $5 options than one $10 option because they feel they are getting more bang for their buck.

What does this do to the demand for OTM options? It increases that demand, which increases the price, which creates volatility skew. These skews are another key to finding profitable option strategy opportunities. Although I strongly recommend using a computer to accurately determine volatility prices, there are a couple of techniques available for people who do not have a computer. One way is to compare the S&P against the Dow Jones Industrial Average. A 1-point movement in the S&P generally corresponds to 8 to 10 points of movement in the Dow. 

For example, if the Dow drops 16 to 20 points, but the S&P is still moving up a point, then S&P volatility is increasing. On the other side, if you are consistently getting 1-point movement in the S&P for more than 10 points movement in the Dow, then volatility on the S&P is decreasing. This is one way to determine volatility. Another way to determine volatility is to check out the range of the markets you wish to trade. The range is the difference between the high for the cycle and the low value for whatever cycle you wish to study (daily, weekly, etc.). For example, try charting the daily range of a stock and then keep a running average of this range. 

Then, compare this range to the range of the Dow or the S&P. If the range of your stock is greater than the Dow/S&P average, then volatility is increasing; if the range is less than the average, then volatility is decreasing. Determining the range or checking out the Dow/S&P relationship are two ways of gauging volatility with or without a computer. You can use these techniques to your advantage to determine whether you should be buying, initiating a trade, or just waiting. Remember, option prices can change quite dramatically between high and low volatility.


In order to stack the odds in your favor when developing options strategies, it is important to clearly distinguish between two types of volatility: implied and historical. Implied volatility (IV) as we have already noted, is the measure of volatility that is embedded in an option’s price. In addition, each options contract will have a unique level of implied volatility that can be computed using an option pricing model. All else being equal, the greater an underlying asset’s volatility, the higher the level of IV. That is, an underlying asset that exhibits a great deal of volatility will command a higher option premium than an underlying asset with low volatility.

To understand why a volatile stock will command a higher option premium, consider buying a call option on XYZ with a strike price of 50 and expiration in January (the XYZ January 50 call) during the month of December. If the stock has been trading between $40 and $45 for the past six weeks, the odds of the option rising above $50 by January are relatively slim. As a result, the XYZ January 50 call option will not carry much value. But say the stock has been trading between $40 and $80 during the past six weeks and sometimes jumps $15 in a single day. In that case, XYZ has exhibited relatively high volatility, and therefore the stock has a better chance of rising above $50 by January. 

A call option, which gives the buyer the right to purchase the stock at $50 a share, will have better odds of being in-the-money and as a result will command a higher price if the stock has been exhibiting higher levels of volatility. Options traders understand that stocks with higher volatility have a greater chance of being in-the-money at expiration than low-volatility stocks. Consequently, all else being equal, a stock with higher volatility will have more expensive option premiums than a low-volatility stock. 

Mathematically, the difference in premiums between the two stocks owes to a difference in implied volatility—which is computed using an option pricing model like the one developed by Fischer Black and Myron Scholes, the Black-Scholes model. Furthermore, IV is generally discussed as a percentage. For example, the IV of the XYZ January 50 call is 25 percent. Implied volatility of 20 percent or less is considered low. Extremely volatile stocks can have IV in the triple digits.Sometimes traders and analysts attempt to gauge whether the implied volatility of an options contract is appropriate. 

For example, if the IV is too high given the underlying asset’s future volatility, the options may be overpriced and worth selling. On the other hand, if IV is too low given the outlook for the underlying asset, the option premiums may be too low, or cheap, and worth buying. One way to determine whether implied volatility is high or low at any given point in time is to compare it to its past levels. For example, if the options of an underlying asset have IV in the 20 to 25 percent range during the past six months and then suddenly spike up to 50 percent, the option premiums have become expensive.

Statistical volatility (SV) can also offer a barometer to determine whether an options contract is cheap (IV too low) or expensive (IV too high). Since SV is computed as the annualized standard deviation of past prices over a period of time (10, 30, 90 days), it is considered a measure of historical volatility because it looks at past prices. SV is a tool for reviewing the past volatility of a stock or index. Like implied volatility, it is discussed in terms of percentages. Comparing the SV to IV can offer indications regarding the appropriateness of the current option premiums. 

If the implied volatility is significantly higher than the statistical volatility, chances are the options are expensive. That is, the option premiums are pricing in the expectations of much higher volatility going forward when compared to the underlying asset’s actual volatility in the past. When implied volatility is low relative to statistical volatility, the options might be cheap. That is, relative to the asset’s historical volatility, the IV and option premiums are high. Savvy traders attempt to take advantage of large differences between historical and implied volatility. 


Now the challenge for the options strategist and particularly the delta neutral trader is to effectively interpret and manage these Greeks not only at trade initiation but also throughout the life of the position. The first thing to realize is that changes in the underlying instrument cause changes in the delta, which then impact all the other Greeks. Keep these three rules in mind when evaluating the Greeks of your position.

1. The deltas of out-of-the-money options are smaller and they continue to decrease as you go further out-of-the-money. When the options strategist purchases options, theta is negative and gamma is positive. In this situation the position would lose money through time decay but price movement has a positive impact.

2. When the trader sells option premium then theta is positive and gamma is negative. This position would make money through time decay but price movement would have a negative effect. Also, theta and gamma both increase, the closer the position gets to expiration. In addition, theta and gamma are larger for at-the-money options.

3. When selling options, vega is negative; if implied volatility rises the position loses money and if it declines the position makes money. Conversely, when a trader purchases options, vega is positive, so a rise in implied volatility is profitable and decreases have a negative impact on the position’s profitability.

To be an effective options strategist, particularly if you want to make delta neutral type adjustments, understanding these Greek basics and being able to apply them to your options strategies is paramount to your success. These very important measurement tools coupled with the position’s risk graph can provide the necessary information that will allow you to consistently execute profitable trades and send your equity curve sharply upward.


Many times when you have conversations with options traders, you will notice that they refer to the delta, gamma, vega, or theta of their positions. This terminology can be confusing and sometimes intimidating to those who have not been exposed to this type of rhetoric. When broken down, all of these terms refer to relatively simple concepts that can help you to more thoroughly understand the risks and potential rewards of option positions. 

Having a comprehensive understanding of these concepts will help you to decrease your risk exposure, reduce your stress levels, and increase your overall profitability as a trader. Learning how to integrate these basic concepts into your own trading programs can have a powerful effect on your success. Since prices are constantly changing, the Greeks provide traders with the means to determine just how sensitive a specific trade is to price fluctuations. Combining an understanding of the Greeks with the powerful insights risk profiles provide can help you take your options trading to another level.

Option pricing is based on a variety of factors. Each of these factors can be used to help determine the correct strategy to be used in a market. Volatility is a vital part of this process. Charting the volatility of your favorite markets will enable you to spot abnormalities that can translate into healthy profits. Since this is such a complicated subject, a great deal of time, money, and energy is spent to explore its daily fluctuations and profitable applications.


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