Question

Why when the stock pays dividends. Give a numerical example (choosing x, k, r, T−t,σ) in which it is obvious (without any formulas) that American put price on

a nondividend paying stock is larger then the corresponding European put price.

Answer 1

American put options have more value than the European put option because of the liquidity and the time value of money. Suppose x is 100 k is 120 then the American put option holder has the right to exercise the option any time before expiry to sell the asset at K. Thus he has more liquidity.

On selling at K he can invest 120$ at r=5% for time T-t= 3 months to get 121.47$ thereby earn an interest of $1.47$. He cannot earn the interest in a European put option where he has to hold it till expiry when x may be greater than k and thus render the put option worthless. In a dividend paying stock though the interest calculated above is the amount of dividend hence the American option’s value is not greater than that of a European option.