Options Education

Options Greeks

In this section, we're going to discuss some more advanced topics as we delve into the mathematics behind options pricing. Don't worry if you're not a fan of complex algebraic formulas and mathematical algorithms, the items we're going to discuss are all calculated for you automatically within the optionsXpress system. Our goal in discussing this information is to help you become familiar with some of the components that help determine the overall price of an option so that you'll be able to better anticipate price movements when they occur.

As a new options trader, you've probably heard people refer to "options Greeks." These are simply a number of different ingredients in the overall "recipe" that determines the value of an option. They're called "Greeks" simply because the different variables are named after letters of the Greek alphabet. Each has a place, and every one contributes to the equation.

There are two ways to locate the options Greeks information. The first is to simply click on the "Pricer" link in the Toolbox menu. The other method is to look up an option chain where you'll find the "Delta" and Implied Volatility values for the particular option strike price and expiration date.

Now, let's discuss the options Greeks individually, and explain how they're used to determine an option's value.

Delta

Delta is the variable that tracks the relationship between the change in the price of an underlying stock and the change in the value of the option contracts associated with it. Delta ranges in value from 0.00 on the low side, to a maximum of 1.00. An option with a Delta of 1.00 means that for every dollar the stock rises in price, the options will increase in value by one dollar per share. This becomes very important when you consider that most option contracts control 100 shares of stock. If the underlying stock rises $1.00 per share in value, the "Delta 1.00" contract just increased in value by $100.

Generally, an "at-the-money" option strike price will have a Delta somewhere between 0.50 and 0.75. For every dollar the stock rises, the at- the-money option will increase by 50 to 75 cents per share, or $50 to $75 per contract on a 100-share contract. The deeper in the money an option becomes, the greater the Delta, until the Delta reaches 1.00. Delta will never exceed 1.00 for a single contract, although the net effect can appear to be more than 1.00. Keep in mind, the higher the Delta, the more expensive the option contract. The reverse is also true - the lower the Delta, the less expensive the option.

Every once in a while, you may hear stories of novice options traders who are enticed to buy deep out-of-the-money options contracts thinking that they can get so much more for their money. While it's true, you can buy several deeper out-of-the-money options for the same price as one or two Delta 1.00 options, the value is nowhere near the same.

Sometimes it helps to think about Delta as an option's "horsepower." Here's an interesting example:

Let's suppose that you're considering buying call options on stock "XYZ." Let's further suppose that the current stock price is $92.35/share. As you consider which option to purchase, you take a look at the "Pricer" link under the Toolbox menu, where you find the following information (among other information) listed:

Notice that if the underlying stock price is $92.35/share, the $80, $85, and $90 strike prices are "in-the-money." This is also indicated by the shaded area of the table.

Suppose that you have $1120 to spend to purchase your options contracts (for the moment, we'll disregard commission costs.) Using the information found in the table above, which option strike price represents the best value for the money?

Well, let's consider the relative purchasing power you now have. You could purchase one contract of the $80 strike price options, (XYZ CP) with a Delta of 1.00. That purchase would cost you $1120. The advantage of this purchase, of course, is that for every dollar the stock price rises, your options contract will increase dollar for dollar (remember, however, that Delta works both ways - your options contract will lose value dollar for dollar with the falling price of the stock).

Let's consider something interesting here. If you can purchase a Delta 1.00 contract by buying the $80 strike price option for $1120, isn't the $85 strike price almost as good?

You could still only buy one contract of the $85 strike with the money you have available, but you still benefit from almost the same Delta, right? For every dollar the stock rises, your options contracts will increase $0.96 per share. One contract of the $85 strike price would cost you only $850, compared to $1120 and you get nearly the same Delta benefit! Let's consider the $90 strike price options.

