Why are American-style options worth more than European-style options?

Question

Why are American-style options worth more than European-style options?

I realize I can exercise American-style options anytime before the expire, but I can only exercise European-style options during their "exercise period" (usually right when they expire, but no earlier).

So it makes sense that an American option is worth at least as much as a European option.

But why is it worth more? If I exercise my American option early, I may make more money than waiting until expiration, but I also may make less. Mathematically, there's no advantage, since I'm equally likely to make as much money by waiting for expiration.

Mathematically speaking, is there ever a good reason to exercise an American option early?

I realize people don't always behave logically, but even the formulas that valuate options show that American options are worth more.

EDIT (answering comments):

• Many of the comments people made would apply to lookback options (http://en.wikipedia.org/wiki/Lookback_option), but not to American options.

• American options can do everything European options can and more. I understand that, and that means American options can't be worth any LESS than European options, but I'm still not clear on how you would mathematically calculate this extra value.

• @Aaronaught: "The difference between an American and European option is the difference between getting N chances to get it right (N being the number of days 'til expiration) and getting just one chance. It should be easy to see why you're more likely to profit with the former, even if you can't accurately predict price movement."

REPLY: I don't think you really get N chances. Once you exercise the option, that's it, no more chances. And, if you decide not to exercise the option today, and the underlying's price falls, you can't go back in time and exercise it yesterday.

• @jdsweet: Think of it this way, if you traveled back through time one month - with perfect knowledge of AAPL's stock price over that period - which happens to peak viciously then return to its old price at the end of the period - wouldn't you pay more for an American option?

REPLY: Well, no. I'd simply buy a cheaper European option that expires right at the spike. Or, I'd buy a European option that expires later, and sell it when the spike occurs. American options don't give you precognition, so you're still guessing as to when to sell/exercise.

• @jdsweet: "one of the more common reasons people exercise (as opposed to sell) an American option before expiration is if an unexpected dividend (larger than remaining time value of the option) was just announced that's going to be paid before the option contract expires"

REPLY: This makes sense. I'm actually trading FOREX options, so there is no dividend. In that scenario, are European and American options worth the same?

Answer 1

According to the book of Hull, american and european calls on non-dividend paying stocks should have the same value. American puts, however, should be equals to, or more valuable than, european puts.

The reason for this is the time value of money. In a put, you get the option to sell a stock at a given strike price. If you exercise this option at t=0, you receive the strike price at t=0 and can invest it at the risk-free rate. Lets imagine the rf rate is 10% and the strike price is 10$. this means at t=1, you would get 11.0517$. If, on the other hand, you did'nt exercise the option early, at t=1 you would simply receive the strike price (10$). Basically, the strike price, which is your payoff for a put option, doesn't earn interest.

Another way to look at this is that an option is composed of two elements: The "insurance" element and the time value of the option. The insurance element is what you pay in order to have the option to buy a stock at a certain price. For put options, it is equals to the payout= max(K-S, 0) where K=Strike Price and St= Stock price. The time value of the option can be thought of as a risk-premium. It's difference between the value of the option and the insurance element.

If the benefits of exercising a put option early (i.e- earning the risk free rate on the proceeds) outweighs the time value of the put option, it should be exercised early.

Yet another way to look at this is by looking at the upper bounds of put options. For a european put, today's value of the option can never be worth more than the present value of the strike price discounted at the risk-free rate. If this rule isn't respected, there would be an arbitrage opportunity by simply investing at the risk-free rate. For an american put, since it can be exercised at any time, the maximum value it can take today is simply equals to the strike price. Therefore, since the PV of the strike price is smaller than the strike price, the american put can have a bigger value.

Bear in mind this is for a non-dividend paying stock. As previously mentioned, if a stock pays a dividend it might also be optimal to exercise just before these are paid.

Answer 2

I'm sorry, but your math is wrong. You are not equally likely to make as much money by waiting for expiration.

Share prices are moving constantly in both directions. Very rarely does any stock go either straight up or straight down. Consider a stock with a share price of $12 today. Perhaps that stock is a bad buy, and in 1 month's time it will be down to $10. But the market hasn't quite wised up to this yet, and over the next week it rallies up to $15.

