Question $2.$ Suppose that the risk$-$free bond price evolves in the following way;

$A(0)=100,A(1)=108,A(2)=116$

Further suppose that a risky asset pays no dividends and has the following possible scenarios.

$2-$a$)$ Draw a tree for the dynamics of the stock. Compute $P^{**}({}_i),i=1,2,3,4.$

$2-$b$)$ Motivated by the fact that under any scenario $S(2)S(1)$ and there are scenarios where

the inequality is strict, investor is trying to create an arbitrage portfolio. Find an arbitrage

portfolio, or argue that it cannot be done.

$2-$c$)$ Find the fair price at time $t=0$ of a European put on the underlying stock with exercise

date $T=2$ and strike price $K=100.$

$2-$d$)$ Deduce the fair price at time $t=0$ of an American call option on the underlying stock

with exercise date $T=2$ and strike price $K=100.$ Should the American option be exercised

early? If so$,$ when?

$2-$e$)$ Find the fair price at time $t=0$ of an American put option on the underlying stock

with exercise date $T=2$ and strike price $K=100.$ Should the American option be exercised

early? If yes, when?

$2-$f$)$ Find the fair price at time $t=0$ for an American derivative where the payment function

is $f(s)=12-\sqrt[\phantom{2}]{s}($that is$,$ if the derivative is exercised at time $n$ and the stock's price is

$S(n),$ the the holder receives $f(S(n)).$ Should the option be exercised early? If yes, when?