(a) Draw a tree.

(b) Use replicating portfolios to calculate the price of a European at-the-money put

option with 1 year left to maturity.

(c) Use "risk neutral" probabilities to price a European at-the-money put option with

1 year left to maturity.

(d) What is the price of an American at-the-money put option with 1 year left to

maturity?

Today (time t=0) the stock price of company Z is S0=50. In 6 months (time t=0.5) it changes (under the true probability measure P ) with probability of 60% to S0.5(u)=65 and with probability 40% to S0.5(d)=40. From time t=0.5 to time t=1 the stock price may increase by 20% with a probability of 80% or decrease by 30% with a probability of 20%. The risk free interest rate is constant and equal to 5% (c.c.). The yield curve is flat. Today (time t=0) the stock price of company Z is S0=50. In 6 months (time t=0.5) it changes (under the true probability measure P ) with probability of 60% to S0.5(u)=65 and with probability 40% to S0.5(d)=40. From time t=0.5 to time t=1 the stock price may increase by 20% with a probability of 80% or decrease by 30% with a probability of 20%. The risk free interest rate is constant and equal to 5% (c.c.). The yield curve is flat