Investments, tenth edition

  Put Option Valuation 

 

We have concentrated so far on call option valuation. We can derive Black-Scholes 

European put option values from call option values using the put-call parity theorem. 

To value the put option, we simply calculate the value of the corresponding call option in 

Equation 21.1 from the Black-Scholes formula, and solve for the put option value as 

   

 P 5 C 1 PV(X) 2 S

0

 

 

 5 C 1 Xe

2rT

2 S

0

  

(21.2)  

We calculate the present value of the exercise price using continuous compounding to be 

consistent with the Black-Scholes formula. 

 Sometimes, it is easier to work with a put option valuation formula directly. If we sub-

stitute the Black-Scholes formula for a call in Equation 21.2, we obtain the value of a 

European put option as 

 

  P 5 Xe

2rT

31 2 N(d

2

)

4 2 S

0

31 2 N(d

1

)

4  

(21.3)   

 Using data from Example 21.4 ( C   5  $13.70,  X   5  $95,  S   5  $100,  r   5  .10,  s   5  .50, and 

 T   5  .25), Equation 21.3 implies that a European put option on that stock with identical 

exercise price and time to expiration is worth 

$95e

2

.10 3 .25

(1 2 .5714) 2 $100(1 2 .6664) 5 $6.35

 Example  21.5 

Black-Scholes Put Valuation 

  

15

 An exact formula for American call valuation on dividend-paying stocks has been developed in Richard Roll, 

“An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends,” 

 Journal of Financial Economics 

 5 (November 1977). The technique has been discussed and revised in 

Robert Geske, “A Note on an Analytical Formula for Unprotected American Call Options on Stocks with Known 

Dividends,”  Journal of Financial Economics  7 (December 1979), and Robert E. Whaley, “On the Valuation of 

American Call Options on Stocks with Known Dividends,”  Journal of Financial Economics  9 (June 1981). 

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  Options, Futures, and Other Derivatives