CHAPTER
4

The Greeks

In This Chapter

• Basic concepts of options pricing
• The calculus of options price changes
• Understanding the role of volatility
• Using the Greeks for insights on valuation

Not only do options draw their value from an underlying security, they also have features that give them extra value. The relationship between the price of the option and the price of the underlying, the time to expiration, and the market rate of interest is complex.

How complex? We’re talking college-level-math complex. I’m not kidding. Many options strategists, market makers, and traders are PhD-level mathematicians, actuaries, and physicists who prefer the excitement of trading to the drama of teaching undergraduates or calculating car insurance rates. These people can develop and use valuation models for complex option positions and just about any market situation.

You don’t need to have taken calculus to be a good options trader, but it helps to have a mathematical brain. There are many traders without formal schooling who do well, but they are nevertheless comfortable with math and with information shown on graphs. The better you can grasp the relationships among all of the different factors that go into an option’s price, the better off you will be.

This chapter is an introduction to the math of options. For some people, this will be all they need. For others, there are more advanced classes, offered by the exchanges and by private vendors. Some people want to see the equations while others just want to watch the prices in action.

The key mathematical variables used in options trading are known collectively as the Greeks. Here’s an introduction to get your started.

Delta

In math, delta is a rate of change. The term is from calculus—it’s used for the first derivative—but you don’t need calculus to understand it. Here are some examples:

• The difference between 1 and 2 is 1. In percentage terms, it is 100 percent.
• The difference between 2 and 3 is also 1, but in percentage terms, it’s 50 percent.
• The difference between 3 and 4 is also 1, but the percentage change is now just 33 percent.

The amount by which the numbers change from 1 to 2, 2 to 3, and 3 to 4 is the same, but the rate at which they change is different. That’s the delta.

In these examples, delta was the percentage relationship between one number and the next. To an options trader, delta is the relationship between the price of the underlying asset and the premium of the option, scaled by dollars. If the premium goes up $1 for every dollar the underlying increases, then the delta is $1. If the premium goes up $0.50 for each dollar increase in the underlying, then the delta is $0.50.

Delta in Pictures

The following graph shows the delta for a call and for a put. The delta is the curve of the line—also known as the slope or the first derivative—and you can see how the size of the delta changes as the price changes.

Delta is very much related to the moneyness of the option, which is how close the option is to being in the money. The closer the option is to being in the in money, or if it is already in the money, the closer the delta will be to 1.00.

OTM stands for “out of the money,” ATM for “at the money,” and ITM for “in the money.”

Delta in Use

Delta is sometimes known as the hedge ratio because it can be used to determine how many options are needed to hedge a share of stock. This makes it useful for many different types of trading strategies.

Here’s how it works. Suppose the delta of a put is 0.75.

• This means one call’s change in price would match the change in price of 0.75 shares of stock.
• If you have 100 options and 75 shares of stock, and the stock price goes up by $1 per share, you now have a profit on the stock of $75.
• The put options will decline in price by $0.75 each, for a loss of $75, so the movements cancel each other out. This is what you would expect from a hedge.
• Likewise, if the stock falls $1 per share, the put options increase by $0.75 each, and once again, the movements cancel each other out.

The opposite would happen with a call option. If the delta is 0.75 and the stock increases by $1, the option would increase by $0.75. Although “hedge” is in the name, the hedge ratio can be used to design speculative positions.

The delta changes as the price of the underlying changes. It’s not a “one and done” number. Hedgers in particular may need to readjust their positions in order to maintain the hedge.

Did You Know?

If the price of an option changed at a consistent rate when the price of the underlying asset changed, then it would be graphed along a straight line, with delta being the slope. However, delta is a curve, which is why calculus comes into play.

Delta is one of the most important measures for those using options to hedge. There’s only one catch: delta doesn’t follow a straight line. It will change as the stock and the option change in price.

In mathematical terms, gamma is the second derivative, or the rate of the rate of change. (Note the rate of is repeated on purpose, as gamma tells you the speed at which the rate of change changes.) It is getting into hard-core calculus.