How many of the $90 strike price (at-the-money) contracts could you buy for $1120? You could actually purchase two of the $90 strike price "at-the-money" contracts with your available funds. What's the Delta? It's 0.75 per share, PER CONTRACT. In other words, while Delta for any single contract will never be more than 1.00, the combined, or "Net Delta," felt by the increase in purchasing power available by buying the $90 strike price contracts is actually a "Net Delta" of 1.50. For every $1.00 the stock moves higher, your options contracts will have a net increase of $1.50 per share.

Keeping in mind that the further we move "out-of-the-money" the less expensive the options become, let's consider the purchasing power we could put behind the $95 strike price options. How many could you buy with $1120? If you take $1120 and divide it by the Ask price of the $95 strike price option ($2.25 per share), we could buy almost 5 contracts. Although you can't buy a partial contract, let's assume that we could buy 5 contracts. What's the "Net Delta?" The Net Delta is only 0.85. (5 contracts times a Delta of 0.17 per share.) Even though you could buy five times the number of contracts as you could if you bought the deep in-the-money $80 strike price options, the total combined Delta for the five $95 contracts is only $.85 on the dollar.

What about the $100 strike price contracts? You could buy 15 contracts of the $100 strike price options, but with a Delta of 0.00, you wouldn't see any increase in options value at all until the stock price crept closer to $100. By that time, the $90 strike price options would be approaching a Net Delta of nearly $2.00 per share for every $1.00 increase in the price of XYZ stock.

How is Delta like horsepower? Imagine if these five option strike prices were drag racers, each with the same potential horsepower. The light turns green, the deep in-the-money Delta 1.00 $80 strike price car leaps off the line instantly, using every ounce of available horsepower. The $85 strike price car is hot on its tail with nearly the same horsepower. Suddenly, the $90 strike price dragster zooms past 1.5 times as fast as the first two cars. The $95 strike price dragster is just pulling off the line, and won't hit its maximum power curve for several seconds. The $100 strike price dragster? It just stalled on the line. That is the power of Delta.

Gamma

Gamma is one of the least understood but most powerful of the options Greeks. Gamma is the variable that tracks the rate of change of Delta. In the previous example, we compared Delta to horsepower. Keeping with that same example, Gamma is the acceleration or rate of change of the dragsters. How quickly can each car make use of its available horsepower? That's Gamma.

Unfortunately, many options investors fail to recognize the effect that Gamma can exert upon the value of an option. In reality, knowing how the Delta changes can make a significant difference in the profitability of an option play. Here's an example:

Suppose that IBM is currently trading for $23.00 per share. Suppose further that a glance at the options Greeks showed that the current Delta for the $20 strike price option was .96. This means that for every dollar per share the stock price rises, the $20 strike price option will increase $0.96 per share. That's the current situation, but a quick check of the Gamma value for the $20 strike Calls is 0.04. This means that if the stock were to increase a dollar per share, the Delta would increase by 0.04, resulting in a Delta of 1.00. Once the stock price reaches the level of $24 per share, the $20 strike price options will increase dollar for dollar in value with any further increase in the value of the stock.

Now, suppose that you were faced with two different option plays. Each of the two plays have the same Delta, and each option costs the same to purchase. But suppose one of the options carried a Gamma of 0.01, and the other carried a Gamma of 0.05. All other things being equal, the option contract with the higher Gamma would generally result in a greater profit for a given period of time, when compared to the option with a lower Gamma. A higher Gamma often equates to greater acceleration in profitability with a given increase in the value of the stock.

Theta

Theta is used to track the rate of decay of the time value of an option. The time value of an option is the variable piece of the option's value. Here's an example: Suppose you identify a stock that is currently trading for $21.35 per share, and you decide to buy the $20 strike price Call options. Before buying the options, you note that the ask price for the $20 strike price Call for the current month's expiration is $2.50 per share.

There are two components in the price of the option at the $2.50 per share ask price. Those components are the intrinsic value and the time value. The intrinsic value of the option is simply the difference between the price of the stock ($21.35 per share) and the strike price of the option ($20.00 per share), using the example above.

