Investments, tenth edition

    Dividends and Put Option Valuation 

 Equation 21.2 or 21.3 is valid for European puts on non-dividend-paying stocks. As we did 

for call options, if the underlying asset pays a dividend, we can find European put values 

by substituting  S  

0

   2  PV(Div) for  S  

0

 . Cell E7 in Spreadsheet 21.1 allows for a continuous 

dividend flow with a dividend yield of  d . In that case  S  

0

   2  PV(Div)  5   S  

0

  e  

 2  d  T 

 . 

 However, listed put options on stocks are American options that offer the opportunity of 

early exercise, and we have seen that the right to exercise puts early can turn out to be valu-

able. This means that an American put option must be worth more than the corresponding 

European option. Therefore, Equation 21.2 or 21.3 describes only the lower bound on the 

true value of the American put. However, in many applications the approximation is very 

accurate.  

16

        

 Notice that this value is consistent with put-call parity: 

P 5 C 1 PV(X ) 2 S

0

 5 13.70 1 95e

2

.10 3 .25

 2 100 5 6.35

 As we noted traders can do, we might then compare this formula value to the actual put 

price as one step in formulating a trading strategy. 

  

16

 For a more complete treatment of American put valuation, see R. Geske and H. E. Johnson, “The American Put 

Valued Analytically,”  Journal of Finance  39 (December 1984), pp. 1511–24. 

    21.5 

Using the Black-Scholes Formula