But what does “rate of the rate of change” mean? Well, go back to the explanation for delta. When the numbers change from 1 to 2, 2 to 3, and 3 to 4, they change by 1. The rate of change—the delta—differs; it is 100 percent, 50 percent, and 33 percent as you move along the line. In other words, delta is the rate of change of the numbers.

Now, the difference between the rate of change from 100 percent to 50 percent is 50 percent, and the rate of change from 50 percent to 33 percent is 34 percent. That rate of change of the rate of change (whew!) is the gamma. (And if you struggled with calculus back in the day, this is a great example of how it’s a really cool, really powerful tool used in many different fields.)

The graph shows gamma, which would be the slope of the relationship between the delta and the stock price. It’s the rate of change in the rate of change—the speed at which the delta changes—in the price of the option. It varies depending on how much the price of the underlying asset changes.

How gamma changes with the price of the underlying asset for a particular asset. When the underlying asset is way out of the money or deep in the money, the rate at which delta changes is very small.

If the gamma is 0, the price won’t change much if delta changes. This happens if the option is deep in the money (so any price change in the underlying asset will affect the price of the option one-to-one) or if the option is so far out of the money the value of the underlying is almost irrelevant. At the other extreme, a gamma of 1 indicates the delta will change really fast when the underlying asset changes price, so the price of the option will change quickly, too.

The gamma will be greatest when the option is at the money, which you may remember is the point at which the price of the underlying asset is the same as the strike price of the option. As the option moves in the money (where it is profitable to exercise) and out of the money (where it is not profitable to exercise), the gamma will be smaller. The greater the gamma, the faster an option price will change when the underlying price changes.

Delta tells you how much an option moves, and gamma tells you how fast it is moving. Consequently, gamma is used in risk management more than it is in actual trading. Someone looking to manage risk needs to know how quickly a given situation could move in the wrong direction. Anyone who wants to prevent an option account from going to 0 should pay attention to the gamma of their open options positions. Traders should close out their options positions if the gamma becomes too large for comfort.

Market Maxim

If you find yourself at the bottom of a hole, the first thing to do is stop digging. Risk management is important because losses can multiply.

Gamma is important for another reason. It is a reminder that the option price relationships are not linear. Actual price movements form a curve, and that means price changes can be more dramatic than you might expect.

Theta

Theta is a measure of time decay, or how much an option’s price falls over time. It’s another calculation made using calculus, and it shows the rate of change of the option over time. It is used to figure out how long to keep a trade in place, especially in combination with delta.

Delta tells you how much an option’s price changes with the price of the underlying asset. Theta tells you how much it changes with time. Another way to think of it is:

• Delta shows you the move in the intrinsic value of an option.
• Theta shows you the move in the time value.

Both affect valuation in different ways.

How option prices fall as time passes. In mathematical terms, the theta of the option declines to 0.

This graph shows the theta for a call and for a put. The theta is the curve of the line—also known as the slope or the first derivative—and you can see how the size of the theta changes as the time to expiration changes. Theta is a negative number because the time value declines over time as the option gets closer to expiration. A share of stock, which never expires, has a theta of 0. An option at the moment of expiration has an infinitely negative theta because it will go to 0 as soon as it expires.

Theta is expressed in dollar terms. A theta of –0.05 means if everything else stayed the same, the option would decline in value by $0.05 in the next day.

Other First Derivatives

Delta and theta are the most important of the Greeks to traders, and they are both first derivatives in calculus. They have company, though, with lesser (but still useful) Greeks that define how much the option price changes relative to other variables.

Vega represents how much an option’s price changes due to a change in volatility. It is also a derivative, just like delta. Some underlying assets gyrate a lot in price, while others are steadier. The less predictable the asset’s price, the more valuable the insurance provisions of an option will be. Also, the less predictable the price for an asset, the more likely a speculator will find the option ends up in the money.

How vega changes with the price of the underlying asset.

This graph shows the vega for a call and for a put. The vega is the curve of the line, and it changes as the price of the underlying asset becomes more volatile. Because vega is discussed in the form of price, it is easy to use. If the amount of volatility increases, the option’s premium will increase by a dollar amount equal to the vega multiplied by the percentage increase. In other words, if the vega is .20 and the option’s volatility increases by 3 percent, the option’s price would increase by .20 × 3 = $0.60.

