Forecasting Realized Variance Using Jumps

Andrey FradkinEcon 201

4/18/2007

Outline

• Theoretical Background• The HAR-RV-CJ Model• Is the HAR-RV-CJ model better than the HAR-RV

model?• Does IV contain more information than RV, C, J?• How does market risk ask measured by the VIX affect

the RV of individual stock?• Future Work

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Variance 2

Formulas Part 1

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Realized Variation:

Realized Bi-Power Variation:

Formulas Part 2

• Tri-Power Quarticity

• Quad-Power Quarticity

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Formulas Part 3

• Z-statistics (max version)

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Original HAR-RV-J Model (Taken from Andersen, Bollerslev, Diebold 2006)

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The HAR-RV-CJ Model

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Summary Statistics

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Summary Statistics for Daily GS Realized Volatilities and Jumps

RVt RVt 1/2 log(RVt ) Jt Jt ½ log(Jt+1) Mean .00025 .0145 -8.63 3.2e-06 .00018 .778 St.Dev. .00028 .0065 0.798 7e-05 .0017 1.53 Min. 9.6e-06 .0031 -11.54 .000 .000 -11.11 Max. .00394 .0628 -5.53 .0024 .049 1

Summary Statistics for Daily SPY Realized Volatilities and Jumps

RVt RVt 1/2 log(RVt ) Jt Jt ½ log(Jt+1) Mean .0001 .009 -9.61 1.4e-06 .00016 .644St.Dev. .0001 .0045 .885 2.3e-05 .0011 2.01 Min. 4.73e-06 .0021 -12.26 .000 .000 -12.62Max. .0016 .0403 -6.422 .0008 .0281 1

Findings

• Jump factors were usually insignificant and added very little to the R^2

• The highest R^2 was obtained by using either log values or square root values

• To keep scaling the same I used square root values in the regressions in this presentation

• Separating the continuous and jump parts of the realized variance did not improve the r^2

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Does IV have more information than RV?

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• Steps to test this• First regress the average realized variance over a month (rv22) on the independent variables with the best adjusted R^2• Then regress RV22 against the predicted values from the previous regression and the implied volatility• See which turns have the highest R^2 and significance

For GS – 1st Regression

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Newey-West

F22.gssqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

gssqc1 0.16 0.03 5.61 0.00 0.11 0.22

gssqc5 0.28 0.06 4.38 0.00 0.15 0.40

gssqc22 0.42 0.06 6.65 0.00 0.29 0.54

gssqj1 0.00 0.03 -0.14 0.89 -0.07 0.06

gssqj5 0.05 0.13 0.40 0.69 -0.21 0.32

gssqj22 -0.32 0.12 -2.71 0.01 -0.55 -0.09

_cons 0.00 0.00 3.07 0.00 0.00 0.00

Adj R-squared = 0.7410

For GS – 2nd Regression

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Newey-WestF22.gssqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

gspredict 0.59 0.15 4.01 0.00 0.30 0.89gssqiv22 0.35 0.11 3.15 0.00 0.13 0.57_cons 0.00 0.00 -0.41 0.68 0.00 0.00

Adj R-squared = .7713Newey-West

F22.gssqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

gssqiv22 0.77 0.10 7.88 0.00 0.58 0.96_cons 0.00 0.00 0.49 0.63 0.00 0.00

Adj R-squared = 0.7205

Newey-WestF22.gssqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

gspredict 1.00 0.07 14.49 0.00 0.86 1.14_cons 0.00 0.00 0.00 1.00 0.00 0.00

Adj R-squared = 0.7421

For Spy – 1st Regression

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Newey-West

F22.sqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

sqc1 0.16 0.04 3.97 0.00 0.08 0.23

sqc5 0.46 0.09 5.32 0.00 0.29 0.62

sqc22 0.18 0.08 2.13 0.03 0.01 0.35

sqj1 -0.02 0.05 -0.48 0.63 -0.12 0.07

sqj5 -0.12 0.09 -1.30 0.19 -0.29 0.06

sqj22 0.23 0.28 0.83 0.41 -0.31 0.77

_cons 0.00 0.00 4.04 0.00 0.00 0.00

Adj R-squared = 0.6606

For SPY – 2nd Regression

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Newey-WestF22.sqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

sqvix 0.54 0.13 4.11 0.00 0.28 0.80spypredict 0.50 0.13 3.82 0.00 0.24 0.75_cons 0.00 0.00 -0.65 0.51 0.00 0.00

Adj R-Squared = 0.7075Newey-West

F22.sqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

sqvix 0.96 0.10 9.70 0.00 0.77 1.15_cons 0.00 0.00 0.50 0.62 0.00 0.00

Adj R-squared = 0.6724Newey-West

F22.sqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

spypredict 1.00 0.07 15.01 0.00 0.87 1.13_cons 0.00 0.00 0.00 1.00 0.00 0.00

Adj R-squared = 0.6621

How Does Market Volatility effect Individual Stock Volatility? (cont)

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Newey-WestF22.gspredict Coef. Std. Err. t P>t [95% Conf. Interval]

sqvix 0.89 0.11 8.12 0.00 0.68 1.11_cons 0.01 0.00 7.05 0.00 0.00 0.01

Adj R-squared = 0.5691Newey-West

F22.gssqrv22 Coef. Std. Err. t P>t [95% Conf. Interval]

sqvix 1.17 0.16 7.27 0.00 0.85 1.48_cons 0.00 0.00 3.25 0.00 0.00 0.01

Adj R-squared = 0.4731

How Does Market Volatility effect Individual Stock Volatility?

• Tried a lot of regression of future realized variance on estimates, vix, rv of market

• Found that the coefficients are not significant for the volatility of the market if the forecasts are included as independent variables

• If future Realized Volatility is just regressed on VIX there is a significant coefficient

• The Adj-R^2 increases by around .07

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Future Work• Analyze more stock• Write up proposal• Try to use more advanced time-series techniques• More work with returns • What happens to implied volatility before and after

jumps? Preliminary results (implied volatility changes much more than average on the day of jump)

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