Question

Rework Problem 24-5 using the Black-Scholes model to estimate the value of the option. The risk-free rate is 6%.

 

Data from problem 24-5:

Fethe's Funny Hats is considering selling trademarked curly purple-haired wigs for University of Western Ontario football games. The purchase cost for a 2-year franchise to sell the wigs is $20,000. If demand is good (40% probability), then the net cash flows will be $28,000 per year for 2 years. If demand is bad (60% probability), then the net cash flows will be $8,000 per year for 2 years. Fethe's cost of capital is 10%.

Answer 1

P = PV as of time zero of all expected future cash flows if the project is repeated starting in year 2. Note it includes both the good cash flows and the bad cash flows since as of now, we don’t know which outcome will result, and P excludes the $20,000 investment in the franchise.  

 

 

Expected PV of cash flows (as of time 0) = 40,161(0.40) + 11,475(0.60) = $22,949 = P.

 

The strike price, X, is the cost to extend the franchise at the end of year 2, and is $20,000.

 

The time to expiration is the time you decide whether or not to extend the franchise, and is at the end of year 2.

 

           P = $22,949

           X = $20,000

            t = 2.

        rRF = 0.06.

            

To calculate the variance of the project’s returns using the indirect method, first calculate the standard deviation of the value at year 2. The value is either 48,595 (probability 40%) or 13,884 (probability 60%). 

 

Expected value at year 2: 48,595(0.4) + 13,884(0.6) = 27,768. 

Standard deviation = [(48,595 – 27,768)2(0.40) + (13,884 – 27,768)2(0.60)]1/2 = 17,005.

 

Then calculate the coefficient of variation:

CV = 17,005/27,768 = 0.6124.

 

σ2 = ln(CV2 + 1)/t = ln(0.61242 + 1)/2 = 0.1592

so σ2 = 0.1592 

 

 

From Excel function NORMSDIST, or approximated from the table in Appendix C:

N(d1) = 0.7699

N(d2) = 0.5692 

 

 

Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:

 

V = P[N(d1)] – [N(d2)]

   = $22.949(0.7699) – $20e(–0.06)(2)(0.5692)

   = $17.67 – $10.10

   = $7.57 thousand.