Homework 1
1 Problems
1 Use the data provided in the file: SP500data.xlsx. Starting as early as possible (given a specific model), estimate the daily variance according to:
• 1-month moving average (i.e. use 21 observations)
• 1-year moving average (i.e. use 252 observations)
• RiskMetrics model (λ = 0.94)
• GARCH model
• HAR model
Annualize all the variance measures.
Hint: For both RiskMetrics and GARCH model set the variance of the first day equal to the squared return of the first day.
Hint: Moving average, RiskMetrics, and GARCH models are all based on daily log-return.
The HAR model is the only one that uses Realized Variances.
2 Compute the sample autocorrelation function of the realized variance (RVt) for lags ranging from 1 to 100 days. What do you notice?
3 Using the 5-minutes Realized Variance (RV) as your benchmark for the true (but unknown variance), which of the above methods (i.e. MA, RM, GARCH, HAR) provide the best forecast of the next day variance? i.e. which method provide the best forecast of tomorrow RVt+1?
Hint 1: You can test this but running the following regression: RVt+1 = a+bP red.V ari t→t+1+ t+1: compare the regression coefficients and the R2
. Interpret the results!
Hint 2: P red.V ari t →t+1 is the predicted variance for day t+1 using information available up to the end of the day at time t for model i (i = MovingAverage, RiskMetrics, GARCH, HAR);
4 Using the provided data on the 1-minute prices for Tesla (TSLA) compute its daily Realized
Variance (Annualized) on each day.
2 Skills required
• Clear understanding of how to use Excel: both basic and advance functions
• Use of Solver and LINEST() function
• Understanding of OLS and Maximum Likelihood
• Understanding of the different models for volatility forecasting along with their strengths
and their limitations:
– Moving Average
– Risk Metrics
– GARCH
– HAR
3 Upload on Blackboard
Please upload only one Homework per group (max three students per group). Make sure that your submission includes:
• Excel spreadsheet in which you show all the work you have done. You can put comments in order to make more clear your procedure. Please, be clean and precise.
• A short report (pdf file) with figures, tables and a concise description of the results. Think as if you had to present your results to your boss.
4 Deadline and Solution
• Deadline: Be sure before the next class!
• The solution will be posted on Blackboard after the next class:
Look for the file VarianceEstimation.xlsx