Here given problem we had.
x0 = $ 1.2
x1 = $ 1.5
S0 = 4
u = 2
d = 1/2
r = 1/4
k = 5
t = 1
then S1(H)= USo ; S1(H)=8
So(T) = d So; S1(T) =1/2 *4,S1(T)=2
Than the hedge the option we may show with the capital x0 = $ 1.2 ,Δ0 = 1/2 shares at the time 0,and the time 0, and that investing (traingle zero s0 is 1.2 - 1 /2*2
= 1.2 - 2 = 0.8
--- At that time 1 the cash position wwould be (1+r) (Xo - Δ0) = -1,and that the stock position would be either
= 1/2 S0 (H) =4 or 1/2 S0 (T) =1
= X1(H) = 1/2 S1(H) + (1+R) X0 - Δ0 S0) =3
= 1//2 * 8 + (1 + 1/4) (0.8)
(OR)
= X1(T) =1/2 S1(T)+(1+R) X0 - Δ0 S0) =0
= 1/2*2 + (1+1/4) (-0.8)
On the other hand pay off of call was ( S1(H)--K)^T = 8-5 =3 IF W =H ;OR ( S1(T)--K)^T = (2-5)^T=0 IF W =T
As we see that the stock and that bond portfolio worth x0 = 1.2 hedges (replicate) the call . By the no arbitrage principle time - 0 value of the call option must be X0 = 1.2
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