Investments, tenth edition

  Early Exercise of American Puts 

 For  American   put options,  the optimality of early exercise is most definitely a possibility. To 

see why, consider a simple example. Suppose that you purchase a put option on a stock. Soon 

the firm goes bankrupt, and the stock price falls to zero. Of course you want to exercise now, 

because the stock price can fall no lower. Immediate exercise gives you immediate receipt of 

the exercise price, which can be invested to start generating income. Delay in exercise means a 

time-value-of-money cost. The right to exercise a put option before expiration must have value. 

 Now suppose instead that the firm is only  nearly  bankrupt, with the stock selling at just 

a few cents. Immediate exercise may still be optimal. After all, the stock price can fall by 

only a very small amount, meaning that the proceeds from future exercise cannot be more 

than a few cents greater than the proceeds from immediate exercise. Against this possibil-

ity of a tiny increase in proceeds must be weighed the time-value-of-money cost of defer-

ring exercise. Clearly, there is some stock price below which early exercise is optimal. 

 This argument also proves that the American put must be worth more than its European 

counterpart. The American put allows you to exercise anytime before expiration. Because 

the right to exercise early may be useful in some circumstances, it will command a 

 premium in the capital market. The American put therefore will sell for a higher price than 

a European put with otherwise identical terms. 

  Figure 21.4 , panel A illustrates the value of an American put option as a function of the 

current stock price,  S  

0

 . Once the stock price drops below a critical value, denoted  S *  in 

the figure, exercise becomes optimal. At that point the option-pricing curve is tangent to the 

straight line depicting the intrinsic value of the option. If and when the stock price reaches 

S *, the put option is exercised and its payoff equals its intrinsic value.   

In contrast, the value of the European put, which is graphed in  Figure 21.4 , panel B, is 

not asymptotic to the intrinsic value line. Because early exercise is prohibited, the maxi-

mum value of the European put is PV( X ), which occurs at the point  S  

0

   5  0. Obviously, for 

a long enough horizon, PV( X ) can be made arbitrarily small.