Intraday Trading Invariants for Equity-Index Futures Torben G. Andersen, Oleg Bondarenko, Albert S. Kyle and Anna Obizhaeva Fields Institute Toronto, Canada January 2015 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 1/29
Hypotheses on Trading – Volatility Interactions Competing Ideas of How Information and Trades Induce Price Changes . . . And How Price Changes Convey News and Induce Trading One Extreme: All Information Incorporated via Trading Transactions or Trading Volume Drive Volatility Transaction Clock (Mandelbrot and Taylor, 1967; Jones, Kaul & Lipson, 1994; An´ e & Geman, 2000) Volume Clock (Clark, 1973) Other Extreme: Information Flow Drives Market Activity Latent Factor Driving both Trading and Volatility Mixture of Distributions (Tauchen & Pitts, 1983; Andersen, 1996) Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 2/29
Daily S&P 500 E-Mini Futures Market Activity Volatility σ Volume V 1.2 5000 1 4000 0.8 3000 0.6 2000 0.4 1000 0.2 0 0 2008 2009 2010 2011 2012 2008 2009 2010 2011 2012 Trade Size Q Number of Trades N 500 15 400 10 300 200 5 100 0 0 2008 2009 2010 2011 2012 2008 2009 2010 2011 2012 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 3/29
Hypotheses on Trading – Volatility Interactions Focus typically on Observations at Daily Level Exception: An´ e & Geman, 2000; Near Tick-by-Tick But Nobody has been Able to Confirm their Results We also Fail to Verify Hypothesis Use High-Frequency Data to Explore Interactions Include Market Microstructure “ Invariance ” in Analysis Empirical Support for Invariance via Diverse Market Phenomena Major Expansion in Realm of Features Covered by Invariance Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 4/29
Why High-Frequency Analysis Fact : Pronounced Intraday Market Activity Patterns News Incorporated into Prices quickly; Trading Fast Huge Systematic Variation over 24-Hour Trading Day Does any Basic Regularity Apply in this Setting? Macroeconomic Announcements particular Challenge Large Price Jump on Impact without (much) Trading Subsequent Price Discovery Process Sudden Market Turmoil: Crisis, Flash Crash Do Same or Different Regularities Apply in this Context? Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 5/29
Intraday Pattern for Market Activity Variables 4 x 10 Volume V Volatility σ 0.8 3 0.7 2.5 0.6 2 0.5 0.4 1.5 0.3 1 0.2 0.5 0.1 0 0 −5 0 5 10 15 −5 0 5 10 15 Trade Size Q Number of Trades N 1200 25 1000 20 800 15 600 10 400 5 200 0 0 −5 0 5 10 15 −5 0 5 10 15 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 6/29
S&P 500 E-Mini Futures Market Sample : BBO Files from CME Group; Jan 4, 2008 – Nov 2, 2011 Among World’s Most Active Markets – Price Discovery for Equities Time Stamped to Second; Sequenced in Actual Order Using Front Month Contract until Week before Expiration (most Liquid) Three Daily Regimes : Asia, Europe, North America Sunday – Thursday 17:15 – 2:00; 2:00-8:30; 8:30 – 15:15 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 7/29
Our Variables D = 959 Trading Days ; T = 1 , 320 1-Minute Intervals per Day N dt = Number of Transactions per Minute; V dt = Volume (Number of Contracts Traded per Minute); Q dt = Average Trade Size (Contracts per Trade over Minute); P dt = Average Price (over Minute); σ dt = Volatility in Minute Annualized (Decimal Form); W dt = Market Speed (dollars at Risk per Minute) = P dt V dt σ dt We define variables for intraday interval t by averaging across all days: D 1 � = n dt , t = 1 , ... , T . n t D · d =1 and variables for trading day d by averaging across all intervals, T 1 � n d = n dt , d = 1 , ... , D . T · t =1 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 8/29
Descriptive Statistics for S&P 500 E-mini Futures Regime 1 Regime 2 Regime 3 All Volatility 0.16 0.25 0.40 0.26 Volume 92.01 600.73 4725.56 1663.97 # Trades 13.87 66.74 360.04 135.70 Notional Value, $Mln 5.25 34.37 265.84 94.02 Trade Size 5.82 8.41 13.30 8.95 Market Depth 54.06 265.12 984.07 401.76 Bid-Ask Spread 26.54 25.69 25.13 25.86 Business Time 26.13 5.43 1.00 2.65 Sample Averages per 1 min. Volatility is Annualized (in Decimal Form). Business Time is proportional to W − 2 / 3 , and it is Normalized to 1 in Regime 3 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 9/29
