Pricing Options

The Key Factors in Determining an Options Price

To anyone just becoming familiar with options, understanding how options are priced can be one of the greatest challenges. Until the early 1970s, only cumbersome mathematical models existed to help traders determine the theoretical value of an option. Since these models involved complex equations and market prices changed constantly, they didn't prove especially practical.

In 1973, two University of Chicago mathematics professors, Fischer Black and Myron Scholes devised a model that even today remains a standard for many option traders. The Black-Scholes Model, as it is known, was such an important advancement that the professors earned a Nobel Prize for their work.

Theoretical Value

The Black-Scholes Model, or any model for that matter, is not always representative of what happens in real life. Models are limited by the numerical inputs used to calculate the theoretical value of an option. They will never be able to take into consideration qualitative factors like market sentiment. For this reason, the theoretical prices and actual market prices may bear little resemblance.

Having said that, let's look at the quantitative factors that impact an option's theoretical value:

• Strike price
• Time remaining until expiration
• Price of the underlying stock
• Volatility of the underlying stock
• Interest rates
• Dividends

Strike Price and Intrinsic Value

The strike price plays a significant role in the market price of an option because it determines whether an option has any intrinsic value. For example, if the underlying stock is trading at $84 an 80 call will have $4 of intrinsic value because it gives the call owner the right to buy the stock for $80. At the same time, the 80 put will have no intrinsic value because it doesn't make sense to sell a stock for $80 when it can be sold for $84 on the open market. In this situation, whatever value the put has will be purely extrinsic (time) value.

As you can see on the table below, the closer an option is to the current stock price, the more extrinsic value it has. Conversely, the further in- or out-of-the-money the option is, the lower its extrinsic value.

AT&T Corporation (NYSE: T)
Stock Price: 19.00

Option	Price	Intrinsic Value	Extrinsic (Time) Value
Calls
Dec 12.50	6.60	6.50	0.10
Dec 15.00	4.20	4.00	0.20
Dec 17.50	1.90	1.50	0.40
Dec 20.00	0.55	-	0.55
Dec 22.50	0.15	-	0.15
Dec 25.00	0.10	-	0.10
Jan 12.50	6.70	6.50	0.20
Jan 15.00	4.40	4.00	0.40
Jan 17.50	2.05	1.50	0.55
Jan 20.00	0.75	-	0.75
Jan 22.50	0.25	-	0.25
Jan 25.00	0.10	-	0.10
Puts
Dec 12.50	0.05	-	0.05
Dec 15.00	0.10	-	0.10
Dec 17.50	0.35	-	0.35
Dec 20.00	1.50	1.00	0.50
Dec 22.50	3.70	3.50	0.20
Dec 25.00	6.10	6.00	0.10
Jan 12.50	0.10	-	0.10
Jan 15.00	0.15	-	0.15
Jan 17.50	0.65	-	0.65
Jan 20.00	1.90	1.00	0.90
Jan 22.50	3.90	3.50	0.40
Jan 25.00	6.30	6.00	0.30

Deep-in-the-money options tend to move on in tandem with the underlying stock. Thus, when the stock moves $1, the option value also changes by $1. With deep out-of-the-money options, the situation is a bit different. Since it would take a significant move in the underlying stock to increase the likelihood that a deep out-of-the-money option finishes in the money, people aren't usually willing to pay much for them. As a result, they tend to have low extrinsic value. At the same time, these options have no intrinsic value.

Time Remaining Until Expiration

Unlike the strike price of an option which remains fixed, the time remaining until expiration changes over the life of the option. As an option approaches expiration, the time value tends to decrease more rapidly. The rate at which an option loses value is often referred to as theta.

Time value is also known as extrinsic value because it represents the premium people are willing to pay above and beyond an option's intrinsic value. For example, in the table below, a December 20 call for AT&T is considered out-of-the-money because the strike price (20) is higher than the current market price ($19.00). As such, the price of the December 20 call ($0.55) consists exclusively of time value.

