1. You are stock price analyst for the company. You are given historical daily stock prices of your company from Day 1 to Day 700 (Assuming stock market trade daily). Assuming the stock price follows the Black-Scholes framework. Determine the historical annual volatility, σˆ of this stock. Next, estimate the expected annual rate of return αˆ. Assuming this is non-dividend paying stock, simulate the daily stock price for the next 90 days with the estimated annual volatility, σˆ and expected annual rate of return, αˆ. Then, simulate the daily stock price again for the next 90 days with 5%, 15%, 25% and 35% deviation of your estimated annual volatility (assuming no change in the estimated rate of return). (20 marks)

2. Suppose the exchange rate between Malaysia Ringgit (MYR) and United State of America Dollar (USD) follows the Black-Scholes framework. You are required to price the 30-day MYR-denominated Asian arithmetic average strike put option on USD. You are required to adopt Monte Carlo valuation for this pricing. Using the appropriate parameters (current exchange rate and volatility) and different sets of interest rates (MYR & USD), simulate daily exchange rate for next 30 days. Next, determine the price of the option. (30 marks)

3. Given the current exchange rate between dollars and euros is $1.50/e. Given also the following: 

• The exchange rate follows the Black-Scholes framework. 

• The continuously compounded risk-free interest rate in dollars is 2%. 

• The continuously compounded risk-free interest rate in euros is 6%. 

• The volatility of the exchange rate is 10%. A 3-year dollar-denominated Asian average price call option on euros has a settlement value based on the arithmetic average exchange rate at the end of each of three years. Its strike price is $1.40/e. By simulating 10,000 trials (each trial has 3 numbers of period), determine the average of the option prices produced by the 10,000 trials. (20 marks)

4. You are the market-maker for the FastOption company. Your company is planning to launch an 120-days Asian Options on Stock XYZ. You are required to simulate a 120- days Asian arithmetic average price option. To reduce the variance of the estimate, you use the control variate method, with a 120-days Asian geometric price option as the control variate. Given the following parameters: 

• The stock price is 55. • The strike price is 58. 

• The continuous dividend yield is 0.015. • The continuously compounded risk-free interest rate is 0.044. 

• The stock’s expected return is 0.25. Using Black-Scholes equation to obtain the price of a call option with the strike price of 58. Next, by simulation, estimate the price of call option with strike price 56, 57, 58, 59 and 60 respectively. (30 marks)