I need to look into a one-period Binomial model $(B_t, S_t)$ with interest rate $r = 0.1$ , $S_0 = 100$ and $$ S_t= 120 \, \text{with probability}\, 0.5 $$ $$ S_t= 60\, \text{with probability}\, 0.5 $$
a) I need to show this market has no arbitrage. b) Find the replicating portfolio for the contingent claim $X = S^2_T$. c) Find the price of $X$ that leads to no arbitrage. d) Find the risk-neutral measure for this market.
Now I guess that in a) I need to show that the state-price vector is positive, but how do I do that? For the rest I have no idea and would welcome your help!
To get you started, let $\omega_1$ and $\omega_2$ denote the two market states at terminal time $T$.
There are two securities, a bond with initial price $B_0 = 1$ and terminal prices $B_T(\omega_1) = B_T(\omega_2) = 1.1$ and, a stock with initial price $S_0 = 100$ and terminal prices $S_T(\omega_1) = 120$ and $S_T(\omega_2) = 60$.
For (a), the state-price vector $(q_1,q_2)'$ is the solution of
$$\pmatrix{B_T(\omega_1) & B_T(\omega_2) \\S_T(\omega_1) & S_T(\omega_2)}\pmatrix{q_1\\q_2 }=\pmatrix{B_0\\S_0 },$$
which reduces to
$$1.1 q_1 + 1.1 q_2 = 1\\ 120 q_1 + 60 q_2 = 100$$
The existence of the solution $q_1 \approx 0.7576, \, q_2 \approx 0.1515$ implies no arbitrage.
For (b), let $\alpha$ and $\beta$ be the holdings of stock and bond in the portfolio that replicates the contingent claim $X = S_T^2$. The value of the portfolio must be identical to the value of the claim in all terminal states, implying
$$\alpha S_T(\omega_1) + \beta B_T(\omega_1) = S_T(\omega_1)^2\\ \alpha S_T(\omega_2) + \beta B_T(\omega_2) = S_T(\omega_2)^2$$
This reduces to
$$120 \alpha +1.1 \beta = (120)^2\\ 60 \alpha + 1.1 \beta = (60)^2$$
with the solution $\alpha = 180$ and $\beta \approx -6545.45$.
From here it should be obvious how to find the fair value of the claim for part (c) as the initial value of the replicating portfolio. I'll let you think about how to find the risk-neutral measure.