Question

Brigham, E. F., & Ehrhardt, M. C. (2017). Financial management: theory and practice. Australia: South-Western.

Chapter 8 Mini Case P. 371

e. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes option pricing model (OPM).

(1) What assumptions underlie the OPM?

(2) Write out the three equations that constitute the model.

(3) According to the OPM, what is the value of a call option with the following characteristics?

Stock price $27.00

Strike price $25.00 Time to expiration 6 month 0.5 years

Risk-free rate 6.0%

Stock return standard deviation 0.49

Answer 1

Part (1)

Assumptions inherent in the OPM are:

• The option is European and can only be exercised at expiration and not any time prior to expiration.
• The stock is a non dividend paying stock
• Markets are efficient and hence, the market movements cannot be anticipated
• Zero transaction costs
• The risk-free rate is known and is constant
• The volatility of the underlying is known and is constant.
• The returns of the underlying stock are normally distributed.

Part (2)

The three equations are:

Value of call = S N (d1) - KetN(d2) where () + (6 + d2 = di-o vt

Part (3)

7 Inputs 8 S 9t 10 O 11 K 12 r 13 Output 14 d1 15 d2 16 N(D1)= 17 N(D2) 18 Value of the call option, C 27 0.50 49.00% 0.49 = 49% 25 6.00% Risk free rate Values Formula Used 0.481946784 =(LN(C8/C11)+(C12+C10^2/2)*C9)/(C10*SQRT(09)) 0.135464461 =C14-C10*SQRT(09) 0.685078125 =NORMSDIST(C14) 0.553877669 =NORMSDIST(C15) 5.06 =C8*C16-C11*EXP(-C12*09)*C17

the value of a call option = $ 5.06