Second Attempt at Jump-Detection and Analysis Mike Schwert ECON201FS 2/13/08

My Approach This Week Last time, found too many jump days. Likely explanations include microstructure noise from using minute-by-minute prices and accidental inclusion of overnight returns in intraday calculations. This time, rewrote code to sample prices at 5 minute frequency and exclude overnight returns. Recalculated summary statistics and number of jump days. Examined effect of sampling frequency on jump detection by calculating z-statistics and counting jump days for 5, 10, 15, and 20 minute sampling frequencies. Examined effect of sampling frequency on volatility calculations by creating volatility signature plots for realized variance and bipower variation.

GE Stock Prices (5 minute frequency) Note: Closed at 39.28 on 9/10/01, opened at 35.50 on 9/17/01, bottomed out at 28.70 on 9/21/01

Statistics Calculated Realized Variance: Bipower Variation: Relative Jump:

Statistics Calculated Tri-Power Quarticity: Quad-Power Quarticity: Z-Statistic:

Summary Statistics variable mean std. dev min max Realized Variance 0.0111 Bipower Variation 2.4719 x 10-4 3.7134 x 10-4 8.3864 x 10-6 0.0092 Relative Jump 0.0601 0.1152 -0.2542 0.6221 Tri-power Quarticity 3.9150 x 10-7 7.7188 x 10-6 7.8711 x 10-11 3.8164 x 10-4 Quad-power 3.2979 x 10-7 5.8122 x 10-6 2.8166 x 10-4 ZQP-max Statistic 0.6545 1.2069 -2.7547 6.9951

ZQP-max Statistics – 5 minute sampling frequency Number of jumps at 1% level of significance: 234 out of 2670 days (8.76%) Number of jumps at 0.1% level of significance: 84 out of 2670 days (3.15%) Number of jumps at 0.01% level of significance: 29 out of 2670 days (1.09%) Note: 1% significance when Z > 2.33, 0.1% significance when Z > 3.09, 0.01% significance when Z > 3.71.

ZQP-max Statistics – 10 minute sampling frequency Number of jumps at 1% level of significance: 186 out of 2670 days (6.97%) Number of jumps at 0.1% level of significance: 60 out of 2670 days (2.25%) Number of jumps at 0.01% level of significance: 17 out of 2670 days (0.64%) Note: 1% significance when Z > 2.33, 0.1% significance when Z > 3.09, 0.01% significance when Z > 3.71.

ZQP-max Statistics – 15 minute sampling frequency Number of jumps at 1% level of significance: 148 out of 2670 days (5.54%) Number of jumps at 0.1% level of significance: 42 out of 2670 days (1.57%) Number of jumps at 0.01% level of significance: 13 out of 2670 days (0.49%) Note: 1% significance when Z > 2.33, 0.1% significance when Z > 3.09, 0.01% significance when Z > 3.71.

ZQP-max Statistics – 20 minute sampling frequency Number of jumps at 1% level of significance: 141 out of 2670 days (5.28%) Number of jumps at 0.1% level of significance: 40 out of 2670 days (1.50%) Number of jumps at 0.01% level of significance: 11 out of 2670 days (0.41%) Note: 1% significance when Z > 2.33, 0.1% significance when Z > 3.09, 0.01% significance when Z > 3.71.

Volatility Signature Plots Used idea introduced by Andersen, Bollerslev, Diebold, and Labys (1999). Calculated mean daily realized variance and bipower variation over the sample period under sampling frequencies of 1 minute, 2 minutes, …, 30 minutes. Plotted mean realized variance and bipower variation on the y-axis with sampling frequency on the x-axis. RV and BV are higher for high-frequency samples because returns are distorted by microstructure noise such as bid-ask bounce. RV and BV decrease as interval between samples increases because microstructure noise is cancelled out. Must be wary of using too low of a sampling frequency, as sampling variation will affect volatility calculations. Balance between sampling variation and microstructure noise appears to be reached around 15 minute sampling frequency.

Volatility Signature Plot – Realized Variance

Volatility Signature Plot – Bipower Variation

Possible Extensions Perform same calculations on S&P 100 index and stocks highly correlated with GE, or those with similar beta, or from a similar industry, etc. Check whether GE jumps on the same days as these other assets. Determine how much jumps are systematic vs. idiosyncratic. Use volatility signature plots from several stocks to determine ideal sampling frequency for jump detection, if possible. Incorporate ARCH, GARCH, or stochastic volatility models somehow?