Today is 10th August 2022, when the BHP Group Ltd (BHP.AX)'sshare is priced at $38.56. Consider an BHP option that expires on 15th December 2022, with a strike price of $42. Assume no dividends are paid on this stock. The annual historical standard deviation of BHP stocks is 33.57%. The risk-free rate is 2.5628% with annual compounding. Ignore transaction costs (e.g. bid-ask spread).

a) Using a three-step (i.e three periods) binomial tree model, calculate theoretical prices today for i) a December American put option, ii) a December European call option, and iii) a December European put option on BHP stock. Draw the trees and show all calculation workings at each node.