I am a student new to this field, so this is probably a stupid question. When reading John Hull's book, I became confused about early exercise of American style options (especially for put options). In theory, the idea of early exercise is clear. You should exercise the option at a (S, t) when the intrinsic value of the option is higher than the expected future value of the option price. But in real life, given the fact that the option price is always higher than or equal to the intrinsic value, why would an investor ever want to exercise an American option before expiration, compare to sell it directly in the market (except in some special cases, like he want the stock of that company, or he already holds the underlying and uses puts as insurance).Thanks.

QuoteOriginally posted by: xl1986I am a student new to this field, so this is probably a stupid question. ... But in real life, given the fact that the option price is always higher than or equal to the intrinsic value, ...This is not one of them true facts. Discounting can falsify it for puts, dividends for calls.

An option involves the exchange of one asset for another. It's true that the option has non-negative value, so the only reason to exercise early is the difference in value from holding the two assets is greater than the time value. The most common textbook example is when you have an in-the-money put option on a stock that pays no dividends, at least not before the option expiry. By exercising early, you can invest the exercise cash between now and expiry. If the put option is sufficiently in-the-money, the extra interest can be worth more than the time value.Another example is a call option on a dividend paying stock, immediately before the dividend date of record. If you exercise now you get the dividend, which can be worth more than the interest lost on the exercise price plus the time value of the option.

Thanks for the quick replies. Maybe I am not clear enough, sorry for that. The book only compared two cases: exercise the option at time t or hold it to some future date. But in real world, there is a third option, you can also sell the option in the market and receive the bid price, which has to be higher than or equal to the current exercise value (K-S for put and S-K for call). Had it not, a trader can purchase the option and exercise it immediately to earn some risk free profit (minus the transaction cost involved). Then my question is, in practice, when will a trader prefer exercising the option over selling it directly in the market.

if we put for simplicity r = 0 and we are at t then optimal exercise american option in theory is the random time on [ t , T ] when it offers max of the return. If S ( u ) is a random function then buyer of the call think that he can sell option for and receives max { S ( u ) - K , 0 } at any u from [ t , T ]. Non markovian moment h of exercise is one that { S ( h ) - K , 0 } / C ( t , x ) - 1 = max { u : S ( u ) - K , 0 } / C ( t , x ) - 1 or { S ( h ) - K , 0 } = max { u : S ( u ) - K , 0 }. This non markovian moment can be approximated by markovian moments which specify return and risk to fail. If market price of the American call is c and S is GBM we can calculate { u : S ( h(q) ) - K , 0 } / c - 1 where h (q) is the moment to reach level q > 0 by S ( u ). We can calculate the chance that investor will exercise call. Having a strategy which specifies upper bound for the admitted risk one can find appropriate level q and make a decision whether to buy or not option. It somewhat alternative approach.

Yes, the option price should be greater than or equal to the intrinsic value. But if it optimal to exercise immediately then the price will equal the expected value. The decision whether to see in the market or exercise will be based on transaction costs. Generally if you want to keep the position created by the exercise, it's cheaper to exercise. If you don't want to keep that position, it will usually be cheaper to sell. For example, if you are exercising a call early to get the stock dividend, if you want to hold the stock afterwards you would likely exercise, if you plan to sell the stock immediately upon exercise, it may be cheaper just to sell the option.In any event the difference between the choices is likely to be small.

Thanks a lot. That clears my confusion

It seems that settings of the problem are different. I am talking about theoretical exercise assuming that : stock is govern by GBM with given mu, sigma and the option price is known 'c' at t. We need to make a decision whether it is reasonable to by American call (or put). In this setting we introduced h and its approximation h (q). This is market pricing we can specify market risk as P { h ( q ) > T }.When we decided to buy call at t we are going along the path for a fixed scenario. We have our information about market risk and understand that along our scenario we might be at the scenario from { h ( q ) > T }. Thus, assume for example that we reach the level q - 1 at some u. We again need to make a decision whether to wait or not depending on the value T - u. The problem looks more complex , more risky than under BS

QuoteOriginally posted by: xl1986I am a student new to this field, so this is probably a stupid question. When reading John Hull's book, I became confused about early exercise of American style options (especially for put options). In theory, the idea of early exercise is clear. You should exercise the option at a (S, t) when the intrinsic value of the option is higher than the expected future value of the option price. But in real life, given the fact that the option price is always higher than or equal to the intrinsic value, why would an investor ever want to exercise an American option before expiration, compare to sell it directly in the market (except in some special cases, like he want the stock of that company, or he already holds the underlying and uses puts as insurance).Thanks.hello, you should read the article "On Exercising American Options. The Risk of Making More Money Than You Expected" (P.Wilmott and al.). it examines the issue of early exercise fom the option buyer's standpoint.