Consider the following data available for an asset under a binomial tree model with two steps: u=1.08, d=0.95, asset price S=160, and risk-free rate r=2%. An American put option on the asset has a strike price K of 165:

a) Estimate the risk-neutral probability.

b) Estimate the value of the American put option and plot the binomial tree with the estimated final and intermediate option prices and payoffs at each node. Is an early exercise optimal?

c) What is the value of a European put option given the same data?

d) Estimate the volatility .

e) Calculate delta at the end of the first time step for the European put option. What is the interpretation of the delta estimate?