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Binomial and Black-Scholes models The current price of United Carbon (UC) stock is$200. The standard deviation is 22.3% a year, and the interest rate is

Binomial and Black-Scholes models The current price of United Carbon (UC) stock is$200. The standard deviation is 22.3% a year, and the interest rate is 21% a year. A one-yearcall option on UC has an exercise price of $180.

a. Use the Black-Scholes model to value the call option on UC. You may find it helpful touse the spreadsheet in Table 21.2 on the book's website,www.mhhe.com/bma.

b. Use the formula given in Section 21-2 to calculate the up and down moves that you woulduse if you valued the UC option with the one-period binomial method. Now value theoption by using that method.

c. Recalculate the up and down moves and revalue the option by using the two-period binomialmethod.

ANSWERS:

a.P= 200EX = 180s= 0.223t= 1.0rf= 0.21

N(d1) =N(1.4388) = 0.9249

N(d2) =N(1.2158) = 0.8880

Call value = [N(d1)P] - [N(d2)PV(EX)]

= [0.9249200] - [0.8880(180/1.21)] = $52.88

b.

Letpequal the probability that the stock price will rise.Then, for a risk-neutral investor:

(p0.25) + (1-p)(-0.20) = 0.21

p= 0.91

In one year, the stock price will be either $250 or $160, and the option values will be $70 or $0, respectively.Therefore, the value of the option is:

c.

Letpequal the probability that the stock price will rise.Then, for a risk-neutral investor:

(p0.171) + (1-p)(-0.146) = 0.10

p= 0.776

Could you explain how they got 0.10 in the answer in part C for the interest rate.

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