Intraday Patterns in Jump Components of Equity Returns

1. Intraday Patterns in Jump Components of Equity Returns Peter Van Tassel Duke University Durham, North Carolina 22 October 2007

2. Agenda • Brief summary of background literature and model considered • Data • The Lee-Mykland Statistic • Intraday patterns in diffusive and jump components of equity volatility • Forecasting realized variance, a preliminary attempt at incorporating intraday patterns in jump components

3. The Model • Consider stock price evolution defined as, with time-varying drift, stochastic volatility, and a Poisson jump process. • Define intraday geometric return as,

4. The Model: Realized Variance • Define the realized variance as, • As cited in Huang and Tauchen (2005), Andersen, Bollerslev, Diebold (2002) note that for an arbitrary day t we have which is a consistent estimator of the integrated variance and the jump component.

5. The Model: Bi-Power Variation • Define the bi-power variation as, • Under what Huang and Tauchen (2005) describe as reasonable assumptions about the model for stock price evolution, Barndorff-Nielsen and Shephard (2004b) implies that making the bi-power variation a consistent estimator for the integrated variance.

6. The Model: Jump Component • As noted in Barndorff-Nielsen Shephard (2004b), the difference between the realized variance and the bi-power variation is a consistent estimator of the jump component in equity volatility. • In particular, • This result is the basis for all of our subsequent work.

7. The Data • Use high frequency data from Trade and Quote Database • Price series is from 1 January 2001 to 31 December 2005 • We will focus on the Standard & Poor’s Depository Receipt as a proxy for the market portfolio. The SPDR is an exchange-traded fund launched in 1993 to track the performance of the S&P500. • We will also consider nine individual stocks: Citigroup, Lowes, Altria Group, Merck, Pepsi, the United Parcel Service, Verizon Wireless, Walmart, and Exxon Mobil.

8. The Data: Sampling Frequency • How finely we sample is limited by the bias market-microstructure noise (MMN) can have on prices. • MMN arises from a variety of sources. Two examples are trading mechanism and discreteness of prices, discussed in Black (1976) and Harris (1991) respectively. • We include volatility signature plots as recommended by Andersen, Bollerselv, Diebold, and Labys (2000) to visually select a suitable frequency. • Each plot contains the average realized variance at various sampling frequencies.

9. The Data: Volatility Signature Plots • The x-axis plots the sampling frequency in minutes. • The y-axis plots the annualized volatility in percentage terms,

10. The Lee-Mykland Statistic • Our motivation comes from Huang and Tauchen (2005). They recommend a Z statistic that flags entire trading days as statistically significant under the null hypothesis that there is no jump over the course of an entire day. • It is defined as, • The Lee-Mykland statistic is a natural extension of the Z statistic. It is a jump detection scheme based on principal of comparing realized variance and bi-power variation. Its purported advantage is that it allows for flagging of individual returns as statistically significant jumps.

11. The Lee-Mykland Statistic: Definition • We will define the Lee-Mykland statistic as, • It is introduced in a working paper by Lee and Mykland (2006) and used in working paper from the St. Louis Federal Reserve Bank by Lahaye, Laurent, and Neely (2007). • We are primarily interested in using the statistic as a method for flagging statistically significant intraday returns as jumps. • This builds on the work of Huang and Tauchen (2005) which presents a jump detection test that flags entire trading days.

12. The Lee-Mykland Statistic: Intraday Patterns • This is annoying • We are interested in the number of statistically significant jumps at different points in the trading day. • In the subsequent slides we consider • K is the window size. It determines the extent to which the bi-power variation is backward looking. • M is the within-day sampling frequency.

