1) Answer this question using the data on daily closing prices provided in the spreadsheet Assignment Data.xlsx for Securities A, B, and C, for the time period from July 2, 2012 to June 30, 2014.1 (20 marks in total as indicated in each part) a) For each security in the data, calculate: (i) the expected daily return (1 mark); (ii) the annualized volatility (1 mark); and (iii) the correlations between stocks (1 mark). Based on your results: (iv) comment on which combination of two securities you would expect to provide the highest diversification potential (1 mark). b) Consider a possible investment comprising Security A and Security B. Assume that no short-selling is allowed. Using Excel: (i) produce a chart showing alternative risk-return combinations from this portfolio (1 mark); and (ii) interpret your results, in comparison to investing in either of the individual securities (2 marks). Also calculate the weights for portfolios consisting of these two securities that yield: (iii) the maximum expected return (1 mark); (iv) the minimum variance (1 mark); and (v) the maximum Sharpe ratio assuming a daily risk-free rate of 0.014% (1 mark). c) Now assume that an investor is interested in combining all three securities into an optimal portfolio. Assume that no short-selling is allowed. Using Excel: (i) construct a spreadsheet to calculate the optimal weights for each of the stocks such that they maximize the expected portfolio return for a given standard deviation of portfolio return (4 marks); and (ii) create a plot of the efficient frontier for the three risky assets (1 mark). Also calculate the weights for portfolios consisting of the three securities that yield: (iii) the maximum expected return (1 mark); (iv) the minimum variance (1 mark); and (v) the maximum Sharpe ratio assuming a daily risk-free rate of 0.014% (1 mark). d) Compare your results in parts (b) and (c) above, with respect to the portfolio providing the maximum expected return, the minimum variance portfolio, and the portfolio with the maximum Sharpe ratio (2 marks) (2) This question refers to the same price data as Question 1. Assume that on December 31, 2013, when the Security A price was $77.80, a trader sold 150,000 European call options on Security A with strike price K=$80 and expiration date June 30, 2014. Suppose that the option premium was $500,000. Further assume that the annualized standard deviation of Security A returns is 15%, the annualized risk-free rate is 3% and that Security A does not pay any dividend during the time period from December 31, 2013 to June 30, 2014. (20 marks in total as indicated in each part) a) Using the DerivaGem software, apply the Black-Scholes formula to calculate: (i) the price of the option (1 mark); (ii) delta (1 mark); (iii) gamma (1 mark); (iv) vega (1 mark); and (v) rho of the option (1 mark). Use weeks as the time unit with respect to time to exercise, i.e., from December 31, 2013 to June 30, 2014 there are 26 weeks corresponding to 0.5 years), and interpret each result. Further, using Excel provide graphs and explanation/interpretation showing: (vi) the relationship between the value of the option and the strike price (1 mark); (vii) the delta of the option as a function of the stock price (1 mark); (viii) the relationship between the gamma of the option and volatility (1 mark); (ix) the relationship between vega of the option and the stock price (1 mark); and (x) the relationship between rho of the option and the stock price (1 mark). b) (i) Explain what delta neutrality means and how the trader can hedge to make the option portfolio delta neutral, including a numerical example in your answer (4 marks). Assume that the trader rebalances the portfolio on a fortnightly basis, i.e., on the following dates to preserve delta neutrality: 14/1/2014, 28/1/2014, 11/2/2014, 25/2/2014, 11/3/2014, 25/3/2014, 8/4/2014, 22/4/2014, 6/5/2014, 20/5/2014, 3/6/2014, 17/6/2014. Provide a table that contains for each of the above dates: (ii) the underlying price (1 mark); (iii) the current delta of the option (1 mark); (iv) the number of units of underlying purchased/sold (1 mark); (v) the cost of underlying purchased/sold (1 mark); (vi) the cost of interest assuming an annualized risk-free rate of 3% (1 mark); and (vii) the cumulative cost of the hedge strategy (1 mark) (3) This question refers to the same price data as Question 1. Assume that an investor is interested in monitoring the volatility of Security B, and estimates a GARCH(1,1) as well as an EWMA model for this security using data from July 2, 2012 to June 30, 2014 ('model estimation period'). (20 marks in total as indicated in each part) a) (i) Using Excel, estimate using maximum likelihood a GARCH(1,1) model at a daily frequency for Security B and report its key parameters (4 marks); (ii) provide a plot of the estimated volatility in the model estimation period based on the estimated GARCH(1,1) model (1 mark); (iii) report the estimated long-run average volatility (1 mark); (iv) report the volatility forecasts for all trading days in the month of July, 2014 (1 mark); and (v) provide