If you bought a European option (let's say an at-the-money call, expiring in 1 month, at $12 on our start date), then you lost. Your option expired worthless.

If you bought an American option, you could have exercised it when the share price was at $15 and made a nice profit.

Keep in mind we are talking about exactly the same stock, with exactly the same history, over exactly the same time period. The only difference is the option contract. The American option could have made you money, if you exercised it at any time during the rally, but not the European option - you would have been forced to hold onto it for a month and finally let it expire worthless.

(Of course that's not strictly true, since the European option itself can be sold while it is in the money - but eventually, somebody is going to end up holding the bag, nobody can exercise it until expiration.)

The difference between an American and European option is the difference between getting N chances to get it right (N being the number of days 'til expiration) and getting just one chance. It should be easy to see why you're more likely to profit with the former, even if you can't accurately predict price movement.

Answer 3

An option is an instrument that gives you the "right" (but not the obligation) to do something (if you are long).

An American option gives you more "rights" (to exercise on more days) than a European option.

The more "rights," the greater the (theoretical) value of the option, all other things being equal, of course. That's just how options work.

You could point to an ex post result, and and say that's not the case. But it is true ex ante.

Answer 4

OK, my fault for not doing more research. Wikipedia explains this well:

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

Basically, there are some cases where it's advantageous to exercise an American option early.

For non-gold currency options, this is only when the carrying cost (interest rate differential aka swap rate or rollover rate) is high.

The slight probability that this may occur makes an American option worth slightly more.

Answer 5

If you're into math, do this thought experiment:

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Consider the outcome X of a random walk process (a stock doesn't behave this way, but for understanding the question you asked, this is useful):

On the first day, X=some integer X1. On each subsequent day, X goes up or down by 1 with probability 1/2.

Let's think of buying a call option on X. A European option with a strike price of S that expires on day N, if held until that day and then exercised if profitable, would yield a value Y = min(X[N]-S, 0). This has an expected value E[Y] that you could actually calculate. (should be related to the binomial distribution, but my probability & statistics hat isn't working too well today) The market value V[k] of that option on day #k, where 1 < k < N, should be V[k] = E[Y]|X[k], which you can also actually calculate. On day #N, V[N] = Y. (the value is known)

An American option, if held until day #k and then exercised if profitable, would yield a value Y[k] = min(X[k]-S, 0).

For the moment, forget about selling the option on the market. (so, the choices are either exercise it on some day #k, or letting it expire)

Let's say it's day k=N-1.

If X[N-1] >= S+1 (in the money), then you have two choices: exercise today, or exercise tomorrow if profitable. The expected value is the same. (Both are equal to X[N-1]-S). So you might as well exercise it and make use of your money elsewhere.

If X[N-1] <= S-1 (out of the money), the expected value is 0, whether you exercise today, when you know it's worthless, or if you wait until tomorrow, when the best case is if X[N-1]=S-1 and X[N] goes up to S, so the option is still worthless.

But if X[N-1] = S (at the money), here's where it gets interesting. If you exercise today, it's worth 0. If wait until tomorrow, there's a 1/2 chance it's worth 0 (X[N]=S-1), and a 1/2 chance it's worth 1 (X[N]=S+1). Aha! So the expected value is 1/2. Therefore you should wait until tomorrow.

Now let's say it's day k=N-2.

Similar situation, but more choices: If X[N-2] >= S+2, you can either sell it today, in which case you know the value = X[N-2]-S, or you can wait until tomorrow, when the expected value is also X[N-2]-S. Again, you might as well exercise it now.

If X[N-2] <= S-2, you know the option is worthless.

If X[N-2] = S-1, it's worth 0 today, whereas if you wait until tomorrow, it's either worth an expected value of 1/2 if it goes up (X[N-1]=S), or 0 if it goes down, for a net expected value of 1/4, so you should wait.

If X[N-2] = S, it's worth 0 today, whereas tomorrow it's either worth an expected value of 1 if it goes up, or 0 if it goes down -> net expected value of 1/2, so you should wait.

