A stock price is currently $100. Over each of the next two six-month periods, it is expected to go up by 10% or down by 10%. The risk-free interest rate is 10% per year with semi-annual compounding.

Part I.

Use the two-steps binomial tree model to calculate the value of a one-year American put option with an exercise price of $101.

Part II.

Is there any early exercise premium contained in price of the above American put option? If there is, what is the early exercise premium?

Part III.

Based on no arbitrage principle and riskless portfolio we can construct along the above binomial tree, briefly discuss how we can hedge risk if we write a European put option with an exercise price of $101 and 1-year maturity.