Week 10 Homework Questions

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Sheet1

Consider the following questions on the pricing of options on the stock of ARB Inc.:

a. A share of ARB stock sells for $75 and has a standard deviation of returns equal to 20 percent per year. The current risk-free rate is 9 percent and the stock pays two dividends: (1) a $2 dividend just prior to the option's expiration day, which is 91 days from now (ie, exactly one-quarter of a year), and (2) a $2 dividend 182 days from now (ie., exactly one-half year). Calculate the Black-Shoules value for a European -style call option with an exercise price of $70.

Current Security price = S1 =

$ 75.00

Share value = S2 =

$ 73.04

S2 = Cum dividend value - present value of dividend

S2 = 75 - 2/en = 75 - 2/e 0.09*3/12 =

75 - 1.96 = 73.04

Exercise price = X =

$ 70.00

Time to expiration 1 = T1

0.25

0.50

Time to expiration 2 = T2

182

0.5

0.71

Risk-free rate = RFR

Security price volitility = Ơ =

0.20

Black-Shoules value for a European call option

d1 =

ln(S2/D) + (r + Ơ2/2)T

Ơ√T

d1 =

ln(73.04/70) + (0.09 + 0.22/2)0.25

N(d) = 0.5 + (d)(4.4-d)/10

V0 = SN(d1)-Ke-rT N(d2)

0.2√0.25

N(d1) = 0.5 + (d1)(4.4-d1)/10

V0 =

73.04*0.7587 - 70*0.98*0.7277

d1 =

ln(1.0434) + (0.09 + 0.22/2)0.25

N(d1) =

0.5 + (.699)(4.4 - 0.699)/10

V0 =

5.495

0.2√0.25

N(d1) =

0.7587

d1 =

.0424 + 0.0275

0.10

N(d2) = 0.5 + (d2)(4.4-d2)/10

d1 =

0.699

N(d2) =

0.5 + (.599)(4.4 - 0.599)/10

N(d2) =

0.7277

d2 =

d1 - Ơ √T

d2 =

0.699 - 0.2*√0.25

d2 =

0.599

b. What would be the price of a 91-day European-style put option on ARB stock having the same exercise price?

S + P - C = PV of E

73.04 + P - 5.495 = PV of 70

P =

68.51 + 5.495 - 73.04

P =

0.965

c. Calculate the change in a call option's value that would occur if ARB's management suddenly decided to suspend dividend payments and this action had no effect on the price of the company's stock.

d1 =

ln(S1/D) + (r + Ơ2/2)T

N(d) = 0.5 + (d)(4.4-d)/10

V0 = SN(d1)-Ke-rT N(d2)

Ơ√T

N(d1) = 0.5 + (d1)(4.4-d1)/10

V0 =

75.00*0.8314 - 70*0.98*0.8057

d1 =

ln(75/70) + (.09 + .022/2)0.25

N(d1) =

0.5 + (.9647)(4.4 - 0.9647)/10

V0 =

7.084

0.2√0.25

N(d1) =

0.8314

d1 =

ln(1.0714) + (0.09 + 0.22/2)0.25

Vo without dividends =

7.084

0.2√0.25

N(d2) = 0.5 + (d2)(4.4-d2)/10

Vo with dividends =

5.495

d1 =

0.6897 + 0.0275

N(d2) =

0.5 + (.8647)(4.4 - 0.8647)/10

Change in value of option =

1.589

0.10

N(d2) =

0.8057

d1 =

0.9647

d2 =

d1 - Ơ √T

d2 =

0.9647 - 0.2*√0.25

d2 =

0.8647

d. Briefly describe (without calculations) how your answer in Part a would differ under the following separate circumstances: (1) volitility of ARB stock increases to 30 percent, (2) the risk-free rate decreases to 8 percent.

1) Security price volitility = Ơ =

0.30

d1 =

ln(S2/D) + (r + Ơ2/2)T

N(d) = 0.5 + (d)(4.4-d)/10

V0 = S2N(d1)-Ke-rT N(d2)

Ơ√T

N(d1) = 0.5 + (d1)(4.4-d1)/10

V0 =

73.04*0.6976 - 70*0.98*0.6446

d1 =

ln(73.04/70) + (0.09 + 0.32/2)0.25

N(d1) =

0.5 + (.5077)(4.4 - 0.5077)/10

V0 =

6.7331

0.30√0.25

N(d1) =

0.6976

d1 =

ln(1.0434) + (0.09 + 0.32/2)0.25

0.30√0.25

N(d2) = 0.5 + (d2)(4.4-d2)/10

d1 =

ln(1.0434 )+ 0.3375

N(d2) =

0.5 + (.3577)(4.4 - 0.3577)/10

0.15

N(d2) =

0.6446

d1 =

0.0424 + 0.03375

0.15

d1 =

0.5076666667

d2 =

d1 - Ơ √T

d2 =

0.5077 - 0.3*√0.25

d2 =

0.3576666667

2) Change in RFR = 8%

0.2

d1 =

ln(S2/D) + (r + Ơ2/2)T

N(d) = 0.5 + (d)(4.4-d)/10

V0 = S2N(d1)-Ke-rT N(d2)

Ơ√T

N(d1) = 0.5 + (d1)(4.4-d1)/10

V0 =

73.04*0.7511 - 70*0.98*0.7196

d1 =

ln(73.04/70) + (0.08 + 0.32/2)0.25

N(d1) =

0.5 + (.674)(4.4 - 0.674)/10

V0 =

5.4958

0.20√0.25

N(d1) =

0.7511

d1 =

ln(1.0434) + (0.08 + 0.32/2)0.25

0.20√0.25

N(d2) = 0.5 + (d2)(4.4-d2)/10

d1 =

ln(1.0434 )+ 0.025

N(d2) =

0.5 + (.574)(4.4 - 0.574)/10

0.1

N(d2) =

0.7196

d1 =

0.0424 + 0.025

0.1

d1 =

0.674

d2 =

d1 - Ơ √T

d2 =

0.674 - 0.2*√0.25

d2 =

0.574