If X[N-2] = S+1, it's worth 1 today, whereas tomorrow it's either worth an expected value of 2 if it goes up, or 1/2 if it goes down (X[N-1]=S) -> net expected value of 1.25, so you should wait.

If it's day k=N-3, and X[N-3] >= S+3 then E[Y] = X[N-3]-S and you should exercise it now; or if X[N-3] <= S-3 then E[Y]=0.

But if X[N-3] = S+2 then there's an expected value E[Y] of (3+1.25)/2 = 2.125 if you wait until tomorrow, vs. exercising it now with a value of 2; if X[N-3] = S+1 then E[Y] = (2+0.5)/2 = 1.25, vs. exercise value of 1; if X[N-3] = S then E[Y] = (1+0.5)/2 = 0.75 vs. exercise value of 0; if X[N-3] = S-1 then E[Y] = (0.5 + 0)/2 = 0.25, vs. exercise value of 0; if X[N-3] = S-2 then E[Y] = (0.25 + 0)/2 = 0.125, vs. exercise value of 0. (In all 5 cases, wait until tomorrow.)

You can keep this up; the recursion formula is E[Y]|X[k]=S+d = {(E[Y]|X[k+1]=S+d+1)/2 + (E[Y]|X[k+1]=S+d-1) for N-k > d > -(N-k), when you should wait and see} or {0 for d <= -(N-k), when it doesn't matter and the option is worthless} or {d for d >= N-k, when you should exercise the option now}.

The market value of the option on day #k should be the same as the expected value to someone who can either exercise it or wait.

It should be possible to show that the expected value of an American option on X is greater than the expected value of a European option on X. The intuitive reason is that if the option is in the money by a large enough amount that it is not possible to be out of the money, the option should be exercised early (or sold), something a European option doesn't allow, whereas if it is nearly at the money, the option should be held, whereas if it is out of the money by a large enough amount that it is not possible to be in the money, the option is definitely worthless.

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As far as real securities go, they're not random walks (or at least, the probabilities are time-varying and more complex), but there should be analogous situations. And if there's ever a high probability a stock will go down, it's time to exercise/sell an in-the-money American option, whereas you can't do that with a European option.

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edit: ...what do you know: the computation I gave above for the random walk isn't too different conceptually from the Binomial options pricing model.

Answer 6

Differences in liquidity explain why American-style options are generally worth more than their European-style counterparts. As far as I can tell, no one mentioned liquidity in their answer to this question, they just introduced needlessly complex math and logic while ignoring basic economic principles. That's not to say the previous answers are all wrong - they just deal with periphery factors instead of the central cause.

Liquidity is a key determinant of pricing/valuation in financial markets. Liquidity simply describes the ease with which an asset can be bought and sold (converted to cash). Without going into the reasons why, treasury bills are one of the most liquid securities - they can be bought or sold almost instantly at any time for an exact price. The near-perfect liquidity of treasuries is one of the major reasons why the price (yield) of a t-bill will always be higher (lower yield) than that of an otherwise identical corporate or municipal bond. Stated in general terms, a relatively liquid asset is always worth more than an relatively illiquid asset, all else being equal.

The value of liquidity is easy to understand - we experience it everyday in real life. If you're buying a house or car, the ability to resell it if needed is an important component of the decision. It's the same for investors - most people would prefer an asset that they can quickly and easily liquidate if the need for cash arises.

It's no different with options. American-style options allow the holder to exercise (liquidate) at any time, whereas the buyer of a European option has his cash tied up until a specific date. Obviously, it rarely makes sense to exercise an option early in terms of net returns, but sometimes an investor has a desperate need for cash and this need outweighs the reduction in net profits from early exercise.

It could be argued that this liquidity advantage is eliminated by the fact that you can trade (sell) either type of option without restriction before expiration, thus closing the long position. This is a valid point, but it ignores the fact that there's always a buyer on the other side of an option trade, meaning the long position, and the right/restriction of early exercise, is never eliminated, it simply changes hands. It follows that the American-style liquidity advantage increases an options market value regardless of one's position (call/put or short/long).