13. The Lee-Mykland Statistic: SPY Data

14. The Lee-Mykland Statistic: Individual Stocks Axes and Parameter Info: • K = 4.54 days • X-axis: Time at NYSE • Y-axis: Statistically significant jumps

15. The Lee-Mykland Statistic: Kernel Density Estimation

16. Redefine realized variance and bi-power variation as estimators for intraday periods that are approximately one hour in length. We consider the average realized volatility and bi-power variation over seven intraday periods. In particular, Intraday Patterns

17. The x-axis plots time at NYSE. On the y-axis both the estimators are annualized, i.e. Intraday Patterns: SPY

18. Note: These are the same plots included on the previous slide for the SPY. Here they are included for individual stocks. Intraday Patterns: Individual Stocks

19. Now we investigate the effect of intraday patterns on the LM statistic. We begin with a toy example. Consider taking the average of a periodic function that exhibits similar patterns to the smiles or smirks observed in diffusive component of equity volatility. In particular, As usual we interpret M to be the sampling frequency and n to be the window size. Intraday Patterns: Theoretical Example

20. X-axis: Time at NYSE. Y-axis: Lagged values sample the true functions 22 times per period at different window sizes. This is analogous to 17.5 minute sampling and different window sizes with the real data. Intraday Patterns: Theoretical Example

21. Intraday Patterns: Theoretical Example 2 • We want to consider the effects of redefining the Lee-Mykland Statistic so the bi-power variation is both forward looking and backward looking such that it is centered at the current time. • In our theoretical example that means computing for even numbers of n. • As well as for odd numbers of n. • In the next slide we include these computations at a sampling frequency of 22 over various window sizes. As usual the x-axis corresponds to the time of day at the NYSE.

22. Intraday Patterns: Theoretical Example 2

23. Intraday Patterns: Implications for LM Statistic • For even values of K we redefine the Lee-Mykland Statistic as, • For odd values of K we make an analogous adjustment as in the case of the toy example. • In the subsequent slides we compare the original Lee-Mykland Statistic with its new definition for the SPY and PEP. For each plot we will have: • X – Axis: Time at the NYSE • Y – Axis: Number of Statistically Significant Jumps • Title: Window Size of Bi-Power Variation

24. Intraday Patterns: LM and LM* applied to SPY

25. Intraday Patterns: LM and LM* applied to PEP

26. Summary of Results • We detect smiles or smirks in the intraday diffusive and jump components of equity volatility. • The LM Statistic also indicates that similar intraday patterns exist in the number of statistically significant jumps. • However, we are suspicious of our results computed from the LM Statistic. • At the recommended window sizes patterns in local volatility may be eliminated, over-exaggerating the number of statistically significant jumps flagged in the morning and late afternoon.

27. Forecasting • In the subsequent slides we consider some simple regressions to see how well a jump component defined as, can forecast realized variance. • We also consider a variation on the heterogeneous autoregressive realized variance (HAR-RV) models developed in Müller et. al (1997) and Corsi (2003). • In these models the averaged future realized variance is the dependent variable whereas the averages of past values are used as the independent variables. • In particular,

28. Forecasting: SPY Data • … • h • Correlation

29. Forecasting: Background on HAR-RV-CJ • Andersen, Bollerslev, Diebold (2006) separates the jump and diffusive component of equity volatility and includes both components in a heterogeneous autoregressive realized variance continuous jump model (HAR-RV-CJ) • They find minimal persistence in the jump component and do not find a large improvement in explanatory power from dividing realized variance into the two components. • As part of his project for the undergraduate seminar Andrey arrived at similar results performing the same regression on a larger data set. • With his help performing the regressions we proceed by modifying the jump component to focus on the first hour of trading.

30. Forecasting: HAR-RV-CJ Regressions • Now define the diffusive component as, • In ABD (2006) and Andrey’s paper the jump component is defined as, • We will consider the jump component defined as,

31. Forecasting: HAR-RV-CJ Regressions • As before define the averages for the jump and diffusive components as follows, • Now we consider the following regression for the new definition of the jump component,

32. Forecasting: Empirical Results

33. Forecasting: Empirical Results

34. Conclusion • We detect intraday smiles or smirks in both the diffusive and jump component of equity volatility. • The LM statistic indicates similar patterns, but we are suspicious of our results implementing the LM statistic as it is currently defined. • Using the jump component during the first hour of trading compared to the entire trading day does not improve the explanatory power of the HAR-RV or HAR-RV-CJ regressions considered in ABD (2006) and in Andrey’s paper.