Forecasting%20Realized%20Variance%20Using%20Jumps

1
Forecasting Realized Variance Using Jumps

• Andrey Fradkin
• Econ 201
• 4/4/2007

2
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

3
Formulas Part 1
Realized Variation
Realized Bi-Power Variation
4
Formulas Part 2

• Tri-Power Quarticity
• Quad-Power Quarticity

5
Formulas Part 3

• Z-statistics (max version)

6
Realized Variance and Jumps
7
Original HAR-RV-J Model (Taken from Andersen,
Bollerslev, Diebold 2006)
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The HAR-RV-CJ Model
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My Regressions 1 day forward
Newey-West R2.4922 rv Coef. Std.
Err. t Pgtt 95 Conf. Interval c1 .3216361
.0778881 4.13 0.000 .168826 .4744461 c5 .3233613
.1008474 3.21 0.001 .1255069 .5212156 c22 .2478666
.0625769 3.96 0.000 .1250959 .3706373 _cons .0000
285 .0000103 2.76 0.006 8.21e-06 .0000488
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Jumps Dont Matter
Newey-West
R2.4985 rv Coef.
Std. Err. t Pgtt 95 Conf. Interval c1 .32
62136 .0755843 4.32 0.000 .177923
.4745042 c5 .3091024 .0975148 3.17 0.002 .1177858
.5004191 c22 .2419664 .0601737 4.02 0.000 .12391
03 .3600226 j1 1.584021 .9718173 1.63 0.103 -.32
26096 3.490652 j5 -.8471169 1.134404 -0.75 0.455
-3.07273 1.378496 j22 3.587264 3.786084 0.95 0
.344 -3.840741 11.01527 _cons .0000261 .0000101
2.59 0.010 6.35e-06 .0000459
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1 day forward using logs
Newey-West
R20.7737 logrv Coef. Std.
Err. t Pgtt 95 Conf.Interval logc1 .24077
42 .041531 5.80 0.000 .1592938 .3222545 logc5 .439
6577 .0592865 7.42 0.000 .3233424 .5559731 logc22
.2749495 .0418261 6.57 0.000 .19289 .357009 _cons
-.4548797.1309848 -3.47 0.001 -.7118613 -.1978982
Jump terms are insignificant if added to this
regression
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Regression 5 days forward
Newey-West F5.rv5 Coef. Std. Err.
t Pgtt 95 Conf.Interval c1 .1902404 .0405
141 4.70 0.000 .1107546 .2697263 c5 .3198168 .1
070031 2.99 0.003 .1098841 .5297494 c22 .296642
8 .0782428 3.79 0.000 .1431358 .4501498 j1 -.08
87148 .4668765 -0.19 0.849 -1.004694
.8272648 j5 3.129752 1.447759
2.16 0.031 .2893476 5.970156 j22 2.996998 5.7388
14 0.52 0.602 -8.26216 14.25616 _cons .000041
9 .0000154 2.71 0.007 .0000116
.0000721
Practically no change in R2 w/o jumps
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My Regressions 22 day

Newey-West
R2.5172
F22.rv22 Coef. Std. Err. t Pgtt 95
Conf.Interval c1 .1216783 .0230143 5.29 0.
000 .0765252 .1668314 c5 .2577073 .1083063 2.38 0.
017 .0452148 .4701998 c22 .2752547 .0909278 3.03 0
.003 .096858 .4536513 j1 .2384794 .2904984 0.82 0.
412 -.3314668 .8084255 j5 1.570385 2.267699 0.69
0.489 -2.878747 6.019518 j22 5.20189 9.937398 0.5
2 0.601 -14.29488 24.69866 _cons .0000799 .000026
3.08 0.002 .000029 .0001308
Practically no change in R2 w/o jumps
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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 IVt,thh-1 (IVt1 IVt2
  IVth)
• Difft IVt-RVt