If the ask price for the option is $2.50 per share, and we know that the intrinsic value of the option is $1.35 per share, then we can easily deduce the time value component of the option (take the ask price of $2.50 per share and subtract the intrinsic value of $1.35 per share). This gives us a time value component of $1.15 per share.

Now, suppose you consider the $20 strike price Call for the same stock, just one month further out in time. In that case the stock price hasn't changed, and it's still trading for $21.35 per share. The strike price is still the $20 strike price option, but now the ask price for the option is $3.50 per share. What's changed? The only difference in the price of the option is the time value. Since the options for next month are further away from expiration, we should expect to pay more for them.

As the option approaches its expiration date, the time value erodes more rapidly. Tracking the Theta as an option approaches expiration reveals the speed of the decay of time value. Try this: Compare the Theta for the at-the-money option for a particular stock as far out as you can go. Now compare the Theta for the at-the-money options for the next expiration month. You should notice that the Theta for the near-term options is much larger than the Theta for the longer-term options.

In addition to the Delta, Gamma, Theta previously discussed, investors also have access to another, lesser-known pair of options Greeks. These variables are primarily used by institutional money managers, but can be helpful in allowing us to forecast the potential return on our option investment, should a given price movement occur in the underlying stock.

Vega

Vega is the Greek letter used to track the change in an option's theoretical value, given a 1% change in the volatility for the underlying stock.

Here's one way that we can actually make use of the information: let's say there is sudden good news in a stock. When the news hits the wires, the market makers for the stock immediately begin to increase the price of the stock in anticipation of a bullish rally. If investors are willing to overlook the increase in price and purchase the stock regardless of its now overvalued Ask price, the historical volatility for the stock likely increases.

Armed with the current historical volatility figure for the underlying stock, we can calculate the effect of this change to the T-Val of the option. A 1% move in the historical volatility of the underlying stock has a very small effect on the theoretical value of the option, which is why this information is usually monitored only by institutional investors with millions of dollars under management. Individual investors seldom carry enough volume in a given option position to be affected by changes in Vega.

Rho

Rho is the Greek letter used to track the expected change in the price of an option given a 1% change in interest rates. Like Vega, changes in the Rho value of an option aren't greatly affected by changes in interest rates, and are used mostly by institutional investors.

Historical Volatility

Historical Volatility is a measurement of the price movement of the underlying stock over a 6-month period of time. This value can tell us what types of price fluctuations we could expect from a stock, helping us to identify stocks that fit our personal investment strategy. Highly volatile stocks are subject to wide, quick price movements. This isn't necessarily a bad thing, if we are prepared for the consequences. Highly volatile stocks have a greater potential for quick gains, however they can lose money just as quickly. Low volatility stocks are often more stable, and less prone to wild price movements. As a result, they aren't as likely to see heavy gains over a short period of time.

Volatility is expressed as a percentage, and represents the variance in the standard deviation of a change in the price of the underlying stock over a six month period. If that sounds like a little more than you would care to know, don't worry about it. The definition is simply included to satisfy the curious.

Here's what the Historical Volatility figures actually mean to us. If we have a stock with a Historical Volatility of 16%, it simply means that we should expect a stock with a price of $100 per share to fluctuate between a low of $84 per share, and a high of $116 per share over a 6-month period. As you can see, it doesn't matter whether the stock is trading at its low range or its high range, the volatility is the same. The volatility figure does not tell us the potential direction of the stock.

T-Val

The term "T-Val" is short for "Theoretical Value." This figure represents the fair market value for the option under ideal conditions. The value is actually calculated for us through the Black-Scholes formula, which we'll discuss in greater detail below. This formula represented a monumental breakthrough when it was published in 1973. It's still in use today as the standard pricing model for options. One way to think about the T-Val is to compare it to the Kelly Blue Book value for a car. The Blue Book value for a used car represents the price that current market conditions should support for the vehicle.

This doesn't mean, however, that we should only expect to pay what the vehicle is worth according to Kelly. For example, let's say that you have a 1972 Toyota Land Cruiser with a new engine, new tires, and new paint. The Blue Book suggests the value is $200 - $500 dollars, but some individuals may consider the vehicle to be a collector's item, willing to pay us $3000 or more. The fair market value isn't necessarily what we would have to pay to buy it, especially if what we're buying is in high demand.

