Options traders often refer to the delta, gamma, vega and theta of their option positions. Collectively, these terms are known as the "Greeks" and they provide a way to measure the sensitivity of an option's price to quantifiable factors. These terms may seem confusing and intimidating to new option traders, but broken down, the Greeks refer to simple concepts that can help you better understand the risk and potential reward of an option position.

**Finding Values for the Greeks**

First, you should understand that the numbers given for each of the Greeks are strictly theoretical. That means the values are projected based on mathematical models. Most of the information you need to trade options - like the bid, ask and last prices, volume and open interest - is factual data received from the various option exchanges and distributed by your data service and/or brokerage firm.

But the Greeks cannot simply be looked up in your everyday option tables. They need to be calculated, and their accuracy is only as good as the model used to compute them. To get them, you will need access to a computerized solution that calculates them for you. All of the best commercial options-analysis packages will do this, and some of the better brokerage sites specializing in options (OptionVue & Optionstar) also provide this information. Naturally, you could learn the math and calculate the Greeks by hand for each option. But given the large number of options available and time constraints, that would be unrealistic. Below is a matrix that shows all the available options from December, January and April, 2005, for a stock that is currently trading at $60. It is formatted to show the market price, delta, gamma, theta, and vega for each option. As we discuss what each of the Greeks mean, you can refer to this illustration to help you understand the concepts.

The top section shows the call options, with the put options in the lower section. Notice that the strike prices are listed vertically on the left side, with the carrot (>) indicating that the $60 strike price is at-the-money. The out-of-the-money options are those above 60 for the calls and below 60 for the puts, while the in-the-money options are below 60 for the calls and above 60 for the puts. As you move from left to right, the time remaining in the life of the option increases through December, January, and April. The actual number of days left until expiration is shown in parentheses in the column header for each month.

The delta, gamma, theta, and vega figures shown above are normalized for dollars.To normalize the Greeks for dollars you simply multiply them by the contract multiplier of the option. The contract multiplier would be 100 (shares) for most stock options. How the various Greeks move as conditions change depends on how far the strike price is from the actual price of the stock and how much time is left until expiration.

**As the Underlying Stock Price Changes - Delta and Gamma**

Delta measures the sensitivity of an option's theoretical value to a change in the price of the underlying asset. It is normally represented as a number between minus one and one, and it indicates how much the value of an option should change when the price of the underlying stock rises by one dollar. As an alternative convention, the delta can also be shown as a value between -100 and +100 to show the total dollar sensitivity on the value 1 option, which comprises of 100 shares of the underlying. So the normalized deltas above show the actual dollar amount you will gain or lose. For example, if you owned the December 60 put with a delta of -45.2, you should lose $45.20 if the stock price goes up by one dollar.

Call options have positive deltas and put options have negative deltas. At-the-money options generally have deltas around 50. Deep-in-the-money options might have a delta of 80 or higher, while out-of-the-money options have deltas as small as 20 or less. As the stock price moves, delta will change as the option becomes further in- or out-of-the-money. When a stock option gets very deep-in-the-money (delta near 100), it will begin to trade like the stock, moving almost dollar for dollar with the stock price. Meanwhile, far-out-of-the-money options won't move much in absolute dollar terms. Delta is also a very important number to consider when constructing combination positions.

Since delta is such an important factor, option traders are also interested in how delta may change as the stock price moves. Gamma measures the rate of change in the delta for each one-point increase in the underlying asset. It is a valuable tool in helping you forecast changes in the delta of an option or an overall position. Gamma will be larger for the at-the-money options, and gets progressively lower for both the in- and out-of-the-money options. Unlike delta, gamma is always positive for both calls and puts. (For further reading on position delta, see the article:

*Going Beyond Simple Delta, Understanding Position Delta*.)

**Changes in Volatility and the Passage of Time - Theta and Vega**

Theta is a measure of the time decay of an option, the dollar amount that an option will lose each day due to the passage of time. For at-the-money options, theta increases as an option approaches the expiration date. For in- and out-of-the-money options, theta decreases as an option approaches expiration.

Theta is one of the most important concepts for a beginning option trader to understand, because it explains the effect of time on the premium of the options that have been purchased or sold. The further out in time you go, the smaller the time decay will be for an option. If you want to own an option, it is advantageous to purchase longer-term contracts. If you want a strategy that profits from time decay, then you will want to short the shorter-term options, so that the loss in value due to time happens quickly.

The final Greek we will look at is vega. Many people confuse vega and volatility. Volatility measures fluctuations in the underlying asset. Vega measures the sensitivity of the price of an option to changes in volatility. A change in volatility will affect both calls and puts the same way. An increase in volatility will increase the prices of all the options on an asset, and a decrease in volatility causes all the options to decrease in value.

However, each individual option has its own vega and will react to volatility changes a bit differently. The impact of volatility changes is greater for at-the-money options than it is for the in- or out-of-the-money options. While vega affects calls and puts similarly, it does seem to affect calls more than puts. Perhaps because of the anticipation of market growth over time, this effect is more pronounced for longer-term options like LEAPS.

**Using the Greeks to Understand Combination Trades**

In addition to getting the Greeks on individual options, you can also get them for positions that combine multiple options. This can help you quantify the various risks of every trade you consider, no matter how complex. Since option positions have a variety of risk exposures, and these risks vary dramatically over time and with market movements, it is important to have an easy way to understand them.

Below is a risk graph that shows the probable profit/loss of a vertical debit spread that combines 10 long January 60 calls with 10 short January 65 calls and 17.5 calls. The horizontal axis shows various prices of XYZ Corp stock, while the vertical axis shows the profit/loss of the position. The stock is currently trading at $60 (at the vertical wand).

The dotted line shows what the position looks like today; the dashed line shows the position in 30 days; and the solid line shows what the position will look like on the January expiration day. Obviously, this is a bullish position (in fact, it is often referred to as a bull call spread) and would be placed only if you expect the stock to go up in price.

The Greeks let you see how sensitive the position is to changes in the stock price, volatility and time. The middle (dashed) 30-day line, halfway between today and the January expiration date, has been chosen, and the table underneath the graph shows what the predicted profit/loss, delta, gamma, theta, and vega for the position will be then.

**Conclusion**

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 risk-exposure measurements. (For further reading on options' price influences, see the article:

*Getting to Know the Greeks*.)

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 help you take your options trading to another level.