Patterns in the Jump Process, and the Relationship Between Jump Detection and Volatility Dynamics Matthew Rognlie Econ 201FS Wednesday, March 18, 2009

Brief discussion of previous efforts Looked at connection between jumps and unusual increases/decreases in realized volatility in the surrounding days Did not find any pattern Likely problem: realized volatility is not a great measure of the volatility process, especially for noisy data on individual stocks. Could use implied volatility data to test at much higher frequency, but quality high-frequency implied volatility data is not available (with reasonable effort, at least) for individual stocks.

Difficulties with jumps on individual stocks Observation: it is very difficult to get any robust result linking jumps detected using BNS to other aspects of price data or financial markets. Maybe this is due to problems in the detection procedure itself? Is there any kind of regularity in the distribution of jumps detected using BNS?

Are detected jumps a Poisson process? Idea: use nonparametric statistical tests on our sample to see whether we can reject the null that jumps are uniformly distributed through the sample period. We use the Kolmogorov-Smirnov test, a classic nonparametric test that allows us to compare an estimated distribution to a hypothesized distribution, or two distributions with each other.

The Kolmogorov-Smirnov test Let F n (x) be the empirical CDF taken from our sample, and F(x) be our hypothesized distribution's CDF. Then the K-S statistic is simply: The Kolmogorov distribution is:

K-M test (continued)‏ If the sample indeed comes from the distribution F(x), then we have the asymptotic result: Then we reject the null at significance level K a if: If we instead want to compare two empirical distributions to see if they are the same, we get the test statistic:

Results for selected stocks P-values of K-M test of whether the distribution of jump dates is uniform (standard BNS test; 0.01 significance level)‏

Stocks with significant departures from the uniform distribution

Stocks without significant departures from the uniform distribution

Thoughts Clearly some stocks displayed temporal variation in jump frequency, while others (at least within our sample) did not. What drives the patterns that do exist? Unclear. Interesting to try the same tests on jumps in an index.

Other tests We can also try a two-sample K-M test on the entire distribution of z-scores, to see whether it differs before and after a certain date. Like many others, we try the date of Lehman Brothers' collapse:

Other tests (continued)‏ Following the hunch that liquidity may play a role in jumps, we consider a very crude metric for liquidity: the minimum size of a nonzero movement in log prices during the trading day. We set an arbitrary threshold, 2E-4, to divide the sample into “liquid” and “illiquid” periods This has the practical effect of splitting our sample period into pre-decimalization and post- decimalization periods.

Results A little stronger than the previous test, but only significant for WMT and KO. Difficult to say anything concrete about decimalization, except that it has no obvious and universal effect.

z-Score distributions where a significant difference is detected

Are jumps associated higher daily absolute log returns? Apparently not! No need to run t-tests: none of these results are remotely statistically significant, certainly not in the direction we'd expect. Whether a stock has a “jump day” as detected by BNS may mean less than we think.

Issues in jump detection The dynamics of the volatility process are important to the small-sample performance of the BNS jump test. For instance, to illustrate the possible importance of the intraday pattern in volatility, we run a Monte Carlo simulation where prices follow a continuous diffusion process, and volatility is a “step” function: one value for half the day, and half that value for the rest of the day. We also run a simulation where volatility is constant all day. We set 30 as our daily sampling frequency, and simulate 100,000 days.

Results of basic Monte Carlo simulation At 1% significance level, we detect jumps in 7.9% of days with “step-function” volatility, but only 3.0% of days with constant volatility. Since there are no jumps, of course, the asymptotic distribution of the BNS tests indicates we should detect 1% of the time; the higher values are due to small-sample properties. Changes in volatility make a big difference!

Effects from jumps in volatility? Huang and Tauchen (2005) do extensive Monte Carlo simulation to examine the finite-sample properties of BNS jump detection, with leverage effects and a feedback term in the diffusion function. They leave open the project of examining the effect of jumps in the volatility process on the test's performance. This may be important, since Todorov and Tauchen (2008) suggest that volatility is best described as a pure jump process.

Relationship between volatility jumps and price jumps The possible small-sample difficulties of the jump test when exposed to volatility jumps may be particularly important to understanding the relationship between volatility jumps and price jumps. Using tests from Jacod and Todorov (2008), Todorov and Tauchen (2008) find that jumps in volatility and price in the S&P 500 often occur together and are strongly dependent. These jumps typically have opposite sign; this is an encouraging finding for the robustness of the result.

The “Leverage Effect” in Simultaneous Jumps Assuming price and volatility jumps have opposite signs, several natural questions arise. Is there an asymmetry? (volatility jumps upward while price jumps downward, but the opposite is rare?)‏ How does this leverage effect compare to the leverage effect in the continuous component? Do the magnitudes differ? What does this tell us about causality and the appropriateness of various models?