S.O.S. Mathematics CyberBoard

[bluesilver]

Using the concept of arbitrage, explain why the price of American call on a no-dividend-paying stock should be the same as the price of a corresponding European call. Why can the price of American call on a dividend-paying stock be higher than the price of European call?

[sean39]

Do you know the difference between an American call and a European call? I'd be more interested in helping you if you at least told me what you know and how you were thinking about the problem.
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"If I have not seen as far as others, it is because giants were standing on my shoulders." - Hal Abelson

[bluesilver]

I know the difference between European and Americal call (which is in the time expiration.) Unlike the holder of European call, the holder of American call can exercise the American option at any time up to the expiration date. I suspect that if the statement in the first question were not true, an arbitrageur could short European call and long American call to make a profit, but I'm not sure. The case that includes dividend is more complex to me. I should show why the arbitrage argument in the first question does not work in the second question.

[sean39]

The first statement actually isn't strictly true. There are rare situations besides dividend paying stocks where it is beneficial to exercise an option prior to the expiration date. Wikipedia lists a few.

http://en.wikipedia.org/wiki/Option_style

Because it is not exactly true, it is the first question that is harder than the second. If the American and European calls are not priced the same, it stands to reason that the American option is priced higher since it gives the holder the right to exercise early whereas the European option does not carry this right. So the arbitrage strategy would be to buy the cheaper European call and write the more expensive American call pocketing the difference. It is possible to create an unlikely scenario where you would lose out even with an ideal arbitrage strategy.

Let's say on 10/1/07, the stock of ABC Corp. is trading at $10, and ABC Nov 10 calls are at $1.00. This means that American call options on ABC Corp. with an exercise price of $10 and an expiration date of 11/15/07 are trading at $1.00. And say that identical European calls are available for $0.95. You write a contract for American calls, buy a contract for 100 European calls and pocket $5.00. Now let's say that on 10/15/07, the stock is at $10.01, and the American call is exercised. You now have to deliver 100 shares of ABC which you don't have. Your broker will buy 100 shares for you with $1,001 from your account and deliver them to the option holder the next day. You then instruct your broker to short 100 shares of ABC at $10.01, putting $1,001 back into your account. You are perfectly hedged because you owe 100 shares which you borrowed at $10.01 and you have the right to purchase 100 shares at $10.00, so you've locked in a profit of $1 no matter what happens. So let's say the price of the stock returns to $10 on 11/15/07. Your European call options expire worthless, but you make a profit of $1 on your short position in addition to the $5 you made on the original option combination.

But you now owe margin interest on the 100 shares that you borrowed for one month. If the interest rate on your margin account is 8% per annum, you owe $1,001 x 8% / 12 = $6.67, wiping out your $5+$1 = $6 profit and generating a $0.67 loss.

This is a convoluted and highly unlikely example, but it shows that American options are intrinsically more valuable, even when the stock pays no dividend. There are two reasons: (1) there are situations where it is rational to exercise early, and (2) there are times when the option holder acts irrationally, and even though it may have been irrational for the option holder to exercise early, it can still work to your disadvantage as the option writer.
_________________
"If I have not seen as far as others, it is because giants were standing on my shoulders." - Hal Abelson

[bluesilver]

Thank you for this very helpful message -- now I understand the first question! I really appreciate this. Can you please explain the situation in the second question? I was told that in the second question it should be shown that the "ideal" arbitrage argument used in the first question does not work in the second question because of dividend, but I do not understand how this is so. (I'm a newbie in this area, so I still have many things to learn, yet my textbook does not deal with the situation as described in the second question in depth.)

[sean39]

I didn't answer the second question because it was clearly explained in the Wikipedia link I provided. Makes me wonder whether your book is as bad as you say.
_________________
"If I have not seen as far as others, it is because giants were standing on my shoulders." - Hal Abelson

[bluesilver]

I looked at the Wikipedia link, but it does not tell why the arbitrage argument does not work in the second question (when there is dividend-paying stock). It talks that it is optimal to exercise American call on a dividend-paying stock before the expiration date. Or, am I looking in the wrong place?

[sean39]

It is not going to spell out the exact arbitrage play that you are looking for. I already gave you the arbitrage trade. If you go with the less than perfect assumption that it is never beneficial to exercise early and investors always act rationally, then the arbitrage play, as I said above, is to buy the cheaper European calls and write the American calls pocketing the difference. Since the American calls will never be exercised early, you can't lose.

Now, what happens if the stock pays a dividend? The Wikipedia article talks about this. A big hint is that a dividend creates a potentially legitimate reason for investors to exercise early. Now all you have to do is reread the article and/or your book and figure out why and how this could eliminate the arbitrage opportunity.
_________________
"If I have not seen as far as others, it is because giants were standing on my shoulders." - Hal Abelson