Assume a finite state economy with three assets whose payoff matrix is

given by

$x=\left[\begin{array}{ccc}110& 100& 48\\ 110& 50& 40\\ 110& 40& 36\end{array}\right]$

a$.$ Is there a risk$-$free asset in the market?

b$.$ Suppose that asset prices are $$100,$$$70,$ and $$40,$ respectively. Is

there an arbitrage opportunity in the market?

c$.$ Suppose there is an asset that hedges the downside risk with $$10$

payoff in the third $($down$)$ state and nothing in other two states.

What should the price of this asset be$?$

d$.$ What are the risk$-$neutral probabilities?

e$.$ Using the risk$-$neutral valuation approach, recalculate the asset that

hedges the downside risk with a $$10$ payoff in the third $($down$)$ state

and nothing in other two states.