Binomial Trees

Consider a stock which currently sells for $40.$ Assume that during each two$-$month period for the next four months this share price is expected to increase by $2\%$ or decrease by $2\%$ and the risk$-$free interest rate is $2\mathrm{.}5\%$ per annum $($cont$.$ comp.$).$ Consider an exotic derivative that has a payoff given by the formula $($max$[(42\mathrm{.}50-$ST$),0])2$ where ST is the stock price in four months.

a$.$ Draw a two$-$step binomial tree and populate the individual nodes with the share price values at

each node.

b$.$ If this derivative is of European$-$style, value the derivative using no$-$arbitrage arguments.

c$.$ If this derivative is of European$-$style, value the derivative using risk$-$neutral valuation.

d$.$ Verify whether both approaches lead to the same result.

e$.$ If the derivative is of American style, should it be exercised early if the payoff at time t is given by the formula $($max$[(42\mathrm{.}50-$ St$),0])2?$

Note: When you use no$-$arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the individual nodes of the binomial tree.