1. Co-jumps in the Oil Industry
2/20/08
Brian Jansen
Co-jumps in the Oil Industry

2. Introducing the Lee-Mykland Test Results for XOM, COP, and CVX
Co-Jumps in Oil
Brian Jansen
Outline
Introducing the Lee-Mykland Test
New Rejection Region method
Results for XOM, COP, and CVX
Problems with the test
Possible Corrections to the test
Extensions

3. Intro to the Lee-Mykland Jump Test
Brian Jansen
Intro to the Lee-Mykland Jump Test
-Creates a statistic L(i), for each price, comparing the change in price on the interval [ ti-1, ti] to an instantaneous volatility measure using the previous 270 returns

4. Lee-Mykland
Brian Jansen
Rejection Region
-The distribution of L(i) is normal under the null hypothesis that no jumps occur over a given set An {1,2,….n}
-The asymptotic distribution of the absolute value of the maximum L(i) in a given day is exponential
-Where Cn and Sn, given n= and c=sqrt(2/pi) :

5. Lee-Mykland
Brian Jansen
Rejection Region
-To actually test whether a return represents a significant jump, (|L(i)|-Cn)/Sn > B, where B= at 1% level of significance
-The size of the BNS test is .1%, where each day has a .001 probability of being a jump day
-Then, using a 1% level of significance for each observation for the Lee-Mykland statistic will yield a much higher size than .1%

6. Results Lee-Mykland Brian Jansen XOM COP CVX # of jumps 288 416 367
% of returns that are jumps
# of days with a jump
223
328
298
% of days with a jump
Mean(L(i))
.0010
.00068
.00067
Mean(L(i)) if return is a jump
-1.17
-.5865
-.6676
Standard deviation of L(i)
1.3023
1.3131
1.3097

7. Lee-Mykland
Brian Jansen
Results

8. Lee-Mykland
Brian Jansen
Problems of the Test
-The window size they suggest for 5-minute data is K=270 observations
-Thus, they calculate the instantaneous volatility going back 2.5 days
-While this accounts for changes in local volatility on a larger scale, it does not adequately correct for intra- and inter-day changes in volatility
-Specifically, inter-day volatility follows a U-shape, with higher volatility in the morning and lower volatility in the afternoon

9. Lee-Mykland
Brian Jansen
Problems of the Test
-Average BVj=(1/K) ∑ |Rt,j-1|^(1/2)*|Rt,j|*|Rt,j+1|^(1/2)

10. Lee-Mykland
Brian Jansen
Problems of the Test
-The average returns appear to be constant around zero throughout the trading day…(reassuring)

11. Corrections to the Test
Lee-Mykland
Brian Jansen
Corrections to the Test
-Let t=day and j=observation number in a given day
-So, R4,5 refers to the return of the 9:55 observation of the 4th day
-If we scale the return Rt,j by the average BVj at time interval j, the resulting return should account for the daily trend in volatility
-Thus, we could try R*= Rt,j/ sqrt(BVj)
-Then, we can re-calculate the instantaneous volatility using the adjusted returns
-Average BVj=(1/K) ∑ |Rt,j-1|^(1/2)*|Rt,j|*|Rt,j+1|^(1/2)

12. Extensions Correcting the Lee-Mykland test
Conclusion
Brian Jansen
Extensions
Correcting the Lee-Mykland test
Factor analysis looking at LM jump statistic across a range of oil and market stocks
Volatility 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?
More familiarity with the practices of the oil industry, especially their trading desk operation to determine how they deal with oil price volatility