a plot of the volatility forecasts for the month of July, 2014 (1 mark). b) Assume that the investor applies two EWMA models with =0.75 and =0.92. For both models, to start the EWMA calculations, set the variance forecast at the end of the first day in the model estimation period equal to the square of the return on that day. Complete the following tasks: (i) a plot of the estimated volatility in the model estimation period for both models (2 marks); (ii) interpret the differences between the two graphs (2 marks); (iii) report the volatility forecasts from each EWMA model for July 1, 2014 (1 mark); and (iv) interpret the EWMA forecasts on July 1, 2014 with respect to the GARCH(1,1) forecast (1 mark). c) (i) On June 30, 2014, what volatility should be used to price a call option on Security B that expires on September 30, 2014 (count the number of trading days to expiry and assume that there are 252 trading days per year) (2 marks); (ii) Using the DerivaGem software and assuming that the annualized risk-free rate is 3%, calculate the price of the option on June 30, 2014 given a strike price K=$62. Further, calculate the delta, gamma and vega of the option (2 marks); and (iii) Using DerivaGem also provide a graph of delta, gamma, and vega with respect to the price of the underlying (2 marks).(4) This question refers to the same price data as Question 1. (20 marks in total as indicated in each part) a) Assume that at the market close on June 30, 2014 an investor holds a portfolio with investments of $1,200,000 in Security A, $500,000 in Security B and $350,000 in Security C. Calculate the one-day 99% Value-at-Risk (VaR) for this portfolio using all observations from July 2, 2012 to June 30, 2014, under Models A1, A2, and A3 below: (i) Model A1: VaR is calculated using basic historical simulation (2 marks); (ii) Model A2: VaR is calculated using the weighting-of-observations extension to basic historical simulation, assuming =0.985 (2 marks); and (iii) Model A3: VaR is calculated using the volatility-updating extension to basic historical simulation, assuming =0.93 (2 marks). (For the remainder of this question assume that returns are independently normally distributed with a daily mean return of zero.) b) Assume that an investor holds a portfolio that consists of 40% in Security A and 60% in Security B. The investor uses the model building approach and implements the following techniques to conduct a VaR backtesting study during the period July 1, 2013 to June 30, 2014. Calculate the one-day 95% and 99% VaR forecasts for the portfolio on each day during the period July 1, 2013 to June 30, 2014, under Models B1, B2, and B3 below. In your answers, also provide a separate plot for the calculated VaR forecasts based on each model in comparison to the actual returns for the portfolio. Interpret your results with respect to the number of VaR exceptions for each model. The models are as follows: (i) Model B1: a static model where all return observations from July 2, 2012 to June 28, 2013 are used to compute the standard deviation of returns and the i) Model B1: a static model where all return observations from July 2, 2012 to June 28, 2013 are used to compute the standard deviation of returns and the correlation between returns. These values for the standard deviation and coefficients of correlation are then used for calculating VaR forecasts during the period July 1, 2013 to June 30, 2014 (2 marks); Model B2: a model where the standard deviation of returns and correlation coefficient is calculated based on a one-year window of observation For this approach, to generate a forecast for the distribution of portfolio returns for July 1, 2013, the investor uses observations from July 2, 2012 to June 28, 2013. Then to generate a forecast for the return distribution of the portfolio for July 2, 2013, the investor uses all observations from July 3, 2012 to July 1, 2013, and so on (3 marks); and(iii) Model B3: an approach where a forecast for the standard deviation of returns and correlation between returns is generated by implementing an EWMA model with λ=0.93. Therefore, forecasts for July 1, 2013 are based on separate EWMA models using observations from July 2, 2012 to June 28, 2013. To generate a forecast for the standard deviation of returns for July 2, 2013, the EWMA models are updated using the actual return observations on July 1, 2013, and so on. Since in each step, one additional observation is included to forecast the volatility, a so-called recursive window approach is applied. For both volatility models, to start the EWMA calculations, set the variance forecast at the end of the first day (July 2, 2012) equal to the square of the return on that day. The coefficient of correlation between returns from Security A and Security B is assumed to be constant with ρ=0.4 (5 marks). c) Using your results for the 95% and 99% VaR forecasts and the actual changes in portfolio value, conduct a one-tailed test (using the Binomial distribution) and a two tailed Kupiec test for the portfolio. Interpret your results. Which model seems to work best? (4 marks)