**Theorem:**It is never optimal to exercise an American option on a non-dividend paying stock for certain values of risk-free rate r.

**Proof:**
We know that at maturity following relation is true, for a stock price process , strike price K.

Discounting the above equations using the risk free rate r to time t, we obtain

.

where is the European option price.

We also know that, where is the continuation price for an American option. Thus,

Thus, if risk free rate is strictly positive, we can prove that for positive paths

Above equation implies that for an American option on a non dividend paying stock it is never optimal to exercise early.

Q.E.D.

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