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

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### 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.

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### 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.

Consider one-period market \( \mathcal{M} = (B_t, S_t^1, S_t^2) \) with prices satisfying \( B_0 = 1, B_T = 1.1; (S_0^1, S_0^2) = (10, 20) \) and \( (S_1^1, S_1^2) \) has the joint distribution:

- \((15, 21)\) with probability 0.5
- \((10, 22.5)\) with probability 0.25
- \((10, 22.2)\) with probability 0.25

(i) Prove this market contains no arbitrage.
(ii) Consider the contingent claim \( X = \sqrt{S_1^1} + \sqrt{S_1^2} \). Show that there is a portfolio \( h^* \) that replicates \( X \) without finding \( h^* \) explicitly.
(iii) Prove that the arbitrage-free price of \( X \) is the value of the portfolio \( h^* \) at time 0.

Solve a finance problem involving a one-period market \( \mathcal{M} = (B_t, S_t^1, S_t^2) \) with given asset prices and joint distribution. Learn to prove no arbitrage, construct a replicating portfolio, and find the arbitrage-free price for a contingent claim \( X = \sqrt{S_1^1} + \sqrt{S_1^2} \). Suitable for undergraduate finance students.

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