Why are American-style options worth more than European-style options?
"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." 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. 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. "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." 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. 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. 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. 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. If you're into math, do this thought experiment: 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. 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. 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.
The value at expiration does not depend on the price path for a plain vanilla European or American option. At expiration, the value would simply be: max[K - S_T, 0], where: K is the strike price, and S_T is the underlying price at expiration. The simple answer: Because you believe every other option can yield greater losses... So the thinking is; Lend it to France you'll get your money back. Put your money in a bank and there's a chance you won't get it back if the bank goes bust. Investors also believe that rates will continue to go more negative, in which case they will actually MAKE money.  It's a momentum play which will eventually reverse but since the Bond market is extremely liquid, it's a bet that can quickly be unwound. Another theory floating around is that if the weaker countries of the euro leave (e.g. Greece, etc) and the core keep the euro, then the value of the euro will actually rise. So by putting your money in French or German bonds you would also be securing cash in French or German euro's. - Thinking the unthinkable on a euro break-up - Gavyn Davies "First, your question contains a couple of false premises: Options in the U.S. do not trade on the NYSE, which is a stock exchange.  You must have been looking at a listing from an options exchange. There are a handful of options exchanges in the U.S., and while two of these have ""NYSE"" in the name, referring to ""NYSE"" by itself still refers to the stock exchange. Companies typically don't decide themselves whether options will trade for their stock.  The exchange and other market participants (market makers) decide whether to create a market for them. The Toronto Stock Exchange (TSX) is also a stock exchange.  It doesn't list any options.  If you want to see Canadian-listed options on equities, you're looking in the wrong place. Next, yes, RY does have listed options in Canada. Here are some.  Did you know about the Montreal Exchange (MX)?  The MX is part of the TMX Group, which owns both the Toronto Stock Exchange (TSX) and the Montreal Exchange. You'll find lots of Canadian equity and index options trading at the MX.  If you have an options trading account with a decent Canadian broker, you should have access to trade options at the MX. Finally, even considering the existence of the MX, you'll still find that a lot of Canadian companies don't have any options listed.  Simply: smaller and/or less liquid stocks don't have enough demand for options, so the options exchange & market makers don't offer any. It isn't cost-effective for them to create a market where there will be very few participants." One broker told me that I have to simply read the ask size and the bid size, seeing what the market makers are offering. This implies that my order would have to match that price exactly, which is unfortunate because options contract spreads can be WIDE. Also, if my planned position size is larger than the best bid/best ask, then I should break up the order, which is also unfortunate because most brokers charge a lot for options orders. "In short, yes. By ""forward selling"", you enter into a futures contract by which you agree to trade Euros for dollars (US or Singapore) at a set rate agreed to by both parties, at some future time. You are basically making a bet; you think that the dollar will gain on the Euro and thus you'd pay a higher rate on the spot than you've locked in with the future. The other party to the contract is betting against you; he thinks the dollar will weaken, and so the dollars he'll sell you will be worth less than the Euros he gets for them at the agreed rate. Now, in a traditional futures contract, you are obligated to execute it, whether it ends up good or bad for you. You can, to avoid this, buy an ""option"". By buying the option, you pay the other party to the deal for the right to say ""no, thanks"". That way, if the dollar weakens and you'd rather pay spot price at time of delivery, you simply let the contract expire un-executed. The tradeoff is that options cost money up-front which is now sunk; whether you exercise the option or not, the other party gets the option price. That basically creates a ""point spread""; you ""win"" if the dollar appreciates against the Euro enough that you still save money even after buying the option, or if the dollar depreciates against the Euro enough that again you still save money after subtracting the option price, while you ""lose"" if the exchange rates are close enough to what was agreed on that it cost you more to buy the option than you gained by being able to choose to use it." Liquidity on dual listed equities is rarely the same on both exchanges. More liquidity means you would typically get a better price assuming you execute the trades using the same order types. It's recommended to trade where the liquidity is greater unless your trading method benefits somehow from it being lower. It's important to remember that some ADRs (some European companies listed in US) have ADR fees which vary. USD/EUR transaction fees are low when using a decent broker but you're obviously participating in the currency risk. Just a few observations within the Black-Scholes framework: Next, you can now use the Black-Scholes framework (stock price is a Geometric Brownian Motion, no transaction costs, single interest rate, etc. etc.) and numerical methods (such as a PDE solver) to price American style options numerically, but not with a simple closed form formula (though there are closed-form approximations). Yes. If I own a call, an American call option can be exercised at my wish. A European call can only be exercised at expiration, by the way.  Your broker doesn't give you anything but a current quote for a given strike price. There are a number of good option related questions here. A bit of searching and reading will help you understand the process. "The CBOE states, in an investor's guide to Interest Rate Options: The Options’ Underlying Values   Underlying values for the option contracts are   10 times the underlying Treasury yields (rates)—   13-week T-bill yield (for IRX), 5-year T-note   yield (for FVX), 10-year T-note yield (for TNX)   and 30-year T-bond yield (for TYX). The Yahoo! rate listed is the actual Treasury yield; the Google Finance and CBOE rates reflect the 10 times value.  