# Chapter 3 - American put option

## Chapter 3 - American put option

I can't seem to figure out the following.

Suppose I have a market for which there exists a unique probability measure P* under whcih the discounted price of the stock is a martingale, then by Theorem 14, my market is complete.

Furthermore, if my market is complete, then by Theorem 16, the prices of contingent claims are their discounted expected values under the risk neutral measure P*.

But when pricing an American put option, you can't just take the discounted expected values, you need to take into account the fact that it can be exercised at any time...

So to close the loop: how can the existence American put option prevent the fact that there exists a unique probability measure P* under which the discounted price of the stock is a martingale???

Suppose I have a market for which there exists a unique probability measure P* under whcih the discounted price of the stock is a martingale, then by Theorem 14, my market is complete.

Furthermore, if my market is complete, then by Theorem 16, the prices of contingent claims are their discounted expected values under the risk neutral measure P*.

But when pricing an American put option, you can't just take the discounted expected values, you need to take into account the fact that it can be exercised at any time...

So to close the loop: how can the existence American put option prevent the fact that there exists a unique probability measure P* under which the discounted price of the stock is a martingale???

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## Re: Chapter 3 - American put option

Answer from Leoni himself:

The market is complete because of the argument you stated. But the fact that an american option can be exercised brings in an extra complexity. The maturity becomes what they call a stochastic stopping time. So the value of the claim at maturity is in fact max( S (tau) - K, 0) exp( T - tau), 0) which is still a stochastic quantity.

The market is complete because of the argument you stated. But the fact that an american option can be exercised brings in an extra complexity. The maturity becomes what they call a stochastic stopping time. So the value of the claim at maturity is in fact max( S (tau) - K, 0) exp( T - tau), 0) which is still a stochastic quantity.

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Registratiedatum : 22-12-10

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