In the American option valuation problem, suppose that at time 0 , the share price is \(x_{1}=\$ 20\). Return rates are \(u=.07\) and \(d=0\). Use the remark in the section to compute the probability \(q\) if there is a non-risky asset whose rate of return is \(r=.02\). For an option that expires in 4 time units and has exercise price \(\$ 23\), draw a tree diagram representing the possible motions of the stock price. If the discount factor is \(\alpha=.995\), find the value of the option at each node of the tree, in particular at time 0 .