The stock price of Heavy Metal (HM) changes only once a month: either it goes up

Question

The stock price of Heavy Metal (HM) changes only once a month: either it goes up by 20% or it falls by 16.7%. Its price now is $40. The interest rate is 1% per month.

a. What is the value of a one-month call option with an exercise price of $40? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. What is the option delta? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

c. The payoffs of the call option can be replicated by buying shares of stock and borrowing. What amount should be invested in stock and what amount must be borrowed? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

d. What is the value of a two-month call option with an exercise price of $40? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

e. What is the option delta of the two-month call over the first one-month period? (Do not round intermediate calculations. Round your answer to 3 decimal places.)

Explanation / Answer

a)

p = (Rf - d) / (u - d)

p = (1% - (-16.7%)) / (20% - (-16.7%) )

p = 0.48

payoff in upside

Share price= 40*(1+20%) = 48

Option value = Max(48- 40,0) = 8

payoff in downside

Share price= 40*(1-16.7%) = 33.32

Option value = Max(33.32- 40,0) = 0

Value of call = (0.48*8 + (1-0.48)*0) / (1+1%) = 3.81

b)

Delta = (8-0) / (48-33.32) = 0.544

c)

Amount needed in 1 month = Delta*Down price = 0.544*33.32 = 18.17

Amount to be borrowed = 18.17 / (1+1%) = 17.98

d)

First find option values in month 1

Downside , upside

P = (1+20%)*33.32 = 39.98

Option = 0

Downside , downside

P = (1-16.7%)*33.32 = 27.76

Option = 0

Option value = (0.48*0 + (1-0.48)*0) / (1+1%) = 0

Upside, upside

P = (1+20%)*48 = 57.6

Option = 57.6 - 40 = 17.6

Upside, downside

P = (1-16.7%)*48 = 39.98

Option = 0

Option value = (0.48*17.6 + (1-0.48)*0) / (1+1%) = 8.4

Now option value in mont zero =(0.48*8.4 + (1-0.48)*0) / (1+1%) = 3.99 = 4

e)

Delta = Spread of option prices / Spread of stock prices

   = (8.4 - 0) / (48 - 33.32) = 0.572

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