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Strategies & Market Trends : The Covered Calls for Dummies Thread -- Ignore unavailable to you. Want to Upgrade?

To: EnricoPalazzo who wrote (485)	5/9/2001 2:05:42 AM
From: tekboy	Read Replies (3) \| Respond to of 5205

I recall a rather odd scenario from one of my problem sets in college where the question was "should you use the arithmetic or the geometric average?". My TA told the section that I was the only person in the section to get it right. I was quite pleased, until I started to think about it. About halfway through class, I pointed out to my TA that I had in fact gotten the question wrong, and that everyone else had gotten it right. He argued back, and ultimately disagreed (the textbook was no help, as it gave contradictory answers in different chapters). About a week later, he emailed the section to say that in fact I was right in saying that I was wrong, and gave them credit for the answer--although I'm not sure if I got points taken off my problem set. Anyway, while I now believe that both answers are somewhat valid, depending on your attitude towards diversification, this certainly indicates that the question is a tricky one. BTW, that was the last time I went to section. 'course, at a decent college you'd have been taught by professors and not grad students... dtb

To: EnricoPalazzo who wrote (485)	5/9/2001 10:26:55 AM
From: Mathemagician	Respond to of 5205

If the stock goes down to $7.5, you get the stock put to you. This requires more cash than you have, so you borrow $500 (short-term, so you'll pay no interest). You then get 100 shares of stock worth $750, and owe $500, so you end up with $250. On the other hand, if the stock ends up at $15 or higher, you get to keep the whole premium, so you end up with $1000. Maybe you weren't considering the $500 you have to pay back if you lose? Nah. I was ignoring the real-world origin of your example and focusing on the abstraction alone. Nasty habit we math people have, you know? It's why we aren't engineers! :) Let's spend a little time clearly defining the game. Aside: This is turning out to be a pretty interesting exercise, so this is as much fun for me to watch as it is for you. :) In this game, my initial cash outlay is arbitrary. Call it X, where 0<X<=$1000. I'll "borrow" the rest of the $1000 - X using margin buying power, if necessary. If the stock finishes at 15, I wind up with X + $500 at the end of the game. If it finishes at 7.5, I have X - $250. In your examples, X=$1000, I either end up with $1500 or $750 for a return of 50% or -25% respectively; or X=$500 so I either end up with $1000 or $250 for a return of 100% or -50% respectively. The expected value of the game where X=$1000 is $15001/2 + $7501/2 = $1125, for a one-year return of 12.5%. The expected value of the game where X=$500 is $10001/2 + $2501/2 = $625, for a one-year return of 25%. The expected value of the game where X=$250 is $7501/2 + $01/2 = $375, for a one-year return of 50%. I believe we are on the same page. Agreed? (I hope so, else the rest of this post is a colossal waste of time.) Let's divide the whole game by $1000 to normalize the game so that 0<X<=1. That way, X represents the percentage of collateral put up as cash. Now X, a number between 0 and 1, is the cost to play. If the stock finishes at 15 (p = 1/2), the payout is X + 1/2. If the stock finishes at 7.5 (p = 1/2), the payout is X - 1/4. Define EVn(X) to be the expected value after n iterations (years). Define ERn(X) to be the expected total return after n iterations (years). EV1(X) = (X + 1/2)/2 + (X - 1/4)/2 = X + 1/8 ER1(X) = (EV1(X) - X)/X = 1/(8X) Now, ER1(X) is a decreasing function of X. (In fact, your expected return becomes arbitrarily large as X decreases to 0.) The conclusion is that the way to maximize your expected return in the first year is to put up as little collateral as possible. Confusion arises when we try to repeat the game in the following year. For example, If X=1 and I win, I have 1.5 to carry over to the next year. Do I play one game with X=1 and one with X=.5? Do I play two games with X=7.5? Do I play one-and-a-half games with X=1? Do I play fifteen games with X=.1? Even worse, what if X=.01 and I lose? How do I deal with a negative X in year 2? I don't know the answer to these questions offhand. (It may not even matter!) I'm very interested to hear your thoughts. Also, we are making several assumptions as we play this game, some of which are rather subtle. It may be worth our while to lay them out. What do you think? dM