Problem 3: Derivatives Valuation

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 S_T is the stock price in six months.

1. Use risk-neutral valuation.
2. Verify whether both approaches lead to the same result.
3. 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?