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Mark Joshi states :

"If the price of the american option equals its intrinsic value, we exercise and it would be an error not to do so. The reason is that once the option has been exercised, we hold some cash which will grow at the risk-free rate whereas the rights granted by the option will decrease with time. So although the values are equal the time derivatives are not.

An alternative way of looking at this is that once one has made the decision not to exercise for a certain very short period of time, then the value of the option need no longer be more than the intrinsic value as one has given up the rights that enforce this no-arbitrage inequality."

My question is, I don't understand how this is true. I think that if we don't exercise, there's still a chance that the stock price woud reach some level that would generate a larger payoff in a future time compared to the current intrinsic value invested at the risk free rate. What's wrong with my reasoning? is there something I don't get?

Also could anyone tell what is precisely meant with an example by :

"once the option has been exercised, we hold some cash which will grow at the risk-free rate whereas the rights granted by the option will decrease with time. So although the values are equal the time derivatives are not"

AND

"the value of the option need no longer be more than the intrinsic value as one has given up the rights that enforce this no-arbitrage inequality"

Thank you

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There is some circularity in the claim that if the price equals intrinsic value, you should exercise. The option's price equals the sum of the intrinsic value and the time value. If the total price equals the intrinsic price, the market is implicitly telling you that time value is zero (ie a holder would exercise).

Setting aside the rules for corporate actions for the sake of illustration,let's say you own a put option and the underlying stock is worthless. If you early exercise today, you can invest the full strike value in a risk free bond. If you wait until expiry, you still get the strike, but you've missed out on interest.

For call options, lets say you own a deep in the money call. The price of a put with the same strike and expiry is \$1 and the stock is going to pay a \$1.5 dividend. If you early exercise and buy the put option, you receive the dividend.

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