Actuarial Outpost > MFE quick question on american calls
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#1
02-20-2012, 01:07 AM
 xiaoxin Member SOA Join Date: Jun 2011 Studying for C College: '11 Favorite beer: Hoegaarden Posts: 490
quick question on american calls

Since an American call option may be exercised at any time, the option must be worth at least as much as its exercise value S-K.

I fail to understand why being exercised at any time makes it more valuable. That sentence isn't intuitively clear to me.

Someone want to help out?
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#2
02-20-2012, 01:23 AM
 Actuarialsuck Member Join Date: Sep 2007 Posts: 5,324

Because your "S" changes, for a European option with expiry time "T", that stock price is fixed at $S_T$. What if there is some $S_t$ such that $S_t > S_T$ (where $t < T$)? Clearly being able to exercise at t makes it "better".
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#3
02-20-2012, 01:42 AM
 xiaoxin Member SOA Join Date: Jun 2011 Studying for C College: '11 Favorite beer: Hoegaarden Posts: 490

They meant S-K as in S_t - K, so the stock at time t. The option expires at time T>t.

I guess it's similar to what you said except in this case, the stock may go up after time t, so that's why the option value is at least S_t - K.
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#4
02-20-2012, 02:57 AM
 Actuarialsuck Member Join Date: Sep 2007 Posts: 5,324

Yes my point was to get you to think about the fact that it's always better to have freedom in case the stock rises beyond what it is currently / what it would be in the case of European options.
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#5
02-20-2012, 08:04 AM
 LifeIsAPoissonProcess Member SOA Join Date: Aug 2010 Studying for C Posts: 3,385

What if there are huge dividends paid at S_t+.0001? You don't own the stock, so you won't get the dividends. You will however, get a near-worthless option since the amount of the dividend is subtracted from the stock price. In that case, you may want to exercise early in order to reap the benefit of the dividend.
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#6
02-21-2012, 02:03 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 1,360

An important result is that if there are no cash flows from the asset, then the option to exercise the call early has no value and you can value an American call as a European call (i.e. using the BS formula).
#7
02-22-2012, 10:15 AM
 XrossTrain Member Join Date: Jul 2011 Location: Vermont Studying for Life College: UConn-Alum Posts: 45

lets say a stock price over the period of years is like 50,65,75,60

A European option can only exercise at the end (when stock is 60), but an American option can be used even before then lets say at stock price 75, making it more useful and valuable.
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#8
02-22-2012, 12:09 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 1,360

Quote:
 Originally Posted by XrossTrain lets say a stock price over the period of years is like 50,65,75,60 A European option can only exercise at the end (when stock is 60), but an American option can be used even before then lets say at stock price 75, making it more useful and valuable.
The problem with your example is that the market does not know what the next price will be. If it knew that the price in the next period was 60, the current price would not be 75. One can only use what is known at the present to value an option.
#9
02-23-2012, 09:52 AM
 XrossTrain Member Join Date: Jul 2011 Location: Vermont Studying for Life College: UConn-Alum Posts: 45

Yes I realize my mistake now.

I guess American options are only more valuable in terms of dividends on the stock > interest on the strike paid
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#10
02-24-2012, 10:23 AM
 BlackSwans Member Join Date: Sep 2011 Posts: 418

what conclusions could be made if the opposite was true- if a European call option (same expiry, strike, etc.) was strictly more expensive then the corresponding American call? (hint:arbitrage. Furthermore, what can you conclude if there exists an arbitrage possibility??)
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