AcF633: Python Programming Final Individual Project 1.2: Write code to compute Realized Volatility (RV), Bipower Variation (BV) and Truncated Realized Volatility (TRV) measures (defined in the lectures) for each trading day in the sample using different sampling frequencies including 1 second (1s), 2s, 5s, 10s, 15s, 20s, 30s, 40s, 50s, 1 minute (1min), 2min, 3min, 4min, 5min, 6min, 7min, 8min, 9min, 10min, 15min, 20min and 30min. The require outputs are 3 data frames RVdf, RVdf and TRVdf (for Realized Volatility, Bipower Variation and Truncated Realized Volatility respectively), each having columns being the above sampling frequencies and row index being the unique dates in the sample. (5 marks) 1.3: Use results in 1.2, write code to produce a 1-by-3 subplot figure that shows the ‘volatility signature plot’ for RV, BV and TRV. Scale (i.e. multiply) the RVs, BVs and TRVs by 104 when making the plots. Your figure should look similar to the following. 0 500 1000 1500 Sampling frequency (secs) 1.0 1.2 1.4 1.6 1.8 Av er ag ed R V (x 10 4 ) RV signature plot 0 500 1000 1500 Sampling frequency (secs) 0.8 1.0 1.2 1.4 1.6 Av er ag ed B V (x 10 4 ) BV signature plot 0 500 1000 1500 Sampling frequency (secs) 0.8 0.9 1.0 1.1 1.2 1.3 Av er ag ed T RV (x 10 4 ) TRV signature plot (5 marks) 1.4: Using a 5min sampling frequency and a 5% significance level, write code to conduct a jump test to test whether or not there are jumps in the prices of AAPL on each date in the sample. Print the test conclusion for each date. Your jump test should be based on the test statistic zt = Jt√ v ar(Jt) , where Jt = max(RVt BVt, 0) is an estimate of the jump variation on day t. If there are no jumps on day t, zt ～ N(0, 1) (see the lecture slides on high-frequency finance for more details). (5 marks) Task 2: Return-Volatility Modelling and Forecasting (Σ = 25 marks) Refer back to the csv data file ‘DowJones-Feb2022.csv’ that lists the con- stituents of the Dow Jones Industrial Average (DJIA) index as of 9 February 2022 that was investigated in the group projects 1 and 2. Import the data file into Python. Using your student ID or library card number (e.g. 12345678) as a random seed, draw a random sample of 2 stocks (i.e. tickers) from the DJIA index excluding stocks DOW and WBA. Import daily Adjusted Close (Adj Close) prices for both stocks between 01/01/2010 and 31/12/2021 from Yahoo Fi- nance. Compute the log daily returns (in %) for both stocks and drop days with NaN returns. Perform the following tasks. 2.1: Using data between 01/01/2010 and 31/12/2018 as in-sample data, write code to find the best-fitted AR(m)-GJR-GARCH(p, o, q) model with Student’s t errors for the log returns of each stock that minimizes AIC, with m, p, q ∈ 3 AcF633: Python Programming Final Individual Project {1, 2, 3} and o ∈ {0, 1}. Print the best-fitted AR(m)-GJR-GARCH(p, o, q) output for each stock, and use it to test for the presence of ‘leverage effects’ (i.e. asymmetric responses of the conditional variance to the positive and negative shocks) in the return series of each stock. Draw and print your test conclusion using a 5% significance level. (10 marks) 2.2: Write code to plot a 2-by-4 subplot figure that includes the following diag- nostics for the best-fitted AR(m)-GJR-GARCH(p, o, q) model found in Task 2.1: Row 1: (i) Time series plot of the standardized residuals, (ii) histogram of the standardized residuals, fitted with a kernel density estimate and the density of a standard normal distribution, (iii) ACF of the standardized residuals, and (iv) ACF of the squared standardized residuals. Row 2: The same subplots for the second stock. Your figure should look similar to the following for your sample of stocks. Comment on what you observe from the plots. (5 marks) 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Date 4 2 0 2 4 6 AR(3)-GJR-GARCH(1,1,1) Standardized residuals-CRM 4 3 2 1 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 De ns ity Distribution of standardized residuals t(df=3.8) 0 5 10 15 20 25 30 35 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 ACF of standardized residuals 0 5 10 15 20 25 30 35 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 ACF of standardized residuals squared 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Date 6 4 2 0 2 4 6 8 AR(3)-GJR-GARCH(1,1,1) Standardized residuals-INTC 4 3 2 1 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 De ns ity Distribution of standardized residuals t(df=4.8) 0 5 10 15 20 25 30 35 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 ACF of standardized residuals 0 5 10 15 20 25 30 35 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 ACF of standardized residuals squared 2.3: Using the remaining data from 01/01/2019 to 31/12/2021 as out-of-sample data, write code to produce one-step analytic forecasts, together with 95% confidence interval, for the returns and conditional volatility of each stock using respective best-fitted AR(m)-GJR-GARCH(p, o, q) model found in Task 2.1. For each stock, the forecast output is a data frame with 4 columns f, fl and fu, and volf corresponding to the one-step return forecasts, 95% CI lower bounds and upper bounds for return forecasts, and one-step conditional volatility forecasts. (5 marks) 2.4: Use results in Task 2.3, write code to plot a 2-by-2 subplot figure showing: Row 1: (i) the one-step return fo
