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Let’s go through each part of the problem step by step.

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

To show that there is no arbitrage in this market, we need to demonstrate that there exists a risk-neutral probability measure under which the discounted price processes are martingales. In other words, we need to find probabilities \( q_1, q_2, \) and \( q_3 \) corresponding to each possible outcome such that:

1. The discounted expected value of \( S^1_1 \) under this probability measure equals \( S^1_0 \).
2. The discounted expected value of \( S^2_1 \) under this probability measure equals \( S^2_0 \).

The given probabilities for each outcome are:
- \((S^1_1, S^2_1) = (15, 21)\) with probability 0.5
- \((S^1_1, S^2_1) = (10, 22.5)\) with probability 0.25
- \((S^1_1, S^2_1) = (10, 22.2)\) with probability 0.25

The risk-free bond has \( B_0 = 1 \) and \( B_1 = 1.1 \), so the discount factor is \( \frac{1}{1.1} \).

For no arbitrage to hold, the expected discounted values should satisfy:
\[
\frac{1}{1.1} \mathbb{E}_q[S^1_1] = S^1_0 \quad \text{and} \quad \frac{1}{1.1} \mathbb{E}_q[S^2_1] = S^2_0.
\]

Solving for the appropriate probabilities will show that arbitrage does not exist.

### Part (ii): Consider the contingent claim \( X = \sqrt{S^1_1} + \sqrt{S^2_1} \). Show that there exists a portfolio \( h^* \) that replicates \( X \).

The goal here is to show that there is a portfolio \( h^* \) such that the payoff at \( t = 1 \) of this portfolio matches \( X \). Since we have a finite state market, and each state has specified values of \( S^1_1 \) and \( S^2_1 \), we can construct a portfolio in terms of \( S^1 \) and \( S^2 \) that matches the payoff \( X \) in each state.

To justify why such a portfolio \( h^* \) exists, note that since we have a complete market (with the given distribution covering all possible outcomes), any contingent claim can be replicated by a portfolio of \( S^1 \) and \( S^2 \).

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

In a no-arbitrage market, the price of a contingent claim is equal to the expected discounted payoff under the risk-neutral measure. Thus, the arbitrage-free price of \( X \) at time 0 is given by:
\[
\text{Price of } X = \frac{1}{1.1} \mathbb{E}_q[X],
\]
where \( \mathbb{E}_q[X] \) is the expectation of \( X \) under the risk-neutral probabilities found in part (i). Since \( h^* \) replicates \( X \), the price of \( X \) is indeed the value of \( h^* \) at time 0.

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Would you like further details on any part of this solution or have any specific questions?

Here are some related questions that could expand on the content:

1. How do we calculate the risk-neutral probabilities explicitly in a one-period binomial model?
2. What is the general approach to determine if a market is arbitrage-free?
3. Why does the existence of a risk-neutral measure imply no-arbitrage in financial markets?
4. How would the pricing change if there were transaction costs in this market?
5. How could we extend this problem to multiple periods?

**Tip:** In one-period models, verifying no-arbitrage often involves checking if we can match asset payoffs with expected values under a risk-neutral probability.

Consider a one-period market \( \mathcal{M} = (B_t, S^1_t, S^2_t) \) with prices satisfying \( B_0 = 1 \), \( B_T = 1.1 \); \((S^1_0, S^2_0) = (10, 20)\) and \((S^1_1, S^2_1)\) has the joint distribution: \((15, 21)\) with probability 0.5, \((10, 22.5)\) with probability 0.25, \((10, 22.2)\) with probability 0.25. The tasks are: (i) Prove this market contains no arbitrage. (ii) Consider the contingent claim \( X = \sqrt{S^1_1} + \sqrt{S^2_1} \) and show that there is a portfolio \( h^* \) that replicates \( X \). (iii) Prove that the arbitrage-free price of \( X \) is the value of the portfolio \( h^* \) at time 0.

Learn how to prove no-arbitrage in a one-period market with a joint distribution of asset prices, replicating a contingent claim \( X = \sqrt{S^1_1} + \sqrt{S^2_1} \). This solution involves risk-neutral pricing, discounted values, and constructing a replicating portfolio. Suitable for university-level financial mathematics students.

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