Question

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 the second period has an exercise price of 40.

1. Find the stock price sequence.

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

3. Find the possible prices of the call at the end of the first period. 4. What is the current price of the call?

Answer 1

1) Calculation of stock price sequence:

Su=45(1.10)=49.50

Su2=45(1.10)2=54.45

Sud=45(1.10)(0.90)=44.55

Sd=45(0.90)=40.50

Sd2=45(0.90)2=36.45

2) Calculation of possible prices of the call at expiration:

Cu2=Max[0,Su2-X]=Max[0,54.45-40]=14.45

Cud=Max[0,Sud-X]=Max[0,44.55-40]=4.55

Cd2=Max[0,Sd2-X]=Max[0,36.45-40]=0

3) Calculation of possible prices of the call at the end of the first period:

p=(1+r-d)/(u-d)= 1+0.05-0.90/1.10-0.90=0.15/0.90=0.75

1 – p = 0.25

Cu = [(14.45)(.75) + (4.55)(.25)] / 1.05 = 11.40

Cd = [(4.55)(.75) + (0.00)(.25)] / 1.05 = 3.25

4) Calculation of current price of the call:

C = [(.75) (11.40)+ (.25)(3.25)] / 1.05 = 0.892