newbie question about options

[crazy canuck]

An american call is worth the same as a european call provided there are no dividends to be paid.


This is not true. In fact under the Black Scholes framework the price of an american option (call or put) can't even be expressed analytically, i.e. there is no formula for it.

[Rotating Rabbit]

You're mistaken.

Sure, an american option cannot be expressed analytically (thats what discrete approximation models are for) but an american call is the same as a european call under geometric brownian motion, the assumption for BS. Its never preferable to exercise an american call before its expiry.

e.g. S(t) = price of security at time t. If an american call (Price C,Strike K,Expiry T) is in the money, at time t1 say, t1<T, you can exercise and realise a time-T gain of:

[S(t1)-K]e^r(T-t1)

If instead you sell short at T1, stick the funds S(t1) into a bank paying continuously compounded interest at rate r (as above), then at time T buy the stock at the minimum of K and S(T) with the bank money you have a time-T gain of:

S(t1)e^(T-t1) - min {K, S(T)}

which is greater. So its never preferable to exercise early, thus its the same expected value as its european sibling, under Black-Scholes and its assumptions.

[Your Mom]

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What on earth is montage?

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Some options are traded on more than one exchange at the same time. In fact, there are 5 options exchanges in the US, and many (most?all?) options are traded on all 5. Every now and then the same exact option is traded on one exchange for one price and another exchange for another price. This opens up a profit opportunity. All you have to do is buy it on the exchange where its being sold for less and turn around and sell it on the other exchange where its trading higher. Insta-profit. This imbalance between exchanges is called montage. (mon-tazh)

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Arbitrage, I think, is what he is talking about.

[jason1990]

Just a point that might be of interest to some. The value of the call option when it is exercised is h(S(t)), where

h(x)=max{x-K,0}.

This function is convex and has h(0)=0. These two properties are all one needs in order to conclude that it is never optimal to exercise the option before expiry. Notice that the corresponding function for the put is still convex, but it is not 0 at 0.

[jason1990]

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There are 2 kinds of options: calls and puts. I’ll describe call options now, and puts in a moment.

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In theory, any function of the stock history up to the time of expiry can be regarded as an option. So, in practice, are these more "exotic" options not bought or sold very often?

[crazy canuck]

These two properties are all one needs in order to conclude that it is never optimal to exercise the option before expiry.

This is wrong. Somtimes it is correct to exercise an option early. If it wouldn't be correct, then it would be easy to price american options.

[jason1990]

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These two properties are all one needs in order to conclude that it is never optimal to exercise the option before expiry.

This is wrong. Somtimes it is correct to exercise an option early. If it wouldn't be correct, then it would be easy to price american options.

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Let me just repeat myself:

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Just a point that might be of interest to some. The value of the call option when it is exercised is h(S(t)), where

h(x)=max{x-K,0}.

This function is convex and has h(0)=0. These two properties are all one needs in order to conclude that it is never optimal to exercise the option before expiry. Notice that the corresponding function for the put is still convex, but it is not 0 at 0.

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Nowhere did I contradict the fact that "somtimes it is correct to exercise an option early." So what are you saying?

[crazy canuck]

Nowhere did I contradict the fact that "somtimes it is correct to exercise an option early." So what are you saying?

Exercising before expiry means exercising early.

[jason1990]

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Nowhere did I contradict the fact that "somtimes it is correct to exercise an option early." So what are you saying?

Exercising before expiry means exercising early.

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I said:

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it is never optimal to exercise the option before expiry.

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This means:

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it is never optimal to exercise the option early.

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What is "the option" in this context? It is an American call option. So I am saying:

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it is never optimal to exercise the American call option early.

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This is a fact and I am offering up an explanation for it, which shows conditions under which the same conclusion can be drawn for certain other options as well.

[crazy canuck]

it is never optimal to exercise the American call option early.

Yes it is.

I'm sure there is a simpler explanation, but this will do:

American option pricing


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To do their approximation, BAW decomposes the American price into the European price and the early exercise premium

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If it would be never optimal to exercise the option early the early exercise premium would be zero.