American vs European option values

Look on the sentence and say if you agree or not:

“The possibility of early exercise is not valuable for calls on non-dividend paying stocks, so American and European calls have exactly the same value”.

I do not agree. An option of exercise is always an option. On the 1st level it was explained that arbitrage makes the floor value of MAX(0,S-X) for American call but MAX[0,S-X/(1+r)^T] for European one. It looks contradicting.

Am I right, and should I just treat above quote to be wrong? Because it is from reliable source…

In theory, early exercise is never optimal for American call option on non-dividend stocks. Hence, early exercise has zero value.

Consider that MAX[0,S-X/(1+r)^T] > MAX[0,S-X] for positive r. So these formulas don’t show that the value of an American call option is greater than the value of a European call option.

To show that early exercise has no value for American call options, consider these two scenarios for an American call option (assuming ITM, r >= 0 and zero volatility):

  1. Exercise today: value is S-K

  2. Exercise at time T: Since forward price of the stock is S*(1+r)^T, value at time T is S*(1+r)^T - K. Value today is S-K/(1+r)^T.

Since S-K/(1+r)^T >= S-K, late exercise is optimal.

What benefit do you get if you exercise early? The only time it makes sense is when option liquidity is a problem…other than that it makes no difference, European or American. In fact, even if it was dividend-paying stock, it makes no advantage, contrary to what the book says.

Wow guys, thanks a lot, it seemed strange to me but you are right.

To self-explain in most simple way is that a call is the right to buy something paying nominal price which is constant in the future. Because of time value of money it is always better to pay the same nominal price later than earlier. So it is better to exercise later.

So it seems American calls on non-dividend paying stocks make no sense if markets are liquid. Their floor of MAX(0,S-X) makes no sense as well as more complicated arbitrage makes a floor of MAX(0,S-X/(1+r)^T) also for American call.

Yeah, I think you got it. That’s pretty good - a lot of people just gloss over this part.

Dreary: In theory, early exercise of American call options can be optimal for dividend paying stocks. Consider the extreme case where a stock pays a high dividend, like 80%, At the ex-date, S falls to zero. So intuitively, S-K will have a very small or negative value on the ex-date. Naturally, you would want to exercise early to avoid the stock price drop on the ex-date.

I’m too lazy to write a formula with a discrete dividend, so consider this modification to my previous post. Now, we assume a annually compounded dividend, q, and we can exercise the option now or at time T.

  1. Exercise today: value is S-K

  2. Exercise at time T: Since forward price of the stock is S*(1+r-q)^T, value at time T is S*(1+r-q)^T - K. Value today is S*[(1+r-q)/(1+r)]^T - K/(1+r)^T.

It’s harder to tell from this formula, but for large values of q, S-K > S*[(1+r-q)/(1+r)]^T - K/(1+r)^T. Early exercise would be optimal in these cases.

ohai, theory is not in line with reality! Do this: If you think the option price is going to drop because the underlying is going to drop by the amount of the dividend, short the call! Reality is that the options market prices in “everything”.

Ignore stuff like exchange adjustments for now. I could just write you an American call option today, and the trade would have to adhere to this theory (American options can be OTC). You would not be able to make money by shorting this call, as the buyer would execute before the ex-date. This is what’s on the test, anyway. So you have to accept it for now.

Not true Ohai. If all buyers execute, open interest would be zero. It never happens that way. I’m not talking about the test…I’m just pointing out that theory is wrong.

Early exercise does happen - just not very often. Plus you are assuming that the theory only applies to exchange options, which have certain adjustment rules. Like I said before, we can enter into an OTC American option contract and there would be no reason for the theory to be wrong.

There is no exchange adjustment, it is the market that makes the adjustment. Exchange adjustments occur for a different reason. OTC options between private parties is no different, plus they don’t take place on market-available options. Why would I enter an options deal with you over IBM, if I can do it in a free market style?

From what I learn in B-school, the ability of excercising your option can only be a positive, since it’s giving you more option when compare to a Euro-option. Everyone pays a premium in order to gain flexiblility in the financial market (no free lunch). As Ohai mention a large dividend (lets say worth 90% of stock value) would make most investor use there american option and excercise early. Also lets say if it was a binominal tree and the expected price of the second period is higher than the third period (no natural probability), then obviously the investor would excercise their option early and use there it in the second period.

If exercising is valuable, then the option itself will be valuable! Yes more flexibility is good, and in that case I agree that having an American-style option is better, but the point I’m arguing is that the call option will not lose value once a large dividend is announced, nor will you miss the dividend if you don’t exercise

  1. Assume a stock is trading at $60, and you hold an option with a strike of $50, expiring in one month, probably worth about $15.

  2. The company announces a dividend of $40 payble in two weeks.

  3. What will happen to the stock price? What will happen to the call option price?

  4. Stock price may go up as some people think they are getting free money, which they are not of course, but let us say the stock jumps to $65.

  5. What will happen to the call option price, trading currently at $15? It will also jump to about $20.

  6. You may choose to exercise today, and hold the stock in your account if you choose to, but what are you gaining? I say nothing.

  7. Let us say you don’t exercise, so you keep on holding the call option, and now it’s almost 4 pm on the day when tomorrow the stock goes ex-dividend. That means if the stock closes at $65, tomorrow it will open at $25.

