Why The Probability Distribution is Needed to Compute Expected Return

Consider the following potential 'investment'.  If you buy 100 shares of almost any company, and then sell a one month call option on that stock at the same price you paid for it  (a strategy commonly referred to as a covered call) and if you assume that the price of the stock does not change, you will receive a premium from the buyer of the option that, if annualized, will typically yield a return in the range of 20% or more.  Sounds too good to be true?  It is too good to be true.  Most people will agree that the best guess for the price of a stock tomorrow, or even one month from now is it's current price.  So is the 'no change' assumption reasonable for estimating the annualized return on a covered call option?  Not really.  Even though the current price of a stock is the stock's most likely price tomorrow or a month from now, it is far more likely that the price will be either more or less than the current price;  however, we are uncertain as to whether it will be more or less.  Furthermore, even if we knew that it would be more or less, we would still need to know how much more or less in order to compute the expected annualized return.  If we knew how the probability was distributed over all possible outcomes (or what is know as the Probability Distribution), we would be able to tell if we were gambling or investing.


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