CHAPTER 2

Greek Philosophy

My wife, Kathleen, is not an options trader. Au contraire. However, she, like just about everyone, uses them from time to time-though without really thinking about it. She was on eBay the other day bidding on a pair of shoes. The bid was $45 with three days left to go. She was concerned about the price rising too much and missing the chance to buy them at what she thought was a good price. She noticed, though, that someone else was selling the same shoes with a buy-it-now price of $49-a good-enough price in her opinion. Kathleen was effectively afforded a call option. She had the opportunity to buy the shoes at (the strike price of) $49, a right she could exercise until the offer expired.

The biggest difference between the option in the eBay scenario and the sort of options discussed in this book is transferability. Actual options are tradable-they can be bought and sold. And it is the contract itself that has value-there is one more iteration of pricing.

For example, imagine the $49 opportunity was a coupon or certificate that guaranteed the price of $49, which could be passed along from one person to another. And there was the chance that the $49-price guarantee could represent a discount on the price paid for the shoes-maybe a big discount-should the price of the shoes rise in the eBay auction. The certificate guaranteeing the $49 would have value. Anyone planning to buy the shoes would want the safety of knowing they were guaranteed not to pay more than $49 for the shoes. In fact, some people would even consider paying to buy the certificate itself if they thought the price of the shoes might rise significantly.

Price vs. Value: How Traders Use Option-Pricing Models

Like in the common-life example just discussed, the right to buy or sell an underlying security-that is, an option-can have value, too. The specific value of an option is determined by supply and demand. There are several variables in an option contract, however, that can influence a trader's willingness to demand (desire to buy) or supply (desire to sell) an option at a given price. For example, a trader would rather own-that is, there would be higher demand for-an option that has more time until expiration than a shorter-dated option, all else held constant. And a trader would rather own a call with a lower strike than a higher strike, all else kept constant, because it would give the right to buy at a lower price.

Several elements contribute to the value of an option. It took academics many years to figure out exactly what those elements are. Fischer Black and Myron Scholes together pioneered research in this area at the University of Chicago. Ultimately, their work led to a Nobel Prize for Myron Scholes. Fischer Black died before he could be honored.

In 1973, Black and Scholes published a paper called "The Pricing of Options and Corporate Liabilities" in the Journal of Political Economy, that introduced the Black-Scholes option-pricing model to the world. The Black-Scholes model values European call options on non-dividend-paying stocks. Here, for the first time, was a widely accepted model illustrating what goes into the pricing of an option. Option prices were no longer wild guesswork. They could now be rationalized. Soon, additional models and alterations to the Black-Scholes model were developed for options on indexes, dividend-paying stocks, bonds, commodities, and other optionable instruments. All the option-pricing models commonly in use today have slightly different means but achieve the same end: the option's theoretical value. For American-exercise equity options, six inputs are entered into any option-pricing model to generate a theoretical value: stock price, strike price, time until expiration, interest rate, dividends, and volatility.

Theoretical value-what a concept! A trader plugs six numbers into a pricing model, and it tells him what the option is worth, right? Well, in practical terms, that's not exactly how it works. An option is worth what the market bears. Economists call this price discovery. The price of an option is determined by the forces of supply and demand working in a free and open market. Herein lies an important concept for option traders: the difference between price and value.

Price can be observed rather easily from any source that offers option quotes (web sites, your broker, quote vendors, and so on). Value is calculated by a pricing model. But, in practice, the theoretical value is really not an output at all. It is already known: the market determines it. The trader rectifies price and value by setting the theoretical value to fall between the bid and the offer of the option by adjusting the inputs to the model. Professional traders often refer to the theoretical value as the fair value of the option.

At this point, please note the absence of the mathematical formula for the Black-Scholes model (or any other pricing model, for that matter). Although the foundation of trading option greeks is mathematical, this book will keep the math to a minimum-which is still quite a bit. The focus of this book is on practical applications, not academic theory. It's about learning to drive the car, not mastering its engineering.

The trader has an equation with six inputs equaling one known output. What good is this equation? An option-pricing model helps a trader understand how market forces affect the value of an option. Five of the six inputs are dynamic; the only constant is the strike price of the option in question. If the price of the option changes, it's because one or more of the five variable inputs has changed. These variables are independent of each other, but they can change in harmony, having either a cumulative or net effect on the option's value. An option trader needs to be concerned with the relationship of these variables (price, time, volatility, interest). This multidimensional view of asset pricing is unique to option traders.

Delta

The five figures commonly used by option traders are represented by Greek letters: delta, gamma, theta, vega, rho. The figures are referred to as option greeks. Vega, of course, is not an actual letter of the greek alphabet, but in the options vernacular, it is considered one of the greeks.

The greeks are a derivation of an option-pricing model, and each Greek letter represents a specific sensitivity to influences on the option's value. To understand concepts represented by these five figures, we'll start with delta, which is defined in four ways:

1. The rate of change of an option value relative to a change in the underlying stock price. 2. The derivative of the graph of an option value in relation to the stock price. 3. The equivalent of underlying shares represented by an option position. 4. The estimate of the likelihood of an option expiring in-the-money.1

Definition 1: Delta (?) is the rate of change of an option's value relative to a change in the price of the underlying security. A trader who is bullish on a particular stock may choose to buy a call instead of buying the underlying security. If the price of the stock rises by $1, the trader would expect to profit on the call-but by how much? To answer that question, the trader must consider the delta of the option.

Delta is stated as a percentage. If an option has a 50 delta, its price will change by 50 percent of the change of the underlying stock price. Delta is generally written as either a whole number, without the percent sign, or as a decimal. So if an option has a 50 percent delta, this will be indicated as 0.50, or 50. For the most part, we'll use the former convention in our discussion.

Call values increase when the underlying stock price increases and vice versa. Because calls have this positive correlation with the underlying, they have positive deltas. Here is a simplified example of the effect of delta on an option:

Consider a $60 stock with a call option that has a 0.50 delta and is trading for 3.00. Considering only the delta, if the stock price increases by $1, the theoretical value of the call will rise by 0.50. That's 50 percent of the stock price change. The new call value will be 3.50. If the stock price decreases by $1, the 0.50 delta will cause the call to decrease in value by 0.50, from 3.00 to 2.50.

Puts have a negative correlation to the underlying. That is, put values decrease when the stock price rises and vice versa. Puts, therefore, have negative deltas. Here is a simplified example of the delta effect on a -0.40-delta put:

As the stock rises from $60 to $61, the delta of -0.40 causes the put value to go from $2.25 to $1.85. The put decreases by 40 percent of the stock price increase. If the stock price instead declined by $1, the put value would increase by $0.40, to $2.65.

Unfortunately, real life is a bit more complicated than the simplified examples of delta used here. In reality, the value of both the call and the put will likely be higher with the stock at $61 than was shown in these examples. We'll expand on this concept later when we tackle the topic of gamma.

Definition 2: Delta can also be described another way. Exhibit 2.1 shows the value of a call option with three months to expiration at a variable stock price. As the stock price rises, the call is worth more; as the stock price declines, the call value moves toward zero. Mathematically, for any given point on the graph, the derivative will show the rate of change of the option price. The...