Without putting an exact number on it, the general interest rate (time value of money) could be used to approximate the additional cost of an American-style option over a similar European-style contract.

Answer 7

Think of it this way, if you traveled back through time one month - with perfect knowledge of AAPL's stock price over that period - which happens to peak viciously then return to its old price at the end of the period - wouldn't you pay more for an American option?

Another way to think about options is as an insurance policy. Wouldn't you pay more for a policy that covered fire and earthquake losses as opposed to just losses from earthquakes?

Lastly - and perhaps most directly - one of the more common reasons people exercise (as opposed to sell) an American option before expiration is if an unexpected dividend (larger than remaining time value of the option) was just announced that's going to be paid before the option contract expires. Because only actual stockholders get the dividends, not options holders. A holder of an American option has the ability to exercise in time to grab that dividend - a European option holder doesn't have that ability.

Less flexibility (what you're paying for really) = lower option premium.

Answer 8

An option gives you an option. That is, you aren't buying any security - you are simply buying an option to buy a security. The sole value of what you buy is the option to buy something.

An American option offers more flexibility - i.e. it offers you more options on buying the stock. Since you have more options, the cost of the option is higher.

Of course, a good example makes sense why this is the case. Consider the VIX. Options on the VIX are European style. Sometimes the VIX spikes like crazy - tripling in value in days. It usually comes back down pretty quick though - within a couple of weeks. So far out options on the VIX aren't worth just a whole lot more, because the VIX will probably be back to normal. However, if the person could have excercised them right when it got to the top, they would have made a fortune many times what their option was worth. Since they are Euroopean style, though, they would have to wait till their option was redeemable, right when the VIX would be about back to normal. In this case, an American style option would be far more valuable - especially for something that is difficult to predict, like the VIX.

Answer 9

The value of an option has 2 components, the extrinsic or time value element and the intrinsic value from the difference in the strike price and the underlying asset price. With either an American or European option the intrinsic value of a call option can be 'locked in' any time by selling the same amount of the underlying asset (whether that be a stock, a future etc).

Further, the time value of any option can be monitised by delta hedging the option, i.e. buying or selling an amount of the underlying asset weighted by the measure of certainty (delta) of the option being in the money at expiry.

Instead, the extra value of the American option comes from the financial benefit of being able to realise the value of the underlying asset early. For a dividend paying stock this will predominantly be the dividend. But for non-dividend paying stocks or futures, the buyer of an in-the-money option can realise their intrinsic gains on the option early and earn interest on the profits today. But what they sacrifice is the timevalue of the option.

However when an option becomes very in the money and the delta approaches 1 or -1, the discounting of the intrinsic value (i.e. the extra amount a future cash flow is worth each day as we draw closer to payment) becomes larger than the 'theta' or time value decay of the option. Then it becomes optimal to early exercise, abandon the optionality and realise the monetary gains upfront.

For a non-dividend paying stock, the value of the American call option is actually the same as the European. The spot price of the stock will be lower than the forward price at expiry discounted by the risk free rate (or your cost of funding). This will exactly offset the monetary gain by exercising early and banking the proceeds. However for an option on a future, the value today of the underlying asset (the future) is the same as at expiry and its possible to fully realise the interest earned on the money received today. Hence the American call option is worth more. For both examples the American put option is worth more, slightly more so for the stock. As the stock's spot price is lower than the forward price, the owner of the put option realises a higher (undiscounted) intrinsic profit from selling the stock at the higher strike price today than waiting till expiry, as well as realising the interest earned.

Liquidity may influence the perceived value of being able to exercise early but its not a tangible factor that is added to the commonly used maths of the option valuation, and isn't really a consideration for most of the assets that have tradeable option markets.

It's also important to remember at any point in the life of the option, you don't know the future price path. You're only modelling the distribution of probable outcomes. What subsequently happens after you early exercise an American option no longer has any bearing on its value; this is now zero! Whether the stock subsequently crashes in price is irrelevent. What is relevant is that when you early exercise a call you 'give up' all potential upside protected by the limit to your downside from the strike price.

Answer 10

Put options

Why are American-style options worth more than European-style options?

