This is a question for Mathematical Finance. Please help to answer all sub-questions. Your help is much appreciated!

= Question 2 (40%) Consider an economy with a risk-free bond and a stock (that does not pay dividends). The initial value of the bond is Bo = 1, the initial value of the stock is so $100. The risk-free interest rate per annum with continuous compounding is r = loge(1.05). Every eight months the value of the stock can either go up by 10% with probability 1/2 or go down by 5% with probability 1/2. The return of the stock over each 8 month period is independent of the return in previous months. (a) Consider an at-the-money American put option with 2 year maturity. What is the price of the option today? Approximate to the 4th decimal digit. (b) Consider a 2-year maturity Asian option with discretely sampled averaging A2 = Li_1 S2-1/3, where St is the stock price in t years, and payoff (A2 S2)+. = i) What is the price of the option today? Approximate to the 4th decimal digit. ii) What's the real-world probability that its payoff is strictly positive? Approximate to the 4th decimal digit. = Question 2 (40%) Consider an economy with a risk-free bond and a stock (that does not pay dividends). The initial value of the bond is Bo = 1, the initial value of the stock is so $100. The risk-free interest rate per annum with continuous compounding is r = loge(1.05). Every eight months the value of the stock can either go up by 10% with probability 1/2 or go down by 5% with probability 1/2. The return of the stock over each 8 month period is independent of the return in previous months. (a) Consider an at-the-money American put option with 2 year maturity. What is the price of the option today? Approximate to the 4th decimal digit. (b) Consider a 2-year maturity Asian option with discretely sampled averaging A2 = Li_1 S2-1/3, where St is the stock price in t years, and payoff (A2 S2)+. = i) What is the price of the option today? Approximate to the 4th decimal digit. ii) What's the real-world probability that its payoff is strictly positive? Approximate to the 4th decimal digit