please show the work ~

3. (30 points) Consider a 2period portfolio optimization problem with t = 0, 1, 2. The utility function to be maximized is the same as in the class notes. Assume A = 2. There are only 2 securities in each period: the 1period riskfree rate, and a stock. Similar to the class notes, we use log returns so all the price below are really log prices and we compute return as difference in price. The one-period riskfree rate = 0 in each period. Solve the following two scenarios independently of each other. (a) Assume there are two possibilities at t = 1: stock price = $1.1 or $0.9, each with 50% probability. If stock price at t = 1 is $1.1, mean price at t = 2 is $1.2; if stock price at t = 1 is $0.9, mean price at t = 2 is $1. The volatility of stock price at t = 2 is 100%. The stock price at t = 0 is $1. What is your optimal allocation to the stock at t = 0'? What is the Markowitz allocation at t = 0? (b) Assume that everything else is the same as the previous part except that: If stock price at t = 1 is $1.1, mean price at t = 2 is $1.2; if stock price at t = 1 is $0.9, mean price at t = 2 is $0.9. What is your optimal allocation to the stock at t = 0? What is the Markowitz allocation at t = 0