Consider the non-dividend-paying stock in Homework 3, No.1. Its current price S(0) = S is still $40, yet the economic situation has changed and after each period, there is a 35% chance that the stock price goes up by 20%, 40% chance that the stock price stays the same, and 25% chance that the stock price drops by 10%. A European call option and a European put option on this stock expire on the same day in 4 months at a $43 strike. The current risk-free interest rate is 6% per annum, compounded monthly. Count a month as one period.

(a) Construct a four-period binomial lattice tree to calculate the stock price after three months.

(b) Find risk-neutral probabilities q1, q2, and q3 that make the expectation of stock price after one period (1+rf )S(0) where rf is the risk-free rate per period, and the standard deviation 0.1. (In other words, find q1, q2, and q3 such that E[X] = (1 + rf )S(0) and Var(X) = 0.01 where X is the random variable that represents the stock price after one period).

(c) Use the above risk-neutral probabilities and a four-period binomial lattice tree to calculate the current (t = 0) call option price.

(d) Use the same method as above to calculate the current (t = 0) put option price.

(e) Use Put-Call Parity to verify your answers from (c) and (d). If there is any error (discrepancy), provide your opinion on what caused the discrepancy.