Draw a 2-period binomial tree for a stock with $800 price and 50% standard deviation.

a) The annual risk-free rate is 5%. Calculate the periodic risk-free rate used for an option expiring in 201 days in a two-period binomial tree.

b) Calculate the up and down parameters as well as the risk-neutral probabilities. Draw the tree.

c) Price a European put option with an exercise price of $820.

d) Calculate the put's hedge ratio at the beginning that makes risk-neutral pricing possible. Demonstrate the value of the hedge portfolio of buying 1000 put options at the beginning and both when the stock price goes up and goes down (once). How many stocks do you have to buy or sell at point d to rebalance the hedge portfolio for 1000 puts?

e) Price an American put with the same strike of $820. f) Calculate the value of the European put in point c) using the Black-Scholes-Merton formula. Don't forget to use the risk-free rate as a log return.