Forecasting Realized Variance Using Jumps Andrey Fradkin Econ 201 4/4/2007

Introduction Theoretical Background Summary Graphs and Statistics for data The HAR-RV-CJ Model and regressions using it. Addition of IV to the regression Analysis of possible benefits to using IV Forecasting IV-RV using jumps, do jumps effect risk premiums? Future Work 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 2

Formulas Part 1 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 3 Realized Variation: Realized Bi-Power Variation:

Formulas Part 2 Tri-Power Quarticity Quad-Power Quarticity 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 4

Formulas Part 3 Z-statistics (max version) 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 5

Realized Variance and Jumps 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 6

Original HAR-RV-J Model (Taken from Andersen, Bollerslev, Diebold 2006) 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 7

The HAR-RV-CJ Model 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 8

My Regressions – 1 day forward 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 9 Newey-WestR^2=.4922 rvCoef.Std. Err.tP>t[95% Conf.Interval] c c c _cons e

Jumps Don’t Matter 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 10 Newey-West R^2=.4985 rvCoef.Std. Err.tP>t[95% Conf. Interval] c c c j j j _cons e

1 day forward using logs 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 11 Newey-West R^2= logrvCoef.Std. Err.tP>t[95% Conf.Interval] logc logc logc _cons Jump terms are insignificant if added to this regression

Regression 5 days forward 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 12 Newey-West F5.rv5Coef.Std. Err. tP>t[95% Conf.Interval] c c c j j j _cons Practically no change in R^2 w/o jumps

My Regressions – 22 day 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 13 Newey-West R^2=.5172 F22.rv22Coef.Std. Err.tP>t[95% Conf.Interval] c c c j j j _cons Practically no change in R^2 w/o jumps

Work on Options Data Code for filtering through the many options Takes the implied volatility of the option that is closest to the average of the starting and closing price, provided volume is high enough. Calculate variables: IV t,t+h =h -1 (IV t+1 + IV t+2 … + IV t+h ) Diff t = IV t -RV t 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 14

Means Observations: 1219 Mean RV= Mean IV= Mean Diff= /4/2007 Andrey Fradkin: Forecasting Realized Variance 15 Diff

Autocorrelation of Diff 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 16

IV is a better predictor than RV of future RV 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 17 R-squared= Root MSE= Robust rvCoef.Std. Err.tP>t[95% Conf.Interval] iv j _cons R-squared= Root MSE= Robust rvCoef.Std. Err.tP>t[95% Conf.Interval] c j _cons

Is Diff Significant in forecasting RV? 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 18 R-squared= Root MSE= Robust rvCoef.Std. Err.tP>t[95% Conf.Interval] rv L1.Diff _cons e-06

Using Diff in HAR-RV-CJ Model 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 19 Newey-West R-squared =.5611 rvCoef. Std. Err. t P>t [95% Conf.Interval] c c c j j j L1.diff _cons e-06 Newey-West R-squared = F5.rv5Coef. Std. Err. t P>t [95% Conf.Interval] c c c j j j L1.Diff _cons e-06

Using Diff in HAR-RV-CJ Model cont. 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 20 Newey-West R-Squared: F22.rv22Coef.Std. Err.tP>t[95% Conf.Interval] c c c j j j L1.diff _cons

Predicting Diff Using Jumps 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 21 Newey-West R-squared = diff Coef. Std. Err. tP>t [95% Conf.Interval] c c c j j j _cons Newey-West R-squared = F5.diff Coef. Std. Err. tP>t [95% Conf.Interval] c c c j j j _cons

Predicting Diff Using Jumps 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 22 Newey-West R-squared = F22.diff Coef. Std. Err. tP>t [95% Conf.Interval] c c c j j j _cons Adding or removing jumps does not effect R-Squared

Jumps matter if regressing Diff on IV and Jumps 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 23 Newey-West R-Squared:.1018 diffCoef.Std. Err.tP>t[95% Conf.Interval] iv iv iv j j j _cons Newey-West R-Squared:.16 diffCoef.Std. Err.tP>t[95% Conf.Interval] L1.diff iv iv iv j j j _cons

Future Work Do same regressions on data for other stocks. Add volatility of SPY to regression terms. See if there are possible applications of GARCH models for these regressions. Experiment with other alphas. 4/4/2007 Andrey Fradkin: Forecasting Realized Variance 24