Consider a two$-$period securities market model, with two risky securities $S^1,S^2.$ The dynamics

of the price processes $(S_t^1,S_t^2)$ are shown in the tree below.

In addition, there is a riskless bank account $S^0.$ The $($discretely compounded$)$ interest rate $r$

which governs the dynamics of $S^0$ is constant.

$($a$)$ Assume temporarily that the model is arbitrage$-$free. Show that $r=20\%.$

$($b$)$ Now show that the model is indeed arbitrage$-$free, by finding an EMM for this model.

$($c$)$ Is this model complete? Why or why not?

$($d$)$ Consider a chooser option $x$ on $S^1,$ which allows the holder to decide, at time $t=1,$ whether

the option is a call on $S^1$ with strike $K=21\mathrm{.}60$ and maturity $T=2,$ or a put with strike

$K=21\mathrm{.}60$ and maturity $T=2.$ What are the $t=1-$ and $t=0-$prices of this option?