recasts against the true values during the out-of-sample period for both stocks in your sample, plus the 95% confidence interval of the return forecasts; and (ii) the one-step conditional volatility forecasts. Row 2: The same subplots for the second stock. 4 AcF633: Python Programming Final Individual Project 201 9-05 201 9-09 202 0-01 202 0-05 202 0-09 202 1-01 202 1-05 202 1-09 Date 15 10 5 0 5 10 15 20 25 AR(3)-GJR-GARCH(1,1,1) One-step return forecasts – CRM Observed returns Forecast returns 95% IC 201 9-05 201 9-09 202 0-01 202 0-05 202 0-09 202 1-01 202 1-05 202 1-09 Date 1 2 3 4 5 6 7 8 AR(3)-GJR-GARCH(1,1,1) One-step volatility forecasts – CRM Forecast volatility 201 9-05 201 9-09 202 0-01 202 0-05 202 0-09 202 1-01 202 1-05 202 1-09 Date 20 15 10 5 0 5 10 15 AR(3)-GJR-GARCH(1,1,1) One-step return forecasts – INTC Observed returns Forecast returns 95% IC 201 9-05 201 9-09 202 0-01 202 0-05 202 0-09 202 1-01 202 1-05 202 1-09 Date 1 2 3 4 5 6 7 8 AR(3)-GJR-GARCH(1,1,1) One-step volatility forecasts – INTC Forecast volatility (5 marks) Task 3: Trading Strategies (Σ = 30 marks) 3.1: You are interested in investing in two stocks in your sample. You have an investment budget of $10,000 and you would like to split equally between the two stocks. Consider a buy and hold strategy where you buy $5,000 worth of each stock on the first trading day of the out-of-sample period (i.e. 02/01/2019) using the ‘Adj Close’ price and hold them until the end of that period (i.e. 31/12/2021), assuming, for simplicity, that you can buy and hold a non-integer unit of shares of each stock. Write code to plot the value of your investment in each stock and the total value of your investment in both stocks during the period. Your figure should look similar to the following. 5 AcF633: Python Programming Final Individual Project 201 9-01 201 9-05 201 9-09 202 0-01 202 0-05 202 0-09 202 1-01 202 1-05 202 1-09 202 2-01 Date 4000 6000 8000 10000 12000 14000 16000 $ Buy-and-Hold Investment Value CRM INTC Total Evaluate the profitability of the buy-and-hold strategy by producing a BuyHoldProf data frame of the following format: Start Value End Value ($) Profit/Loss ($) Profit/Loss (%) CRM 5000 INTC 5000 Total 10000 (10 marks) 3.2: Now instead of following a buy-and-hold strategy, you perform a more so- phisticated strategy based on technical analysis. In particular, you use the one- step return forecasts produced by the best-fitted AR(m)-GJR-GARCH(p, o, q) model found in Task 2.3 to decide whether you should buy or sell stocks, as detailed below – On a particular trading day, if the return forecast of a stock for the next day is positive (i.e. the stock price is forecast to increase on the next day), you will purchase the stock at the end of that trading day at the ‘Adj Close’ price using all the remaining cash balance (if any) for the stock. If there is no cash balance for the stock, you do nothing and carry on your stock holdings to the next day. – If the return forecast is negative (i.e. the stock price is forecast to decrease on the next day), you will sell all your holdings of that stock (if any) for cash. If your stock holdings are none, you do nothing and carry on the cash balance for the stock to the next day. – The value of your investment in each stock on a particular day equals the value of your holdings of the stock (if any) plus the remaining cash balance for that stock (if any) on that day. 6 AcF633: Python Programming Final Individual Project Assume, for simplicity, that you keep the cash balances for the two stocks in your sample separate so you only purchase a stock using the remaining cash balance for that stock and not using the cash balance for the other stock. Also assume that (1) there are no transaction costs or fees associated with buying and selling stocks, (2) there is no interest earned on cash balances, and (3) you can buy and sell a non-integer unit of shares of each stock. Write code to plot the value of your investment in each stock and the total value of your investment in both stocks during the out-of-sample period based on the best-fitted AR(m)-GJR-GARCH(p, o, q) model found in Task 2.1. Your figure should look similar to the following. Evaluate the profitability of the GARCH-based investment strategy by producing GARCHProf data frame of the same format as the BuyHoldProf in Task 3.1. 201 9-01 201 9-05 201 9-09 202 0-01 202 0-05 202 0-09 202 1-01 202 1-05 202 1-09 202 2-01 Date 5000 7500 10000 12500 15000 17500 20000 22500 $ AR-GJR-GARCH-based Investment Value CRM INTC Total (10 marks) 3.3: Now assume that there are fixed transaction costs (including commission fees, exchange fess and bid-ask spread) of c = 0.1% for each buy and sell order, regardless of the trading volumes. If you buy $V worth of shares at price p, you only receive V/(p × (1 + c)) unit of shares instead of V/p as in the case of no transaction costs. Similarly, if you sell A units of shares at price p, you only receive $A×p× (1 c). Plot the value of your GARCH-based investment in each stock and the total value of your investment in this case during the out-of-sample period. Re-evaluate the profitability of your investment strategy now by producing GARCHProfWithcost data frame of the same format as the BuyHoldProf in Task 3.1. (10 marks) 7