Implied Volatility

As we discussed earlier, historical volatility is a useful measurement of the price movement of a stock over a given period of time. Historical volatility can be thought of as a sort of baseline standard for the price movement of a stock. The implied volatility, by contrast, is a kind of measurement of the expected price movement above or below that baseline standard.

Think of it this way: suppose you identify a stock with an average annual growth of 10% per year for the last five years. The stock has demonstrated a pattern of growth that forms the basis for an expected 10% annual return. Suppose that there is a very positive news announcement for the stock that may result in significant additional corporate revenues. Investors are likely to consider buying the stock, with the thought that the new opportunity may drive the stock price higher.

As you can imagine, investors aren't the only ones expecting the stock to increase in value over the next few months. The market makers for both the stock and the options will typically raise the price of their securities with the expectation that the bullish news will encourage folks to buy the stock or options, even if they're a bit overvalued. The market makers will seek to raise the price of their securities for as long as investors express a willingness to buy them. This increase in price is reflected in a higher implied volatility measurement.

By comparing the historical and implied volatility, we can see at a glance whether or not the market believes that the underlying stock is likely to increase in price, remain about the same, or drop in price.

Generally speaking, anytime the implied volatility is 20% or more above the historical volatility, the options are considered overvalued. Conversely, an option is usually considered undervalued if the implied volatility is 20% or more below the historical volatility. Many serious options traders will rely heavily on the implied volatility and even track it on a daily basis for a particular stock. If the implied volatility starts to increase, they may begin to look for bullish trades, believing the increased volatility may indicate a change in the market's perception of the stock.

Introduction to the Black-Scholes Model

In 1973, a talented economist named Robert C. Merton published a peer-reviewed paper discussing methods by which stock warrants and other financial instruments could be equitably priced. This paper presented the work of two mathematicians, Fischer Black and Myron Scholes. In what would thereafter be known as the "Black-Scholes" model, Merton described the relationship between the price-movement of the underlying stock, interest rates, and all of the "option Greeks," variables discussed previously.

In 2000, PBS station WGBH-Boston aired a fascinating program that featured the Black-Scholes model. The NOVA documentary entitled "Trillion Dollar Bet" discusses how the formula came to be, and its application within the world of finance. If you'd like to know more about the Black-Scholes model, and about the advantages and risks of financial models like it, you would likely enjoy finding a copy at your local library, or searching for it online through retailers like Amazon.com.

Here's the theory behind the formula: When a call option on a particular stock expires, its intrinsic value is either zero (if the stock price is less than the option's strike price) or the intrinsic value is the difference between the stock price and the strike price of the option. For example, say you buy a call option on XYZ stock with an option strike price of $100. If at the option's expiration date the price of XYZ stock is less than $100, the option is worthless. However, if the stock price is greater than $100—say $120, then the call option is worth $20. The higher the stock price, the more the option contract is worth. The difference between the stock price and the option strike price is the option's "payoff."

Obviously, it's very simple to figure out how much an option is worth at expiration. It's simply a question of subtracting the option strike price from the stock price at expiration. What if I want to know what my option contract is worth right now, or one or two weeks prior to expiration? Remember the two components to an option's value, the "Intrinsic Value" (stock price minus strike price) and the "Time Value."

The Black-Scholes model was created to calculate that "Time Value" component of an option's value at any time prior to the option's expiration date. Add together the "Time Value" and the "Intrinsic Value" and you know precisely what an option is worth.

In order to work, the model depends on a number of input variables. These include the price of the underlying stock, the exercise price of the option, the risk-free interest rate (the annualized, continuously compounded rate on a safe asset with the same maturity as the option) and the time to maturity of the option. The only unobservable is the volatility of the underlying stock price.

Armed with that information, the formula quickly identifies what the option is worth.