Motivational Mathematics (skip) Data Information (skip) Graphing prices

2/20/08 Brian Jansen

Co-jumps in the Oil Industry

• Motivational Mathematics (skip)• Data Information (skip)• Graphing prices • Motivation for my research

– Correlation in stock prices– Correlation in returns

• Factor Analysis– Z-stats– RV– RV-BV

• Extensions

Co-Jumps in Oil Brian Jansen

Outline

-rt,j is log return, M is total # of observations per day

• Realized Variance

• Realized Bi-Power Variation

Motivational Maths Brian Jansen

Realized and Bi-Power Variation

Motivational Maths Brian Jansen

Asymptotic Properties of RV and BV

Motivational Maths Brian Jansen

Tri-power, Max Verison BN-S

• Sampled at the 5-minute frequency• Sampled from 9/3/2002 to 12/31/07 for 1323 total

observed days• Oil futures data at the 5-min frequency, from 1987

– Changing number of observations per day– Different trading days than equity stocks

• Ticker Symbols– XOM—Exxon Mobile– CVX—Chevron Oil– COP—Conoco Phillips

Co-Jumps in Oil Brian Jansen

Data Used

RV, Ztp Statistics Summary Brian Jansen

Statistics Summary

Variable Mean Min Max

COP

RV .2185(ann. vol.) 1.8591e-05 0.0015

Ztp .4849 -3.357 9.4655

XOM

RV .1935(ann. vol.) 1.409e-05 .0014

Ztp .4494 -2.7796 4.7739

CVX

RV .1982(ann. Vol.) 1.5489e-05 .0016

Ztp .4682 -3.001 9.9190

Jump Analysis Brian Jansen

Z-test Graphs

XOM:29 CVX:41 COP:38

Motivational Graphs Brian Jansen

XOM, CVX, COP

Motivational Graphs Brian Jansen

XOM, CVX, COP (close up!)

Motivation Brian Jansen

Stock Price/Jump Correlation

PtCOP PtXOM

PtXOM .9708 1PtCVX .9647 .9921

-Correlation between 5-minute prices

-CVX had 41 jumps out of 1343 days observed; 4 of which were shared by either XOM or COP

-XOM had 29 jumps out of 1343 days observed; 6 of which were shared by either CVX or COP

-COP had 38 jumps out of 1343 days observed; 6 of which were shared by either CVX or XOM

Graphs Brian Jansen

Stock Returns

Graphs Brian Jansen

Stock Returns (Zoom)

Statistics Brian Jansen

Stock Returns

RtCOP RtCVX

RtCVX .974 1

RtXOM .857 .843

-High degree of correlation between the stock returns, especially between CVX and COP

Graphs Brian Jansen

Oil Futures vs. XOM Prices

Graphs Brian Jansen

Oil Futures vs. COP Returns

Statistics Brian Jansen

Stock Returns

RtCOP RtCVX RtXOM RtOIL

RtCOP 1

RtCVX .974 1

RtXOM .857 .843 1

RtOIL .341 .322 .222 1

-Not great correlation between any of the stocks and oil returns

-Questionable return for oil given the nature of the data

Factor Analysis Brian Jansen

Z-Statistic Analysis

z_oil 0.0473 0.1526 0.9745 z_xom 0.2443 0.1462 0.9190 z_cvx 0.9742 -0.0336 0.0498 z_cop 0.9751 -0.0105 0.0491 Variable Factor1 Factor2 Uniqueness

Factor loadings (pattern matrix) and unique variances

LR test: independent vs. saturated: chi2(6) = 3671.73 Prob>chi2 = 0.0000 Factor4 -0.04283 . -0.0222 1.0000 Factor3 -0.03276 0.01007 -0.0170 1.0222 Factor2 0.04590 0.07866 0.0238 1.0391 Factor1 1.96175 1.91585 1.0154 1.0154 Factor Eigenvalue Difference Proportion Cumulative

Rotation: (unrotated) Number of params = 6 Method: principal factors Retained factors = 2Factor analysis/correlation Number of obs = 1323

(obs=1323). factor z_cop z_cvx z_xom z_oil

-For both COP and CVX, Factor1 is loads positively and most variance is explained by common factors (high communality)

Factor Analysis Brian Jansen

Z-Statistic Analysis w/ PCF

z_oil 0.0953 0.9544 0.0801 z_xom 0.4321 0.2697 0.7406 z_cvx 0.9689 -0.1111 0.0489 z_cop 0.9703 -0.1029 0.0479 Variable Factor1 Factor2 Uniqueness

