What's a Jump? Exploring the relationship between jumps and volatility, and a technical issue in jump detection

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1 What's a Jump? Exploring the relationship between jumps and volatility, and a technical issue in jump detection Matthew Rognlie Econ 201FS February 18, 2009

2 Idea: Different Kinds of Jumps Unexpected jumps reflect news that comes at an unexpected time, and may take the market some time to fully understand and process Expected jumps reflect announcements or other information anticipated to come at a particular time Unexpected jumps may lead to higher volatility afterward, either because information is being processed or because the jump indicates new uncertainty about a stock Expected jumps may lead to lower volatility, by quelling speculation about some outcome, or higher volatility

3 Basic Test Any connection between jumps and unusual increases or decreases in volatility the next day? Identify jumps (at 99.9%) and unusual increases/decreases in RV over two day increments (more than 1 log point) MSFT: 6.13% jumps, 12.09% strong RV increases,.55% both; 12.23% strong RV decreases,.86% both JPM: 5.82% jumps, 8.76% strong RV increases,.48% both; 8.80% strong RV decreases,.48% both

4 First impressions These values seem almost independent! No obvious connection between jumps and either strong decreases or increases in RV the next day. Results similar for other stocks, jump sensitivities, thresholds for increases and decreases, etc. RV varies a lot on its own; difficult to come up with significant results Too many jumps maybe informational models don't explain most of them (but then what does?)

5 Other simple tests Correlation between flagged jumps on day t and log of RV difference from day t-1 to t+1: for MSFT,.0021 for JPM Correlation with absolute value of log RV difference:.0004 for MSFT, for JPM Negligible relationships

6 Another direction Could test implied volatility from option data. Advantages: a much finer test of the market perception of jumps' effects on volatility Downside: no easy source of high-frequency options data to pinpoint sudden changes. Need to use daily data, which may have highmagnitude changes from day to day that do not reflect any single jump in implied volatility. Like before, many jumps may be too small to have a significant impact

7 Question: How meaningful are most jumps detected by BNS on individual stocks, anyway? Are many jumps just byproducts of microstructure noise, or problems with the small-sample properties of our test statistics? Jumps seem to occur at similar frequencies throughout very different market environments Does the jump component act like a more-orless random slice of RV in terms of predictivity?

8 Jump components of RV: separated with significance threshold

9 Investigating predictivity: autoregression Regress jump and nonjump components against daily, weekly, and monthly averages of past jump and nonjump components of RV Look at coefficients: how predictive are jumps? JumpD NonjumpD JumpW NonjumpW JumpM NonjumpM Constant MSFT Nonjump MSFT Jump JPM Nonjump JPM Jump (Standard errors not estimated because of the need to account for serial correlation)

10 HAR estimation of nonjump component

11 Thoughts Analysis is extremely preliminary: need to use more stocks, more functional forms (i.e. take square roots to find volatility), alternative significance thresholds, and most importantly find serial-correlation robust standard errors. Idea is to see the extent to which the jump component resembles a random slice of RV for predictive purposes Errors are presumably high since jump days are so uncommon; taking lots of stocks may give a better indication

12 Other topic: U-shaped intraday distribution of returns

13 What's the problem? Z-scores from BNS jump tests may be artificially biased downward in magnitude by this consistent pattern in intraday results The typical expression for the z-statistic contains, in its denominator, the square root of the tripower or quadpower quarticity divided by the bipower variation (possibly maxadjusted). If the day contains multiple periods where returns are scaled higher or lower, but other characteristics are the same (just scaled), this denominator will be higher than if we examine each period separately.

14 Estimating degree of this bias for a simple example Say that the second half of a day has returns that are (on average) exactly a fraction 'x' of those in the first half of the day, but otherwise obey a similar distribution. Then the upward bias in the denominator will be, on average:

15 Further Exploration How does this reflect on the BNS jump test? Is a correction possible? Are the z-scores we calculate really artificially low in magnitude?