Problem 3: Derivatives Valuation (6 marks) 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-ST),0])2 where ST 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 (ST in the payoff function refers to the stock price when the option is exercised), should it be exercised early?