I don't think there's a specific advantage to ""being contrary"", more likely it's a mistake, or just different." Think about it this way. If the strike price is $200, and cost of the option is $0.05. $200 + $0.05 is $200.05. That does not mean that the price of buying the option is more. Neither is the option writer going to pay you $70 to buy the contract.  When you are buying options, you can only have a limited downside and that is the premium that you pay for it.  In case of the $115 contract, your total loss could be a maximum of $19.3.  In case of the $130 contract, your total loss could be a maximum of $9.3. This is due to the fact that the chances of AAPL going to hit $130 is less than the chance of AAPL hitting $115. Therefore, option writers offer the lower probability contracts at a lower price.  Long story short, you do not pay for the Strike price. You only pay the premium and that premium keeps getting lower with and increase in Strike price(Or decrease if it is a put option). Strike price is just a number that you expect the stock or index to break. I would suggest you to read up a little more on pricing from here "The basic idea is that money's worth is dependent on what it can be used to buy. The principal driver of monetary exchange (using one type of currency to ""buy"" another) is that usually, transactions for goods or services in a particular country must be made using that country's official currency. So, if the U.S. has something very valuable (let's say iPhones) that people in other countries want to buy, they have to buy dollars and then use those dollars to buy the consumer electronics from sellers in the U.S. Each country has a ""basket"" of things they produce that another country will want, and a ""shopping list"" of things of value they want from that other country. The net difference in value between the basket and shopping list determines the relative demand for one currency over another; the dollar might gain value relative to the Euro (and thus a Euro will buy fewer dollars) because Europeans want iPhones more than Americans want BMWs, or conversely the Euro can gain strength against the dollar because Americans want BMWs more than Europeans want iPhones. The fact that iPhones are actually made in China kind of plays into it, kind of not; Apple pays the Chinese in Yuan to make them, then receives dollars from international buyers and ships the iPhones to them, making both the Yuan and the dollar more valuable than the Euro or other currencies. The total amount of a currency in circulation can also affect relative prices. Right now the American Fed is pumping billions of dollars a day into the U.S. economy. This means there's a lot of dollars floating around, so they're easy to get and thus demand for them decreases. It's more complex than that (for instance, the dollar is also used as the international standard for trade in oil; you want oil, you pay for it in dollars, increasing demand for dollars even when the United States doesn't actually put any oil on the market to sell), but basically think of different currencies as having value in and of themselves, and that value is affected by how much the market wants that currency." Good questions. I can only add that it may be valuable if the company is bought, they may buy the options. Happened to me in previous company. SPX options are cash settled European style.  You cannot exercise European style options before the expiration date.  Assuming it is the day of expiration and you own 2,000 strike puts and the index settlement value is 1,950 - you would exercise and receive cash for the in the money amount times the contract multiplier.  If instead you owned put options on the S&amp;P 500 SPDR ETF (symbol SPY) those are American style, physically settled options.  You can exercise a long American style option anytime between when your purchase it and when it expires.    If you exercised SPY puts without owning shares of SPY you would end up short stock at the strike price. "All else being equal, you should look for more volatile (riskier) stocks. Technically, it was all time value - the entire value of an ""out of the money"" option is time value.   What's confusing is that time value is affected by numerous variables, only one of which is time. The reason volatility is the one to look at is that all the rest are likely already intuitive to you, or are too minor an influence to worry about: Current risk-free interest rates and a stock's dividend payout during the life of the option affect the value of the call, but are usually minor infulences.   (Higher interest rates makes call values higher, and higher dividend yield makes call values lower.) Longer time to expiration will increase the value of the call, but you're pretty likely already focused on that. The strike price's proximity to the current price affects the call's value - agreeing to sell a stock 5% above current levels will pay more than agreeing to sell it 10% above current levels - but again, this is likely obvious to you. Volatility, or the percent by which the stock is likely to move up or down on a given day, is almost certainly the variable that's not already obvious. Stocks that jump all over the place have higher volatility than those that move more predictably.     The reason that options (calls and puts) cost more on higher volatility names is that options' payout is asymmetrical.  In the case of calls, the option holder gets all the upside, but none of the downside, other than what they paid for the opotion.     If one stock goes up or down $5 every day, and another goes up or down $20 every day and you could pay some fixed amount to get that stock's upside, but not have any exposure to its downside, other than that fixed amount, you'd pay more for the one that pays you $20 or $0 than you would for the one that pays you $5 or $0.    That's why higher volatility (meaning larger daily moves) makes optimum prices higher." So would it be more sensible to park the money in 3 separate fixed deposits created each month or to invest this money as usual in debt and mutual funds and instead just liquidate the recurring deposit and use it to pay my taxes in July? Most taxes would already have been paid by March 31. So what you are talking here is less than Rs 10,000/- of taxes. The optimization you are looking at is very small and you are better of choosing one that is more convenient.