a. The arbitrage-free price of a 1-year forward contract can be derived using the following formula:

Forward Price = Stock Price + Dividend Yield + Risk-Free Rate

Given that the stock doesn't pay dividends in the next two years, the dividend yield is zero. Therefore, the forward price F can be calculated as:

F = S * (1 + r)^T

Where: - S is the current stock price, which is $50 per share. - r is the risk-free rate, which is 5% (annualized). - T is the time to expiration, which is 1 year.

F = 50 * (1 + 0.05)^1 F = 50 * 1.05 F ≈ 52.50

So, the arbitrage-free price of the 1-year forward contract is approximately $52.50 per share.

b. There seem to be violations of the put-call parity and the no-arbitrage principle based on the provided call and put premiums:

1. Put-Call Parity Violation: According to put-call parity, the price of a call option should be equal to the price of the corresponding put option, plus the stock price minus the strike price discounted at the risk-free rate. In this case, the call premium is $2.83 and the put premium is only $0.45 for a strike price of $50. The parity suggests that the call premium should be approximately equal to the put premium plus the difference between the stock price and the strike price discounted at the risk-free rate. However, the call premium is significantly higher than what we would expect given the low put premium, indicating a violation of the put-call parity.

Arbitrage Opportunity: An investor could sell the call option for $2.83, buy the put option for $0.45, and borrow money at the risk-free rate to buy the stock for $50. At the end of the year, the investor can exercise the put option to sell the stock back to the writer for $50, repay the loan, and pocket the difference between the call premium received and the cost of buying the stock and borrowing the money, which is approximately ($2.83 - $0.45 - $2.50) * (1 + 0.05)^1 ≈ $2.10.

1. No-Arbitrage Principle Violation: The no-arbitrage principle states that there should be no riskless profit opportunity in the market. The high call premium and low put premium suggest that there might be such an opportunity.

Arbitrage Opportunity: An investor could sell the call option for $2.83, sell the stock short at the current market price of $50, and invest the proceeds at the risk-free rate. At the end of the year, the investor can buy the stock back at the forward price of $52.50, repay the borrowed stock, and pocket the difference between the proceeds from selling the stock short and the investment at the risk-free rate, which is approximately ($50 - $52.50) * (1 + 0.05)^1 ≈ -$2.25.

These arbitrage opportunities indicate pricing inefficiencies in the options market, suggesting that the market is not fully efficient.