April 29, 2009
American Option on a non-dividend paying stock
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.
^+ "S_T \leq K + (K - S_T)^+")
Discounting the above equations using the risk free rate r to time t, we obtain
} + (K - S_t)^+ "S_t \leq Ke^{r(T-t)} + (K - S_t)^+")
.
} + C_t "\Rightarrow S_t \leq Ke^{r(T-t)} + C_t")
where 
is the European option price.
We also know that, 
where 
is the continuation price for an American option. Thus,
})^+ "\overline{C_t} \geq C_t \geq (S_t - Ke^{r(T-t)})^+")
Thus, if risk free rate is strictly positive, we can prove that for positive paths
})^+ \geq (S_t - K)^+ "\overline{C_t} \geq (S_t - Ke^{r(T-t)})^+ \geq (S_t - K)^+")
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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