Question

1.2. You have a stock in the one-period binomial model such that So and r= 4, S1(H) = 8, S, (T) = 2, 1.5. (a) Show that this setup violates the no-arbitrage assumption. (b) Show that there is a portfolio in the one-period model such that Xo X1 > 0. Such a portfolio is an example of arbitrage extraction. 0, Ao #0, and %3D 1.3. With the same stock as in problem 1.2 but with r = 0.25, suppose that you have found a derivative security that has value V(H) portfolio such that X1 = V This portfolio is an example of a replicating portfolio for the 2 and Vi(T) = 0. Calculate Xo and Ao for a %3D %3D security.

Answer 1

1.2

a. S₀ = 4;  S₁(H) = 8; S₁(T)=2; r = 1.5

Prove that this setup violates the no-arbitrage assumption. i.e. No-arbitrage assumption- 0 < d < 1+r < u

We have the beginning of the period time 0 and the end of period time 1.

At time 0, we have a positive stock price S₀=4

At time 1, the Stock price will either be S₁(H) or S1(T) (H=Heads and T=Tails) Note:  Probability of Heads is p and Tails is q=(1-p).

S₀may increase by the ratio u with probability p, or decrease by the ratio d with probability (1-p)

We introduce two positive numbers u = S₁(H)/S₀ and d = S₁(T)/S₀

u = 8/4 = 2

d = 2/4 = 0.5

No arbitrage assumption = 0 < d < 1+r < u = 0 < 0.5 < 1+1.5 < 2 = 0 < 0.5 < 2.5 < 2, But 2.5 > 2, which is violating the assumption.

Therefore, this setup violates the no-arbitrage assumption.

b. Not sure about the answer.

1.3   r=0.25

V₁(H) = 2; V₁(T) = 0  

Δ₀ = (V₁(H) - V₁(T)) / (S₁(H)/S₁(T))

Δ₀ = (2-0)/(8-2) = 2/6 =1/3

X₀ + Δ₀(((1/1+r)*S₁(H)) - S₀) = (1/1+r) * V₁(H)

= X₀ +1/3((8/1.25) - 4) = 2/(1+0.25)

Therefore, computing the value of X₀, we get X₀ = 0.8 and Δ₀ = 1/3