AT&T Corporation (NYSE: T)
Stock Price: 19.00

Option	Price	Intrinsic Value	Extrinsic (Time) Value
Calls
Dec 12.50	6.60	6.50	0.10
Dec 15.00	4.20	4.00	0.20
Dec 17.50	1.90	1.50	0.40
Dec 20.00	0.55	-	0.55
Dec 22.50	0.15	-	0.15
Dec 25.00	0.10	-	0.10
Jan 12.50	6.70	6.50	0.20
Jan 15.00	4.40	4.00	0.40
Jan 17.50	2.05	1.50	0.55
Jan 20.00	0.75	-	0.75
Jan 22.50	0.25	-	0.25
Jan 25.00	0.10	-	0.10
Puts
Dec 12.50	0.05	-	0.05
Dec 15.00	0.10	-	0.10
Dec 17.50	0.35	-	0.35
Dec 20.00	1.50	1.00	0.50
Dec 22.50	3.70	3.50	0.20
Dec 25.00	6.10	6.00	0.10
Jan 12.50	0.10	-	0.10
Jan 15.00	0.15	-	0.15
Jan 17.50	0.65	-	0.65
Jan 20.00	1.90	1.00	0.90
Jan 22.50	3.90	3.50	0.40
Jan 25.00	6.30	6.00	0.30

Not surprisingly, the January 20 call is worth more than the December 20 call because it has an additional month of time value before expiration. In other words, because the stock has an extra month to move beyond the 20 strike price, people are willing to pay more for the January 20 call than for the December 20 call.

At-the-Money vs. Out-of-the-Money

Another important point to notice on the table above is the relationship between the strike price and the time value of the options. With the stock at 19.00, the 20 strike is considered the most at-the-money. For puts and calls alike, this strike has the highest time value. As we move further away from the current stock price, in either direction, the time value decreases. For example, the deep in-the-money December 12.50 calls have only 0.10 cents in time value while the at-the-money December 20 calls have 0.55 cents.

This makes sense when you consider that the time value is nothing more than the price that people are willing to pay for the chance that an option will finish in-the-money. An option that is far out-of-the-money with almost no chance of finishing in-the-money won't command a particularly high price. Similarly, an option that is already deep-in-the-money can be readily exercised and converted to stock. For these reasons, the contracts that tend to trade most frequently are the options that are at or near-the-money. In a sense, these strike prices command a higher extrinsic value because there is more uncertainty as to whether or not the options will finish in-the-money.

Price of the Underlying Stock

The price of the underlying stock impacts option prices in a number of ways. First, and most basic, the relationship between the stock price and a given strike price determines whether an option has intrinsic value, extrinsic value, or both.

When the strike price is below the current market price, calls will have intrinsic value and puts will not. In the table above, the January 12.50 calls have 6.50 points of intrinsic value while the January 12.50 puts have no intrinsic value at all. Like the January 12.50 calls, the December 12.50 calls have 6.50 of intrinsic value (19.00 - 12.50), but less extrinsic value (0.10 vs.0.20) because there is less time remaining on the contract. In some cases, deep-in-the-money options have intrinsic value but no extrinsic value. In this situation, the options are said to be trading at parity. For example, if the December 12.50 calls were priced at 6.50 rather than 6.60, they would be at parity.

Hedging and Theoretical Value

When using a theoretical model like Black-Scholes, the exact price of the underlying is important because it impacts the price traders are willing to pay for any given option. For example, it isn't enough to know that the stock is trading at $45.50 because a trader may need to buy or sell stock to offset option contracts. For example, if the market for the stock is really 45.25 - 45.50, a trader will only receive 45.25 selling the stock. Thus, the option values should be based on a stock price of 45.25 rather than 45.50. This seemingly small spread can make the difference between a profitable and an unprofitable trade.

Volatility of the Underlying Stock

The volatility of the underlying stock may be the most important factor in pricing options because unlike the other numerical inputs which have an exact value, volatility can only be known with certainty from a historical standpoint. At any given moment, the strike price, the current market price, and a few relatively minor factors we haven't examined yet (i.e., prevailing interest rates, stock dividends) are all exact numbers that people know and agree upon. With volatility, that isn't the case.

What is volatility?

