Homework 2 Derivatives

Fin 441 -

Homework # 2 

Do not use Excel Spreadsheet for homework. You must show calculations in detail to earn credit. Homework must be typed (except binomial tree and table showing hedge) with one-inch margin on all sides.  

1. Consider a stock currently trading at 25 that can go up or down by 15 percent per period. The risk-free rate is 10 percent. Use one-period binomial model.

1. Determine the two possible stock prices for the next period.

Su= $25 (1+ rate of return of stock going up) = $25 (1.15) = $28.75

Sd= $25 (1- rate of return of stock going down) = $25 (1 - .15)

= $25 (.85) = $21.25

1. Determine the intrinsic values at expiration of a European call with can exercise price of 25.

Cu= Max (0, Su – X) = Max (0, $28.75 - $25) = $3.75

Cd= Max (0, Sd – X) = Max (o, $21.25 - $25) = $0

1. Find the value of the option today.

p = (Cu-Cd)/(u-d) = (1 + .10 - .15)/ (1.15 - .85) = (1.10 - .85) / (1.15 - .85) = .8333        

C = (p Cu + (1-p) Cd) / (1+r)

   = (.8333 ($3.75) + (1 - .8333)0)/ (1.10) = $3.12 / $1.10

   = $2.84

1. Construct a hedge by combining a position in stock with a position in the call. Calculate the hedge ratio and show that the return on the hedge portfolio is the risk-free rate regardless of the outcome, assuming that the call trading at the price obtained in part c.

h = (Cu-Cd) / (Su-Sd) = ($3.75 – 0) / ($28.75 - $21.25) = $3.75 / $7.50 = .50 * 1000 = 500

V = hS – C = 500($25) – 1000($2.84) = $9,660

Vu = hSu – Cu = 500($28.75) – 1000($3.75) = $10,625

                rh= ($10625 / $9660) – 1 = .10

Vd = hSd – Cd = 500($21.25) – 1000($0) = $10,625

                rh = ($10,625 / $9.660) – 1 = .10

1. Determine the rate of return from a risk-free hedge if the call is trading at 3.50 when the hedge is initiated.

V = hS – C = 500($25) – 1000($3.50) = $9,000

Vu = hSu – Cu = 500($28.75) – 1000($3.75) = $10,625

                rh = ($10,625 / $9,000) – 1 = .1810

Vd = hSd – Cd = 500($21.25) – 1000($0) = $10,625

                rh = ($10,625 / $9,000) – 1 = .1810

1. Consider a two-period, two-state world. Let the current stock price be 45 and the risk-free rate be 5 percent. Each period the stock price can go either up by 10 percent or down by 10 percent. A call option expiring at the end of second period has an exercise price of 40.

1. Find the stock price sequence.

S = $45        Su =$45(1.10) = $49.50        Sd = $45(.90) = $40.50

Su^2 = $45(1.10)^2 = $54.45         Sud = $45(1.10)(.90) = $44.55        Sd^2 = $45(.90)^2 = $36.45

1. Determine the possible prices of the call at expiration.

Cu^2 = Max(0, Su^2 – x) = Max(0, $54.45 - $40) = $14.45

Cud = Max(0, Cud – x) = Max(0, $44.55 - $40) = $4.55

Cd^2 = Max(0, Cd^2 – x) = Max(0, $36.45 - $40) = $0

1. Find the possible prices of the call at the end of the first period.

p = (1 + r -d) / (u-d) = (1.05 - .90) / (1.10 - .90) = .75