15
Means

• Observations 1219 Mean RV.0002635
• Mean IV.0003173 Mean Diff.0000523

Diff
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Autocorrelation of Diff
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IV is a better predictor than RV of future RV
R-squared 0.5023 Root MSE
.00026 Robust rv Coef. Std. Err. t Pgtt 95
Conf.Interval iv1 1.050039 .0962552 10.91
0.000 .8611945 1.238884 j1 .6298041 .90921
65 0.69 0.489 -1.154003
2.413611 _cons -.0000698.0000254 -2.74 0.006 -.000
1197 -.0000199
R-squared 0.4271 Root MSE
.00028 Robust rv Coef. Std. Err. t Pgtt 95
Conf. Interval c1 .6478913 .1001823 6.47 0
.000 .4513421 .8444406 j1 1.897893 .8402938 2.26 0
.024 .2493062 3.546479 _cons .0000913 .0000223 4.1
0 0.000 .0000476 .0001351
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Is Diff Significant in forecasting RV?
R-squared 0.5465 Root MSE
.00025 Robust rv Coef. Std.
Err. t Pgtt 95 Conf.Interval rv1 1.0396
44 .0941392 11.04 0.000 .8549505 1.224337 L1.Diff
.7441405 .1072339 6.94 0.000 .5337562 .9545247 _c
ons -.0000496.0000239 -2.07 0.038 -.0000966
-2.64e-06
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Using Diff in HAR-RV-CJ Model
Newey-West
R-squared .5611
rv Coef. Std. Err. t Pgtt 95
Conf. Interval c1 .8782383
.1949678 4.50 0.000 .4957259
1.260751 c5 .1978388 .0789141
2.51 0.012 .0430151 .3526624 c22
-.0109185 .1064608 -0.10 0.918
-.2197868 .1979499 j1 2.379697
.984771 2.42 0.016 .4476485
4.311745 j5 -4.892927 1.876258
-2.61 0.009 -8.574008 -1.211847 j22
3.648466 3.529547 1.03 0.301
-3.276246 10.57318 L1.diff
.6761671 .2257157 3.00 0.003 .2333295
1.119005 _cons -.000053 .0000262
-2.02 0.044 -.0001044 -1.55e-06 Newey-W
est
R-squared 0.6447
F5.rv5 Coef. Std. Err. t Pgtt 95
Conf. Interval c1 .6181182
.1238336 4.99 0.000 .3751648
.8610715 c5 .2326215 .107413
2.17 0.031 .0218843 .4433588 c22
.1019241 .0628666 1.62 0.105
-.021416 .2252642 j1 .5261682
.5163181 1.02 0.308 -.4868141
1.53915 j5 -.0505589 1.846144
-0.03 0.978 -3.672573 3.571455 j22
3.228812 5.368064 0.60 0.548
-7.302979 13.7606 L1.Diff
.5242109 .143786 3.65 0.000 .2421122
.8063096 _cons -.0000199 .0000132
-1.51 0.131 -.0000457 5.96e-06
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Using Diff in HAR-RV-CJ Model cont.
Newey-West
R-Squared 0.5676
F22.rv22 Coef. Std. Err. t Pgtt 95
Conf. Interval c1 .4739452 .0803304 5.90
0.000 .31634 .6315504 c5 .1862154 .1068758 1.74
0.082 -.0234709 .3959018 c22 .115742 .0742476 1.5
6 0.119 -.0299291 .2614131 j1 .7448536 .3328171 2
.24 0.025 .0918788 1.397828 j5 -1.032086 2.406812
-0.43 0.668 -5.754162 3.689989 j22 5.355446 10.2
8448 0.52 0.603 -14.82233 25.53322 L1.diff .43145
11 .0938983 4.59 0.000 .247226 .6156761 _cons .00
00285 .000021 1.36 0.175 -.0000127 .0000696
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Predicting Diff Using Jumps
Newey-West R-squared
0.1235 diff Coef. Std. Err. t Pgtt
95 Conf. Interval c1 -.1973631
.0706515 -2.79 0.005 -.3359759 -.0587504 c5
-.1441686 .063503 -2.27 0.023
-.2687566 -.0195806 c22 .1650903 .0995486
1.66 0.097 -.0302165 .3603971 j1 -1.591713
.8951938 -1.78 0.076 -3.348014 .1645889 j5
7.162149 1.46073 4.90 0.000
4.296309 10.02799 j22 -3.263902 2.958828
-1.10 0.270 -9.068895 2.541092 _cons
.0000949 .0000215 4.42 0.000
.0000528 .0001371 Newey-West
R-squared 0.0548 F5.diff Coef.
Std. Err. t Pgtt 95 Conf. Interval
c1 .025571 .0435225
0.59 0.557 -.0598172 .1109593 c5
-.3173051 .1317263 -2.41 0.016
-.5757431 -.0588671 c22 .2137709
.1007057 2.12 0.034 .0161933 .4113484 j1
-.6373502 .8629953 -0.74 0.460
-2.330488 1.055787 j5 -1.319912
1.440435 -0.92 0.360 -4.145946 1.506122 j22
-2.634389 3.527186 -0.75 0.455
-9.554485 4.285707 _cons .0000781
.0000198 3.940.000 .0000392 .0001169
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Predicting Diff Using Jumps
Newey-West

R-squared 0.0072 F22.diff
Coef. Std. Err. t Pgtt 95
Conf. Interval c1 .0278554
.029698 0.94 0.348 -.0304108 .0861216 c5
-.0189465 .0709693 -0.27 0.790
-.1581855 .1202924 c22 .0304386
.0706686 0.43 0.667 -.1082103 .1690875 j1
.7447953 .23193 3.21 0.001
.289758 1.199833 j5 -2.931345
2.05406 -1.43 0.154 -6.961327 1.098638 j22
.6472574 5.335948 0.12 0.903
-9.821655 11.11617 _cons .0000405
.0000126 3.21 0.001 .0000158 .0000653
Adding or removing jumps does not effect R-Squared
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Jumps matter if regressing Diff on IV and Jumps
Newey-West
R-Squared
.1018 diff Coef. Std. Err. t Pgtt 95
Conf. Interval iv1 -.5454239 .2838641 -1.9
2 0.055 -1.102344 .0114957 iv5 -.0509571 .1503595
-0.34 0.735 -.3459508 .2440366 iv22 .5652849 .1773
301 3.19 0.001 .217377 .9131929 j1 -1.557493 .9155
883 -1.70 0.089 -3.353807 .2388207 j5 10.23682 2.0
56567 4.98 0.000 6.201993 14.27165 j22 -9.462402 3
.553551 -2.66 0.008 -16.4342 -2.490609 _cons .0000
605 .0000219 2.76 0.006 .0000175 .0001036
Newey-West
R-Squared .16 diff Coef. Std.
Err. t Pgtt 95 Conf. Interval L1.diff .
2575236 .0986853 2.61 0.009 .0639104 .4511368 iv1
-.5944392 .2546254 -2.33 0.020 -1.093995 -.094883
iv5 .1370913 .200045 0.69 0.493 -.2553824 .52956
5 iv22 .4336471 .1536846 2.82 0.005 .1321293 .735
1649 j1 -1.37075 .946662 -1.45 0.148 -3.228031 .4
865313 j5 8.862341 1.982412 4.47 0.000 4.972993 1
2.75169 j22 -9.133631 2.995484 -3.05 0.002 -15.01
055 -3.256713 _cons .0000459 .000015 3.06 0.002 .
0000165 .0000752
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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.