So What's it Worth?
Yesterday afternoon, IBM closed at $98.33. Let's say I'm bullish on IBM and I want to buy some of the April 100 Call options. I check with my online broker and I see that the Calls are offered at $6.00. Well what kind of a deal is that? Just because they're selling at $6, doesn't mean they're worth $6. In fact, maybe the market makers have too much excess inventory of these options and are having a sale, selling them at $6 when they think they are worth $6.25 or $6.50. Alternatively, those rotten market makers (I can say that, I was one for 8 years) might think that they're worth $5.75 or $5.50 and they are trying to sell them for $6.
So the question I want to know is what is the fair or theoretical value of these options? At $6, am I getting a bargain or am I overpaying, relative to the theoretical value? The very fact that we have a way to answer this question is what makes it possible for traders to trade options successfully as a business, and to make consistent profits over time. It sounds like the key to our success is going to be in buying underpriced options and selling overpriced options.
Unfortunately, it's not quite that simple, or else we'd all be rich. Just because you buy an underpriced option or sell an overpriced option, doesn't guaranty a profit. Let's say that IBM option is worth $6.50 and I buy it at $6. Unless IBM is over $106 at expiration, I'm going to lose on that trade. Conversely, if it's really worth $5.50 and I overpay by buying at $6. I will make money if IBM is over $106 at expiration. Of course, over the long run buying underpriced options and selling overpriced options is the way to go. However, in addition to buying and/or selling options at favorable prices, you may also need to hedge the position and make adjustments as conditions warrant. We'll discuss those concepts at a later date. For now, let's see how to determine fair value.
Until 1973, there was no generally accepted method of calculating the fair value of an option. In that year, Fischer Black and Myron Scholes produced a model and a set of formulas for calculating the value of Puts and Calls in an article titled "The Pricing of Options and Corporate Liabilities" in the Journal of Political Economy. This work was so monumental that in 1997, Scholes received the Nobel Prize in Economics. Black would have shared in the award, but he died 2 years earlier.
The model was developed for European style options and assumes that the direction of the stock follows a random pattern and can not be predicted, although the distribution of the logarithms of the stock returns is assumed to conform to the Normal Distribution. Interest rates are assumed to be constant and commissions and other transaction fees are not taken into account.
The model uses a formula that takes 6 variables into account and calculates the Put and Call option values. The formulas are somewhat complicated although they can be easily programmed on almost any computer. Since the mathematics is beyond the scope of this article, I'm not going to reproduce the formulas here. They can be found in many options books, or if you prefer, send me an email and I'll send them right off to you. Professional Black-Scholes calculators are readily available in addition to the many free ones that are available online: see www.ivolatility.com, for example .
The variables that are needed as inputs to the Black-Scholes formula are:
1) The price of the stock.
2) The exercise price.
3) The time to expiration (expressed as a percentage of a year.)
4) The projected volatility of the stock from the date of calculation until expiration.
5) The risk free rate of return.
6) The annual dividend rate.*
*The original Black-Scholes formula did not take dividends into account. However, it was modified later in 1973 by Robert Merton in his paper titled "Theory of Rational Option Pricing."
If you look at these variables, you'll see that at a given point in time, anyone applying the formula would have the same values for the stock, exercise price, time to expiration, and the expected annual dividend rate. The risk free rate may vary slightly, some might use government bond rates, or CD rates, etc. But the one variable that is not "obvious" and the one that makes options trading an art as well as a science is in the determination of item number 4, the projected volatility. We haven't yet discussed volatility in this set of articles, but it's very important and will be discussed at length in a future article. For now, let's think about it intuitively. Stocks that have big swings and are consistently making large percentage moves, like RIMM for example, have high volatility, while stocks like JNJ which generally don't move very much are considered less volatile.
Choosing the right volatility assumption for use in the Black-Scholes formula is the subject of many articles and books and we'll also talk more about it in the future.
As a way to further our understanding of how a change in these variables would impact the Put and Call prices, let's figure out what would happen if we change one variable at a time, and left the others unchanged. The results are in the table that follows, but you'll get more out of it if you think about it first. Okay, start thinking.
These are the results you should have come up with. Some are pretty obvious and others may require some explanation.
As the: |
Call Values will |
Put Values will |
Stock price increases |
Increase |
Decrease |
Exercise Price increases |
Decrease |
Increase |
Time to Expire decreases |
Decrease |
Decrease |
Projected Volatility increases |
Increase |
Increase |
Risk Free Rate increases |
Increase |
Decrease |
Divided Rate increases |
Decrease |
Increase |
As stocks increase in value, Calls go up and Puts go down, assuming nothing else changes. When the exercise price increases, Calls become more out of the money and therefore decrease in value, while Puts become more in the money and hence increase in value. Long term options, Puts and Calls are worth more that shorter term options, again, all other things being equal. As projected volatility increases there is a higher probability of large moves in the stock price which gets reflected in the increase in the options values, both Puts and Calls.
Probably the hardest variable to think about changing is the risk free rate. Let me try to explain it like this. The Black-Scholes model assumes that stock prices will increase by the risk free rate. The reason is that a rational trader would not be trading if he didn't assume that he could earn at least that rate of return. Of course, in practice the trader may not earn the risk free rate and, in fact, could actually lose money. The key point is that he wouldn't be trading unless he expected to make the risk free rate. So if the risk free rate increases, then the stock price at any time is expected to be higher and we know that a higher stock price will increase Calls and decrease Puts. For example, if the stock is 100 and the risk free rate is 5%, then the model assumes the stock will be 105 after one year. If the rate increases to 6%, the expected stock price is now projected to be 106.
Finally, regarding dividends, options are generally priced to reflect expected dividends. However, since option owners don't receive any dividends, the impact of a dividend increase is reflected in the stock price opening lower on the ex-dividend date. That will decrease Call prices and increase Put prices.
Since the Black-Scholes model was introduced, many other models and methodologies have been produced to calculate option values. It has however passed the test of time and is still used, in a popular modified form, by many market makers and professional traders.
So the next time you want to buy or sell an option, run the numbers through a Black-Scholes calculator. Like I said before, it's not a guaranty, but it's a big step toward making your options trading profitable. By the way, I priced that IBM option and found the fair value to be $5.97, so the $6 offer seems like a theoretically fair price.
As always, if you have any questions about my articles, have suggestions for future topics, or want more information about our options mentoring program, feel free to email me at: SFreifeld@tradingacademy.com or call me at: (888) OTA-2580 ext. 2010.
11. Know Thy Options!
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Stan Freifeld comes to us from the Floor of the American Stock Exchange where he traded options for his own account from 1994-2001. He was a Market Maker for the options on several popular equities including Dupont, Schering Plough, Walgreen's, CBS, U.S. Surgical and Biovail.When he is not trading or thinking about trading, Stan relieves his stress by playing competitive squash, competing in local road rallies with his Ferrari Cabriolet and tutoring local high school students for the SAT's. The bottom line is that Stan, a long time MENSA member, is an engaging teacher with an extraordinary background in options trading and risk management. He is helpful and patient by nature and equally at ease with all levels of traders, from complete novices to advanced pros and academics. He'll be happy to teach you to trade!