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.