Factor loadings (pattern matrix) and unique variances

LR test: independent vs. saturated: chi2(6) = 3671.73 Prob>chi2 = 0.0000 Factor4 0.03359 . 0.0084 1.0000 Factor3 0.88397 0.85038 0.2210 0.9916 Factor2 1.00645 0.12248 0.2516 0.7706 Factor1 2.07598 1.06953 0.5190 0.5190 Factor Eigenvalue Difference Proportion Cumulative

Rotation: (unrotated) Number of params = 6 Method: principal-component factors Retained factors = 2Factor analysis/correlation Number of obs = 1323

(obs=1323). factor z_cop z_cvx z_xom z_oil, pcf

-Principle-Component Factors: treating the communalities (1-uniqueness) as 1, thus allowing for no unique factors

• Interesting: With RV, we see Factor1 explaining COP and XOM, with a high degree of communality

Factor Analysis Brian Jansen

RV Analysis

rv_oil -0.0293 0.1001 0.9891 rv_xom 0.8959 0.0071 0.1973 rv_cvx -0.0146 0.0957 0.9906 rv_cop 0.8962 -0.0023 0.1967 Variable Factor1 Factor2 Uniqueness

Factor loadings (pattern matrix) and unique variances

LR test: independent vs. saturated: chi2(6) = 1799.69 Prob>chi2 = 0.0000 Factor4 -0.11968 . -0.0803 1.0000 Factor3 -0.01667 0.10301 -0.0112 1.0803 Factor2 0.01923 0.03590 0.0129 1.0915 Factor1 1.60696 1.58773 1.0786 1.0786 Factor Eigenvalue Difference Proportion Cumulative

Rotation: (unrotated) Number of params = 6 Method: principal factors Retained factors = 2Factor analysis/correlation Number of obs = 1323

(obs=1323). factor rv_cop rv_cvx rv_xom rv_oil

rv_oil -0.0589 0.7013 0.5047 rv_xom 0.9641 0.0377 0.0691 rv_cvx -0.0298 0.7231 0.4762 rv_cop 0.9646 0.0275 0.0688 Variable Factor1 Factor2 Uniqueness

Factor loadings (pattern matrix) and unique variances

LR test: independent vs. saturated: chi2(6) = 1799.69 Prob>chi2 = 0.0000 Factor4 0.13763 . 0.0344 1.0000 Factor3 0.98128 0.84365 0.2453 0.9656 Factor2 1.01688 0.03561 0.2542 0.7203 Factor1 1.86421 0.84733 0.4661 0.4661 Factor Eigenvalue Difference Proportion Cumulative

Rotation: (unrotated) Number of params = 6 Method: principal-component factors Retained factors = 2Factor analysis/correlation Number of obs = 1323

(obs=1323). factor rv_cop rv_cvx rv_xom rv_oil, pcf

Factor Analysis Brian Jansen

RV Analysis w/ PCF

-When communality is forced to be 1, Factor1 explains COP and XOM while Factor 2 explains CVX and OIL

• Pretty terrible results for RV-BV

Factor Analysis Brian Jansen

RV-BV Analysis

rvbv_oil 0.0866 -0.0829 0.9856 rvbv_xom 0.3702 0.0135 0.8628 rvbv_cvx -0.0239 0.0971 0.9900 rvbv_cop 0.3697 0.0122 0.8632 Variable Factor1 Factor2 Uniqueness

Factor loadings (pattern matrix) and unique variances

LR test: independent vs. saturated: chi2(6) = 70.49 Prob>chi2 = 0.0000 Factor4 -0.17293 . -1.6677 1.0000 Factor3 -0.02180 0.15113 -0.2103 2.6677 Factor2 0.01663 0.03843 0.1604 2.8779 Factor1 0.28180 0.26518 2.7176 2.7176 Factor Eigenvalue Difference Proportion Cumulative

Rotation: (unrotated) Number of params = 6 Method: principal factors Retained factors = 2Factor analysis/correlation Number of obs = 1323

(obs=1323). factor rvbv_cop rvbv_cvx rvbv_xom rvbv_oil

• More familiarity with the practices of the oil industry, especially their trading desk operation to determine how they deal with oil price volatility

• Introducing a new jump test that can detect multiple jumps per day and time of jump. Lee-Mykland (2008)? Dobrev et. al (2007)

• Auto correlation with small lag times• Can we use the implied volatility of same industry

companies and oil futures to forecast volatility using the HAR-RV-CJ model?

Conclusion Brian Jansen

Extensions