## 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 $\inline S_t$, strike price K.

$\inline S_T \leq K + (K - S_T)^+$

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

$\inline S_t \leq Ke^{r(T-t)} + (K - S_t)^+$.

$\inline \Rightarrow S_t \leq Ke^{r(T-t)} + C_t$

where $\inline C_t$ is the European option price.

We also know that, $C_t \geq \overline{C_t} \geq 0$ where $\inline \overline{C_t}$ is the continuation price for an American option. Thus,

$\inline \overline{C_t} \geq C_t \geq (S_t - Ke^{r(T-t)})^+$

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

$\inline \overline{C_t} \geq (S_t - Ke^{r(T-t)})^+ \geq (S_t - K)^+$

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

Q.E.D.