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-S_t),0])^2

where S_t = stock price in six months.

a. What are the payoffs at the final nodes of the tree? [1 mark]

b. Use no-arbitrage arguments (you need to show how to set up the riskless portfolios at the different nodes of the binomial tree). [2 mark]

c. Use risk-neutral valuation. [1 mark]

d. Verify whether both approaches lead to the same result. [1 mark]

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? [1 mark]