In simplest terms, volatility is the tendency of a stock to fluctuate and the likelihood that it will be within a particular price range at a specific moment in time. The higher the volatility, the more prone a stock is to large price swings. Conversely, low volatility stocks tend to show a history of stable prices.

Low Volatility

Some stocks, like utilities, tend to be relatively stable over time because their earnings are relatively predictable. People who invest in these stocks often do so for the slow, steady growth and consistent dividends. At the same time, they want secure investments they don't have to monitor everyday. With these low volatility stocks, the daily price changes are generally fractional. While the long-term trend may be up, the short term trend may even appear to be sideways. A good example of this is Ameren (NYSE: AEE), a large Midwestern utility. Looking at Ameren from the point-of-view of an options trader, we see a 52-week range between $46.50 and $37.43. Given this level of stability, even an untrained chart reader could predict the price range over the subsequent months with a high degree of accuracy. Considering the low likelihood that the stock would deviate from this pattern of low volatility, the market for options on this stock was quite small in terms of volume and price. This is confirmed by the options chain:

Ameren (NYSE: AEE)
Stock price: 44.67

Calls	Price	Vol.		Puts	Price	Vol.
Dec 40	4.90	0		Dec 40	0.25	0
Dec 45	0.80	0		Dec 45	1.50	0
Dec 50	0.20	0		Dec 50	6.10	0
Mar 40	5.00	0		Mar 40	0.65	0
Mar 45	1.15	0		Mar 45	2.60	0
Jun 40	5.00	0		Jun 40	1.15	0
Jun 45	1.55	10		Jun 45	3.30	0
Jun 50	0.30	0		Jun 50	7.30	0

High Volatility

Other stocks, like Internet and biotechs tend to have larger daily price swings. The bigger the price swings, the more volatile the stock. When assessing stock volatility, traders look at a particular period of time (e.g., 90 days). However, it may be necessary to look at volatility over a shorter period, particularly when recent developments change the long-term outlook for a company.

From an options trading standpoint, it makes sense that people would be willing to pay more for options on a stock that has a higher likelihood of making a profitable move during the life of the option. As we can see, that's exactly what happens.

Let's take EBAY, Inc. (Nasdaq: EBAY) as an example. Here's a stock that had some fairly significant price swings in a relatively short period of time. The 52-week range on this stock was from $30.88 to $61.60. Given this level of volatility, it stands to reason that options on this stock would be significantly more expensive than they would for a stable utility like Ameren. Again, this is confirmed by the option chain:

EBAY, Inc. (Nasdaq: EBAY)
Stock price: 56.35

Calls	Price	Vol.		Puts	Price	Vol.
Dec 40	16.60	149		Dec 40	0.15	120
Dec 45	11.70	173		Dec 45	0.25	233
Dec 50	7.20	413		Dec 50	0.80	177
Dec 55	3.60	517		Dec 55	2.20	34
Dec 60	1.45	103		Dec 60	5.00	0
Dec 65	0.50	477		Dec 65	9.10	0
Dec 70	0.15	9		Dec 70	13.80	13
Jan 40	16.80	9		Jan 40	0.35	0
Jan 45	12.20	47		Jan 45	0.65	223
Jan 50	8.00	10		Jan 50	1.50	100
Jan 55	4.60	47		Jan 55	3.10	67
Jan 60	2.30	149		Jan 60	5.90	16
Jan 65	1.05	14		Jan 65	9.70	10
Jan 70	0.50	30		Jan 70	14.00	7

As you might imagine, there are several advantages to trading options on volatile stocks. As we've already discussed, there is a greater likelihood that the options will finish in-the-money. Although the options tend to be more expensive, they also tend to be more liquid. This is an important consideration because whether you are getting in or out of the market, you want to get the best price. The more frequently the contracts trade, the more likely that market competition will maintain a tight bid-ask spread.

If for some reason, the actual volatility of the options decreases, the options will lose value faster than their less volatile counterparts. However, that's a known risk most traders are willing to take. In fact, many traders make their fortunes selling options when volatility is high and covering their positions when the market becomes less volatile.