Question $4$: A one$-$step binomial market has assets $x$ and $Y$ with initial values $x_0$ and $Y_0.$ Terminal

values in the up$-$state are $x_u$ and $Y_u.$ Terminal values in the down$-$state are $x_d$ and $Y_d.$ Both branching

probabilities are strictly positive.

a$)$ Derive the conditions which guarantee that there is no arbitrage in this market.

b$)$ A third asset has values $Z_u$ in the up$-$state and $Z_d$ in the down$-$state. Find the value of $Z_0$ such that

there is no arbitrage in the market.