Black-Scholes (and Merton) solved this unpromising looking equation
with the formula: 
which calculates the expected value of a call.
Graphically, the expected value can be visualized as the weight on a see-saw that would just balance the weight of a distribution of outcomes for the price of the stock above the Strike price
Before I can say "that's all there is to it" there are a couple of more points.
First, the expected value the Black-Scholes model calculates is for a distribution which is modified to eliminate expected appreciation of the underlying equity. Why? That's more than I want to explain. For a start, consider just buying a future on a stock. What should be the price? If you think it has anything to do with expectations of stock value appreciation, WRONG! If you borrow the money, buy the stock today, and then repay the loan on the date of the future, you would only need to borrow the price of todays stock plus interest to make that transaction. The price of the future is NOT what you expect, it is governed by arbitrage equilibrium. The same applies in a more complex way to pricing options. The price of an option is govened by the price a market maker could set if they chose to synthesize the option through a pattern of borrowing, buying the equity, and continuously hedging. The outcome of their process is predicted by Black-Scholes and is the price of the option.
Second, the Black-Scholes model assumes the distribution of outcomes of the price of the underlying equity is log-normal. The Black-Scholes formula itself is really just a standard closed form expression for the integral from Strike price to infinity of price times the probability distribution of price.
While useful as a check to see if the market makers are being fair, an investors interest is not in calculating the price of an option, that is set by the market. The investors real concern should be expected return. Except for the fact that Black-Scholes has added drift compensation terms, expected return is exactly what Black-Scholes is calculating. The investor can use the Black-Scholes equation to determine a real life, non-risk adjusted expectation by simply replacing the factors which are used to set the drift compensation.
This is what you get when you make the substitutions. 
The substitutions are: use R_ new= mu + 1/2)sigma_squared in place of r, the risk free bond rate and use k*exp(-r*T + r_new*T) as a strike price. (mu is the growth rate of the equity, sigma is the variance, T is the time to expiry.) With these substitutions, any of the Black-Scholes calculations should give the real measure expected payoff for the option.
Alternately, it is possible to calculate the return either numerically, or in closed form, by finding the integral from Strike to infinity of price times your favorite probability distribution, with mean and sigma reflecting your outlook on the performance of the underlying equity.
I find the results are suprising. In spite of many claims, covered calls do not produce an expectation of profits for the seller. In fact, it requires considerable optimism to create scenarios in which there are any significant (amount of) expectations at all, and generally there is a very small loss expected for strikes above spot. I conclude that most covered calls have expectation characteristics similar to gambling in a casino which keeps a small, fair edge for the house.
References
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Crafted by John Bailey June 16, 1998 updated December 15, 1998 All rights reserved. |