  8. What will you do seconds before 4 pm? What do you think the option price is at around that time?

Food for thought!

Well if you had an European option and it expires at 5 pm then it would be worth $0 compare to a price of $15 for the American option with a same expiry time. Your American call option that expires at 5 is worth something, but your European option is going to be worthless, assuming that people know that the price of the stock will be less than 50 after the dividend payment. It also proved the point that the ability to excercise an option earlier when it’s in the money is something that’s worth something. The whole point is if all else is the same American option has to be equal or great due to the added flexibility. Put call parrity uses So. and we don’t know what will happen in the future, so the price of stocks going up or down in the future has no effect on the current price of the call option (obviously this is unrealistic).

Also let’s say you have an American option with 3 periods. In period 1 your stock is 15 and excercise price is 15. Next period there is 2 different payoffs. One is for $30 the other is for 10. Let say in period 2 it was 30 dollars and the payoff for period 3 is 40 with a 10% probability and 5 dollars with a 90% probability (no natural proability). Since the expected payoff is 9.5 wouldn’t you want to excercise your American option in period 2? If you had an European option you could only excercise it in period 3. Wouldn’t the European option be worth less than the American option.

I’m not arguing against teh benefit of flexibility in some ways, and I don’t really know first hand how European options work. My only argument is when the book says it’s good to exercise early in order to get the dividend, implying that if you don’t exercise you lose! That I don’t believe, and I have seen it in reality, it makes no difference. The option you’re holding is still worth the correct amount and you can always “sell” it and not lose anything.

Sorry had to sleep for a little (5 -6 am when I was typing yesterday).

Anyways an option is like a Binomial tree where you can expand (call) or sell (put) the business.

*Next part is absolutely useless for CFA, but food for thought.*

If you have an American option it means that you have more than one oportunity to expand your business. You can expand it at period 1, 2, 3. Had to do this for my undergrad MT. Bascially you first calculate the business without expansion. After that you go to period 3 and see what happens to the firm value if you excercised your “expand” (call) option. The difference in price would be the value of an option at period 3. You then use both the options. The difference between the value of 1 option and 2 option is the value of an option in period 2. The is equal to the difference between an American and an European option. Obviously it could be 0 and the option are worthless (out of the money).

The Black Scholes is basically makes the Binomial tree into a model with infinite branches (that’s why we use e^). From what I heard from my prof who went to Booth, in real life people will just use C++ to make an infinite period model. A lot more robust because they’re allowed to control and change variables in each of the infinite periods.

Also using an option is just increase the leverage for a stock. Let say you pay 3 dollar for a call at a strike of 15. If it’s less than 15 you have all of your capital -100% return. If it goes to 20 then your return would be 67% (5/3). If you just bough the stock at $15 and the stock dropped by a $1 your return would be 6.7% (-100% for option). If it were to increase to 20 your return is only 33%.

Also I think what you saw was an European option. They expire automatically and once it expires you just get the difference between strike place and stock price automatically. You don’t have to “use” it since it’s automated. Most of the option places I have seen don’t actually give you the stock, but instead just give you a difference in strike and the market price. Easier to manage for these websites, they don’t want to be the one liable to storing the stock and strike prices for it’s cusomter.

I hope you can see it from my POV. Not trying to go against at al (arguing with you in two treads). If I offended you in anyway water under the bridge man. WATER UNDER THE BRIDGE.

for call, there’s no difference since their lower bound is the same.

but for put, American options are worth more cos u can exercise it earlier and recieve X.

Okay if we’re going strikely by formula then. P + (So - PVD) = C + E/(1+r)^t. = C = P + (S-PVD) - E/(1+r)^t

If we excercise early, E(1+r)^t increase in value. since t is smaller, but we might be able to get the dividend so PVD will be zero. Therefore the option might be worth more if PVD > delta of E/(1+r)^t. Our upper bound for a call would be different. Assuming you excercise early and got the dividend.

In the formula C = P + (S-PVD) - E/(1+r)^t , you did not realize that P will increase by the amount of the dividend!

Anyway, this might be fun to discuss after the exam. Cheers.

Notice I bracketed (S-PVD). I am counting it as one single item. An increase in D decrease the value of the stock. A decrease in price leads to a decrease in Call value or increase in Put value. The could share it equally, but without going through the actual binominal we wouldn’t know how this value would be distributed. You’re assuming call price is constant and I am assuming the put price is constant. Since we’re trying to calculate a call price, we’re assuming all other factors are constant other than call price and whatever value we try to change (P, PVD, So, Rf or E).

Pretty much what your trying to tell me would be equal to me telling you that if excercise price decrease call value will increase or put price would increase. Nothing to do with American or European call option. I am just trying to tell you timing matters, since you might be able to get that dividend you otherwise wouldn’t.

On the other hand, If you want to talk about a put option you should talk about excercise price. Since you can excercise earlier, the E would be worth more so your American put option is worth more than your European option. All else equal. An American put option with dividend is worth the same as an European option because you wouldn’t want to excercise early. Lower stock price higher return. If they pay a dividend then it would reduce stock price, which increases return.