Changes in volatility have a larger effect on the value of an option than do changes in time, but both matter. The difference is that time changes constantly; every moment, the time value of an option declines. Volatility doesn’t change as fast, but when it does change, the effect on the option’s price can be significant. That’s in part because changes in volatility are usually caused by unusual events; the change in time is predictable.

Did You Know?

In the stock market, volatility is the primary source of risk. It is hardly the only source of risk, though. In the options market, it is the change in the value of the underlying asset, which might or might not be related to volatility.

Rho

Rho is another first derivative, but it is based on changes in interest rates rather than changes in underlying prices. The interest rate that matters here is the rate that is used to finance the option position. These are the rates brokerage firms and clearing firms charge traders. These rates are keyed off of the interest rates in the economy, but they aren’t the same at every firm.

The assumption behind rho is the option is being used to hedge an underlying position. This means the trader will either give up a return on cash in order to buy the stock, or she will borrow from the broker in order to short sell it. (To sell short, the trader will borrow shares of the stock from the broker and then sell the shares in the open market. The bet is the price will fall. Later, she will buy the shares back in the future at a lower price in order to repay the loan.) When she sells the stock short, she can invest the cash received and earn interest on it until the position is closed.

This graph shows the rho for a call and for a put. The rho is the curve of the line, and it changes as interest rates change. An increase in interest rates makes call options more valuable because of the interest earned on the cash from the short position. It makes put options less valuable because of the interest expense involved in holding the underlying asset longer rather than investing the money elsewhere.

The value of rho is the dollar effect on option premiums on a 1 percentage point change in interest rates. In other words, a rho of .25 means if interest rates change from 2 to 3 percent, the option price will change by $0.25.

Rho will be greater the longer the time to expiration and the higher the price of the underlying security.

How an option’s price changes with a change in interest rates.

Every option has exposure to interest rates, but options written on interest rates themselves and on different economic factors have even more exposure. For these, interest rates affect both the value of the underlying asset as well as the price on their options.

The following table gives you a summary of the Greek letters used in options trading.

Did You Know?

The Greek alphabet begins with the letters alpha and beta. Those letters are skipped over in options trading, as the Greeks go straight to gamma and delta because alpha and beta already have use in trading. Beta is a measurement of volatility relative to the stock market. The market has a beta of 1. A stock as volatile as the market has a beta of 1; if it is twice as volatile, it has a beta of 2. Alpha refers to the portion of an investment’s return that is not explained by risk. If a stock with a beta of 1 returned 10 percent more than the market, that additional 10 percent would be considered alpha.

How to Use the Greeks

The math is well and good, but it has to be used, yes? And how often have you used algebra since high school?

Well, you’ve used algebra more often than you probably like to admit and more often than you probably realize. Do you ever stand at the grocery store and figure out how many boxes of sale-priced cereal you can buy and still have enough money left over for milk? Then you have used algebra.

As with algebra, traders use the Greeks every day. They sometimes don’t realize it because they aren’t running the calculations themselves; there are computers that do that. Instead, they are interpreting the information the computer spits out.

And so it is with the Greeks. The primary factors affecting an option’s valuation are the price of the underlying asset and the time to expiration—in other words, the delta and the theta. Both of these change constantly. Volatility and interest rates change all the time, too, but these have a lesser effect on the price.

Options valuation models, covered in detail in Chapter 5, will generate the value of the Greeks at any one point in time. Traders then use those results to forecast how an option’s price might change with different changes in price, interest rates, volatility, or time. If, for example, something happens that will make underlying prices more volatile, which options will benefit the most? How should positions be managed accordingly?

Delta and theta are always the starting points, but the other Greeks are in the back of every trader’s mind.

The Least You Need to Know

• Delta is a measure of how much an option’s price changes with a change in the price of the underlying asset.
• Theta is a measure of how much an option’s price changes as time passes.
• Options prices are also affected by the volatility of the underlying asset and interest rates.
• Different computer models will generate these figures, but they are also second nature in an option trader’s brain.