Problem 3: Derivatives Valuation

Could you please help me with this homework question and show working too please.

A stock price is currently $36. During each three-month period for the next six months it is expected to increase by 9% or decrease by 8%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off (max[(40-S_T),0])² where is the stock price in six months.

a.What are the payoffs at the final nodes of the tree?

b.Use no-arbitrage arguments (you need to show how to set up the riskless portfolios at the different nodes of the binomial tree).

c.Use risk-neutral valuation.

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

e.If the derivative is of American style (S_T in the payoff function refers to the stock price when the option is exercised), should it be exercised early?