There are two companies, A and S$,$ whose stocks are selling at $$9$ and $$7$ respectively. Company A decided to sue company S over patent infringements. When the outcome of the lawsuit is announced, if A wins, its stock will rise to $$15,$ and S$$s stock will decrease to $$5.$ If A loses, its stock will be worth $$6$ and S$$s stock will jump to $$10.$ Suppose stocks pay no dividends between now and the announcement.

$($a$)$ Suppose the two states of the world are represented by whether company A wins or loses the lawsuit. Calculate the prices of Arrow$-$Debreu securities in this economy, assuming no$-$arbitrage.

$($b$)$ Suppose there is a riskless asset in the economy that pays $$1$ in both states. Calculate the price of such an asset, assuming no$-$arbitrage. What is the riskless interest rate in this case?

$($c$)$ Now suppose a riskless bond $($say T$-$bills$)$ yields an interest rate of $5\%$ in the next period when the dispute is solved. Is there an arbitrage opportunity? If the answer is yes, outline how you would trade to exploit it assuming no costs of short$-$selling.