Question

A stock price is currently $42. Its stock price will be either $45 or $38 one year from now. The risk-free rate is 5%. A one-year call on the stock has an exercise price of $40.

(a) What are its intrinsic values at stock prices of $45 and $38, respectively?

(b) What should be the hedge ratio?

(c) What should be the value of the hedged portfolio at expiration?

(d) What is the value of the call today?

(e) Verify your answer using the risk-neutral approach—do not just say that you have the same answer; you will need to show the work that the two approaches give the same answer

Answer 1

intrinsic value

= current stock price - exercise price

= 45-40=5

=38-40=-2

hedge ratio

H = (Cu - Cd)/(uSo - dSo)

=5-0/45-38=5:7

value of the hedged portfolio=(1+r-d/u-d)(c1/1+r)+(u-1-r/u-d)(c2/1+r)

c1=5

c2=-2 it means 0

d=38/42=0.905

u=45/42=1.0714

(1+0.05/1.0714-0.905)(5/1+0.05)+0

=(6.31)(4.762)=30.048

value of the call=(1+r-d/u-d)(c1/1+r)+(u-1-r/u-d)(c2/1+r)

c1=5

c2=-2 it means 0

d=38/42=0.905

u=45/42=1.0714

(1+0.05/1.0714-0.905)(5/1+0.05)+0

=(6.31)(4.762)=30.048