Value Is In The Eye Of The Beholder
Many option traders are befuddled by the notion of “theoretical value.” The term “theoretical” is imposing, as are the calculations needed to derive the theoretical value. Many traders ask how the theoretical value can be so low (or so high) if the underlying is poised, under some hypothesis they may embrace, to make a large move up (or down).
Suppose stock XYZ is selling for $50, a call on XYZ with strike price 50 is expiring in five trading days, and the historical volatility of XYZ is 30%. How much is that call worth? Using the Black-Scholes formula, you will find that it is not worth very much, around $0.87.
Suppose, however, you have knowledge of a bulletproof takeover offer paying $70 a share for XYZ stock that will be made public two days from now. How much is the call worth to you?
The call obviously is worth far more than .87 cents–in fact, its value is close to $20. Assuming you are absolutely confident about your information, you would certainly pay $10 for the right to purchase XYZ anytime in the next week for $50, and would probably pay $15, perhaps even $17 or $18, or maybe even $19.
So you would assign a value to this call far greater than would Fisher Black and Myron Scholes. Who would be right? Assuming your information is correct, you would certainly be right. But given the information available to them, Black and Scholes would also be right. Let’s see how that can be.
Say I have just tossed a coin and looked at the result. I see heads. I ask you to tell me the chance that the coin has come up heads, but I don’t allow you to look at the coin. Your best estimate of the chance the coin is showing heads is 1/2, meaning that if you guess heads every time we perform this little experiment you will be right about half the time.
However, for me the probability of heads for is not 1/2, but rather 1; and if my toss had turned up tails, it would be 0. A probability is merely a numerical measure of your uncertainty; your measure equals 1/2 and mine equals 1. And we are both right, meaning that if we perform this little experiment many times, in the long run the results will be consistent with these numbers–I will always be right, and you will be right 50% of the time.
The Black-Scholes model, which is widely used to evaluate options, is a market-neutral model. It does not presume to know what the market is going to do, and so it takes a neutral stance. And it assigns a very low probability to the event that XYZ will close at or above $70 in one week. And the conglomerate of low probabilities assigned to prices far away from today’s price of $50 causes the model to arrive at a value of only $0.87. Value is in the eye of the beholder and is a function of his or her information.