• The maximum price for American put options is higher than the maximum price for European put options.
• The minimum price for American put options is higher than the minimum price for European put options.

To illustrate:

Suppose a stock is selling at $40, and we have a put option with strike price of $50.

American puts

• The minimum price of an American put option is the difference between the strike price and the stock price (i.e. the intrinsic value). In our case, the minimum value of the American put option is $10 (i.e. $50 - $40). If this wasn't the case (i.e. if the put's price is less than $10), it would be possible to do an arbitrage by buying puts and immediately exercise them. For example, if puts cost $7, we could buy the stock (-$40), buy a put (-$7), and immediately exercise the put (+$50), giving a profit of $3.

• The maximum price of an American put option is the strike price. In our case, the maximum value of the put is $50. The put's price cannot be greater than the strike price. Who's going to pay, say, $52 for the right to sell at $50? No one.

European puts

• In contrast with American options, the minimum price of a European put option can be less than the difference between the strike price and the stock price (i.e. less than intrinsic value). For example, if the current price of the stock is $40, a put option with strike price of $50 can sell at $9. Why?

  1. There is no early exercise, so this price discrepancy cannot be immediately exploited. This is unlike American options, which has the ability to immediately exploit the difference to gain a profit of $1.

  2. In the meantime, suppose the stock is at $40, a European put with strike price of $50 costs $10 (which isn't the minimum, as we will soon see). Suppose we currently have currently have $50 in cash. We use the cash to buy the stock (-$40) and a put (-$10). When the option expires, we will be able exercise the option to sell the stock at $50. Net profit = $0. This net profit of $0 is less than what one could earn if one had placed the $50 in an interest-earning bank account. Therefore, the theoretical minimum price of the put option is not $10 (i.e. the intrinsic value), but a cheaper amount that will give the same amount of profit as could have been obtained if the $50 had been placed in the bank.

• In contrast with American options, the maximum price of a European put option is less than the strike price. Why? Suppose this wasn't the case and the price of a put option is exactly equal to the strike price (e.g. $50). In this case, one can sell the put option (+$50), and place the proceeds in a bank account to earn interest. For example, one would be able to sell, lots of puts (e.g. +$1 billion), buy enough stock to cover the puts (e.g. -$800 million), and place the proceeds ($200 million) in a bank account to earn interest. Free loans are not supposed to happen. Hence, the maximum price of a European put option has to be less than the strike price.

Call options

The maximum and minimum prices are the same for both European and American options. To illustrate:

Suppose a stock is selling at $40, and we have a call option with strike price of $30.

• Whether European or American, the minimum price of a call option is more than the difference between the stock price and the strike price (i.e. more than intrinsic value). If this wasn't the case (i.e. the call's price is exactly $10), it would be possible to short the stock (+$40), buy the call (-$10), and place the proceeds ($30) in an interest-earning bank account. For example, one would be able to short lots of stock (e.g. +$1 billion), buy enough puts to cover the short (e.g. -$250 million) and place the cash proceeds ($750 million) in a bank account to earn interest on the cash. Hence, the minimum price of a call option has to be just high enough to negate such a possibility (i.e. the minimum must be greater than the difference between the stock price and strike price).

• Whether European or American, the maximum price of a call option is the price of underlying stock itself. If this wasn't the case (i.e. the call option costs more than the stock), one can just buy the stock rather than bothering with options. After all, why would anyone pay for the right to buy a stock if the right itself costs more than the stock?

Mathematically speaking, is there ever a good reason to exercise an American option early?

• From the foregoing illustrations, it should be clear that an American option never sells below its intrinsic value (if an American option were to sell below its intrinsic value, one would be able to profit by buying the option, and exercising it immediately).
• The price of an option price consists of intrinsic value + time value. Since the price of an American option is never below the intrinsic value, one can conclude that the time value component of an American option is never negative.
• There are two ways to dispose of an option before expiry:
  • Early exercise: If one exercises an American option early, one's gain equals to the intrinsic value only.
  • Sale: If one sells the American option, one's gain includes both the intrinsic value and the time value (which is never negative).
  • Since one will always gain more from a sale than from an early exercise, early exercise does not make sense.