$1.(50$ points$)$ Consider a binomial model with three dates $\backslash ($ t$=0,1,2\backslash ),$ two events $"$up$"$ and "down" at date $1,$ and two date$-2$ successors $("$up$"$ and "down"$)$ of each date$-1$ event. The price of the risky asset is $\backslash ($ S$\_\{0\}=50\backslash )$ at date $0$ and moves up by factor $\backslash ($ u$=1\mathrm{.}2\backslash )$ or down by factor $\backslash ($ d$=0\mathrm{.}8\backslash )$ each period. One$-$period risk$-$free rate of return is $\backslash ($ r$=0\mathrm{.}1\backslash ),$ or $\backslash (10\backslash \%\backslash ),$ the same at dates $1$ and $2.($i$)$ Find strictly positive state prices and risk$-$neutral probabilities. It is enough to provide one$-$period or conditional $\backslash ($ q $\backslash )\text{'}$s and $\backslash (\backslash$pi $\backslash )\text{'}$s$.$ Are asset prices arbitrage free? Explain. $($ii$)$ Find date$-0$ price of European call option on the risky asset with expiration date $2$ and exercise price $46.$ Find date$-1$ prices of the option in events $"$up$"$ and "down." $($iii$)$ Suppose that the call option is American so that it can be exercised at date $1.$ Is it better to exercise the option or sell it at date $1$ in event $"$up$"?$ Do the same comparison for event "down." $($iv$)$ Find date$-0$ price of European put option on the risky asset with expiration date $2$ and exercise price $46$ using the put$-$call parity relationship. What is the payoff of this put option at date $\backslash (2?\backslash )$