A Beautiful Equation
Many mathematicians would agree that one of the most beautiful equations in mathematics is:
It ties together several of the basic symbols upon which the foundation of mathematics is built:
Of course, if you're a physicist, then E = M C2 might be more appealing.
These are both truly awesome equations!
Well, we're options traders and we have a favorite equation as well! It's called the equation of Put/Call parity. Here it is:
C – P = S –X + i – d
Where:
C = Call option price
P = Put option Price
S = Stock price
X = eXercise price
i = cost of carry
d = present value of dividends
This relationship between the Put, Call and stock prices originally appeared in a paper by Hans Stoll entitled "The Relationship Between Put and Call Prices" in 1969. The formula was developed for European options (those are the ones that can't be exercised early) but it also applies pretty well to American options. The reason is because unless one of the options is really deep in the money or the dividend is extremely large, the value of the early exercise component in the American style options is generally small and doesn't have much impact on the formula.
In my opinion, it is very important that you understand this basic relationship between Puts, Calls and Stock. I am constantly amazed when I talk to people who have been trading options for any significant length of time and I realize that they don't know there is a connection between the prices. I guess on one level that's okay, it helps me pay my son's college tuition!
Let's see how to calculate the i and d terms, and then we will apply the formula to some examples. The cost of carry, i, is calculated by multiplying the risk free rate of return by the exercise price times the number of days to expiration divided by 365. For our purposes, this risk free rate is what you can get on an investment with only a minute possibility of default, such as short term CD's. Traders, I know have recently been using about 5%, although after the last 2 rate cuts, perhaps a lower rate like 4.5% may be more appropriate.
To calculate the d, we must take into account any dividends that are being paid prior to expiration and calculate their present value. All that means is we must divide the dividend amount by (1 + risk free interest rate) raised to a power. That power is the time to expiration divided by 365. By the way, since the dividend amount is usually small relative to the stock price and with interest rates being as low as they currently are, most traders just use the actual dividend amount instead of the present value of the dividend, in the formula. It has a minimal effect on the prices.
I know that this all sounds complicated, but wait until you see how easy it is to apply the formulas!
Let's say we know the following about XYZ stock and options:
It's trading @ $52, and doesn't pay a dividend
There are 37 days to Jan expiration
The Jan 50 Call is trading @ 5.50
My risk free rate of return is 5%
The question is, how much should the Jan 50 Put be trading for?
First, calculate i as .05 x 50 x (37/365) = .25 We also know that d = 0, so our equation becomes;
C – P = S – X + i - d
Or 5.50 – P = 52 – 50 + .25 – 0
5.50 – P = 2.25
Finally, we get P = 3.25
Just for fun, let's assume that XYZ did have a 40¢ dividend that was payable in 25 days. I would calculate the present value of the dividend as .40/(1.05)^(25/365) = .399, i.e. .40 (Now it should be apparent why traders would just use the value of the dividend.) Substituting .40 for d in the above equation and solving, we get P = 3.65. In a like manner, we could calculate the Call if we knew the Put price, or for that matter the Stock price, if we knew where the Put and Call were trading.
Okay, this is all very nice, but now that I know it, how do I use it in my trading? That's the $497,097 question ($64,000 increased for inflation.) First, you should feel good knowing that you know and understand something that probably 80% of non-professional options traders don't know. Second, if the prices are out of line, then something may be going on. You may not be able to short the stock, or it may be hard to borrow, or some corporate action such as a merger or takeover may be pending. Finally, if none of these things apply, and the prices are out of line, there is a way to capture the difference.
If the Put is trading for less than the equation says it should be, relative to the Call, we can buy the Put, sell the Call and then lock up our profit by buying stock. Similarly, if the Call is undervalued relative to where the Put is trading, we can buy the Call, sell the Put, and again, lock in the profit by shorting the stock. These positions are called Conversions and Reverse Conversions (or Reversals), respectively. A future article will describe in detail how these positions lock in the profit, but for now it would be educational for you to think about it. Oh, before you get too excited about finding these mis-pricings and making lots of money, you should realize that they don't come along all that often anymore. Also, when they do appear, it's only for a short time, so you have to react quickly, since selling the overpriced options and buying the underpriced options brings the prices back into line rather quickly.
Ten years ago, when I was trading on the floor of the American Stock Exchange, I was able to do them quite frequently, but alas, that was then.
Until next time, good luck with your trading. "Be careful, it's a jungle out there."
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!