Let's tackle each part of this problem step-by-step.

Part (i): Prove this market contains no arbitrage.

To prove there is no arbitrage, we need to check if there exists a risk-neutral probability measure under which the discounted price processes of all assets are martingales.

1. Risk-Free Asset: Given $B_0 = 1$ and $B_T = 1.1$, the risk-free rate $r$ can be computed as: $r = \frac{B_T}{B_0} - 1 = 0.1$

2. Asset Price Dynamics: We have two assets with initial prices $S_0^1 = 10$ and $S_0^2 = 20$. Their prices at time $T$ follow the distribution:

   • $(S_1^1, S_1^2) = (15, 21)$ with probability 0.5
   • $(S_1^1, S_1^2) = (10, 22.5)$ with probability 0.25
   • $(S_1^1, S_1^2) = (10, 22.2)$ with probability 0.25

3. Check for Arbitrage: We need to verify if there exists a set of probabilities $q_1, q_2, q_3$ for the possible outcomes such that the expected discounted price at $T$ is equal to the price at $t = 0$.

   Define the probabilities of the outcomes as $q_1, q_2, q_3$ such that: $0.5q_1 + 0.25q_2 + 0.25q_3 = 1$ Then calculate the expected values to ensure they satisfy the no-arbitrage condition. If such probabilities exist, then the market contains no arbitrage.

Part (ii): Show that there is a portfolio $h^*$ that replicates $X = \sqrt{S_1^1} + \sqrt{S_1^2}$.

To prove the existence of a replicating portfolio $h^*$, we need to show that the contingent claim $X$ can be expressed as a linear combination of the asset prices at time $T$.

1. Formulate $X$ in Terms of $S_1^1$ and $S_1^2$: The claim $X = \sqrt{S_1^1} + \sqrt{S_1^2}$ is dependent on the terminal values of $S_1^1$ and $S_1^2$.

2. Linear Combination: Since the market has two assets and one risk-free asset, a self-financing portfolio $h^*$ can potentially replicate $X$. By the completeness of the market (assuming it is complete), there exists a portfolio $h^*$ that can replicate the claim $X$.

3. Justification: Since $X$ can be expressed as a function of $S_1^1$ and $S_1^2$, the market’s completeness implies the existence of a replicating portfolio.

Part (iii): Prove that the arbitrage-free price of $X$ is the value of the portfolio $h^*$ at time 0.

1. Arbitrage-Free Price: In a market with no arbitrage, the price of a contingent claim $X$ at time 0 is the expected value of its payoff under the risk-neutral measure, discounted by the risk-free rate.

2. Value of $X$ at Time 0: The arbitrage-free price of $X$, denoted by $\pi(X)$, is: $\pi(X) = \mathbb{E}^Q \left[ \frac{X}{1.1} \right]$ where $Q$ is the risk-neutral probability measure.

3. Portfolio Value: Since $h^*$ replicates $X$, the value of $h^*$ at time 0 must equal $\pi(X)$, as any deviation would imply an arbitrage opportunity.

Additional Questions

1. What are the steps to calculate the risk-neutral probabilities $q_1, q_2, q_3$?
2. How does the presence of a risk-free asset influence the arbitrage conditions in this market?
3. Can you construct an explicit replicating portfolio $h^*$ for $X$?
4. How would the solution change if the market were incomplete?
5. What assumptions are necessary for the existence of a risk-neutral measure?

Tip: To determine no-arbitrage conditions, focus on ensuring that expected discounted prices align with initial values under some probability distribution.