Binomial Tree Model for Derivative Pricing and Risk Hedging

Part I: To determine the fair price of the derivative contract, we can use a one-step binomial tree model.

First, let's calculate the two possible stock prices at the end of three months. If the stock price increases by 10%, it will be $20 * (1 + 0.10) = $22. If the stock price decreases by 10%, it will be $20 * (1 - 0.10) = $18.

Next, let's calculate the expected payoff of the derivative contract at maturity for each of the two possible stock prices. If the stock price is $22, the payoff is $22^2 = $484. If the stock price is $18, the payoff is $18^2 = $324.

Now, let's calculate the risk-neutral probabilities for the two possible stock prices. The risk-neutral probability of the stock price increasing is (1 + r - d) / (u - d), where r is the risk-free interest rate and u and d are the up and down factors of the binomial tree model. In this case, u = 1.10, d = 0.90, and r = 0.10/4 = 0.025 (quarterly compounding).

The risk-neutral probability of the stock price increasing is (1 + 0.025 - 0.90) / (1.10 - 0.90) = 0.525. The risk-neutral probability of the stock price decreasing is 1 - 0.525 = 0.475.

Finally, let's calculate the fair price of the derivative contract using the risk-neutral probabilities and expected payoffs. The fair price is the present value of the expected payoff, discounted at the risk-free interest rate.

The fair price of the derivative contract = (0.525 * $484 + 0.475 * $324) / (1 + 0.025) = $403.80.

Therefore, the fair price of the derivative contract with a payoff given by S_T^2 after three months is $403.80.

Part II: To hedge against the risk of selling the contract and ensure no loss when the stock price turns out to be $22 at the end of three months, we need to replicate the payoff of the derivative contract.

Since the derivative contract has a payoff of S_T^2, we need to replicate this payoff using a combination of stocks and bonds.

To replicate the payoff, we need to own a certain number of stocks and borrow or lend an amount of money to balance the payoff.

Since the stock price is expected to be $22, we need to own S_T^2 / S = $22^2 / $22 = $22 worth of stocks.

To finance this stock position, we can borrow the present value of $22 at the risk-free interest rate. The present value of $22 is $22 / (1 + r)^3, where r is the quarterly risk-free interest rate.

Therefore, the risk-hedging approach would involve buying $22 worth of stocks and borrowing $22 / (1 + r)^3.

Part III: If a trader agrees to buy or sell the above contract at a price of $500, the strategy to generate a profit would depend on the fair price of the contract determined in Part I.

If the fair price of the contract is higher than $500, there is an opportunity to generate a profit by selling the contract for $500 and then buying it back at a lower price.

If the fair price of the contract is lower than $500, there is an opportunity to generate a profit by buying the contract for $500 and then selling it at a higher price.

However, without knowing the actual fair price of the contract, it is not possible to determine the specific strategy to generate a profit.