Consider a three$-$period economy $(t=0,1,2)$ with two assets, stock and bond. The stock only pays at

$t=2.$ Its cum$-$dividend price $($i$.$e$.$ before dividend$)$ follows the process $S_{t+1}=S_t{H}_{t+1},$ where $H_{t+1}$ takes

values $u>1$ and $\frac{1}{u}$ with equal probabilities. The bond also pays at $t=2$ only. Its price $($also before

payment$)$ follows the process $B_{t+1}=B_{t}R,$ where $R$ is a constant. Without loss of generality, you can

assume that $S_0=B_0=1.$

$($a$)$ Derive the prices of all state$-$contingent claims. What are conditions on $R$ and $u$ implied by no

arbitrage?