Market Microstructure Invariance Theoretical Motivation Invariance based on Strategic Implementation of Trading Ideas or “Bets” Bets not Observable and Meta Orders Shredded into individual Orders Invariance Principle applies across Time and Assets: Dollar-Risk Transfer per Bet in Business Time is i.i.d. I = P · Q B · σ · N − 1 / 2 B P is price, Q B is bet size, σ is volatility, and N B (Business Time) is number of bets. For empirical work, we write in in logs: log I = p + q B + 1 2 s − 1 2 n B . Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 10/29
Market Microstructure Invariance Generating Testable Hypotheses Define the Quantity “Bet Activity” or “Market Activity” W , W = P · V · σ Under simplifying Assumptions, W (essentially) Observable ( w = logW = p + q B + n B ) Invariance Principle across Time and Assets now implies: n B ∼ 2 q B ∼ 1 3 w − [ p + 1 3 w and 2 s ] For Same ( σ, P ), Variation in Volume: 2 / 3 from N B , 1 / 3 from Q B . For Varying ( σ, P ), specific Power Relations are Implied Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 11/29
Market Microstructure Invariance Auxiliary Hypotheses invoked for Testable Hypotheses – still Success ◮ Kyle and Obizhaeva (2014) find invariance relationships in portfolio transition orders; ◮ Kyle, Obizhaeva, and Tuzun (2012) find invariance relationships in print sizes in TAQ data; ◮ Kyle, Obizhaeva, Sinha, and Tuzun (2014) find invariance relationships in number of news articles; ◮ Bae, Kyle, Lee, and Obizhaeva (2014) find invariance relationships in Korean data. Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 12/29
Invariance Inspired Predications Motivated by invariance, we stipulate a similar relation between intraday variables: I dt = P dt · Q dt · σ dt · N − 1 / 2 dt This implies in logs: n dt = c + 2q dt + s dt + u dt . Or, in terms of log of trading activity w = logW = p + q B + n B : = c + 2 3 · w j + u n n j j , q j = c + 1 3 w j − 1 2 s j + u q j . Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 13/29
Suggestive Test for Alternative Theories V and Q implicitly included within W ( = PV σ = PQN σ ). σ 2 Ignoring P , Relation dt ∼ V dt implies s dt = c + n dt + q dt . σ 2 Ignoring P , Relation dt ∼ N dt implies s dt = c + n dt . 2 3 + u n � � n j = c + w j − 2 q j j . [Clark] 3 and 2 3 [ w j − q j ] + u n n j = c + j . [An´ e & Geman] and 2 3 [ w j ] + u n n j = c + j . [Invariance] In terms of the nested specification: c + β · q j + u q s j − n j = j . where β = 1 (Clark), β = 0 (An´ e-Geman), or β = − 2 (invariance). Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 14/29
Suggestive Test for Alternative Theories W This table reports on intraday OLS regressions of log N onto log Q 3 / 2 (Clark), log W Q (An´ e & Geman), and log W (Invariance). All models predict β = 2 / 3. For each regression there are T = 1 , 320 observations. Table : OLS Regression of log N onto scaled log W ¯ R 2 α β se( α ) se( β ) Clark 2.4119 0.9757 0.0031 0.0016 0.9965 An´ e & Geman 1.7538 0.8490 0.0018 0.0006 0.9993 Invariance 0.8579 0.6708 0.0033 0.0007 0.9986 Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 15/29
Suggestive Test for Alternative Theories log N vs log W/Q 3 / 2 log N vs log W/Q 7 7 6 6 5 5 4 4 3 3 2 2 −1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 log N vs log W 7 6 5 4 3 2 0 2 4 6 8 10 Figure : OLS Regression Line (solid) and Model Predicted (dashed). Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 16/29
Trading Intensity and Trade Size Table : Indraday OLS Regression of log σ 2 N onto log Q ¯ R 2 se( c ) se( β ) c β -2.6275 -2.0002 0.0199 0.0099 0.9687 This table reports the results of the intraday OLS regression, s t − n t = c + β · q t + u t . Clark, An´ e & Geman and invariance predict β = 1, β = 0, and β = − 2, respectively. The regression exploits T = 1 , 320 observations. Invariance yields vastly Superior Fit to Intraday Activity Patterns Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 17/29
Trading Intensity and Trade Size log σ 2 /N vs log Q log σ 2 /N vs log Q −5 −5 −6 −6 −7 −7 −8 −8 −9 −9 1 1.5 2 2.5 3 3.5 1 1.5 2 2.5 3 3.5 Figure : The figures provide intraday scatter plots of log σ 2 N t versus log Q t . Also shown are OLS Regression line (solid) and the predicted line according to invariance (dashed). The right panel is the same as the left panel, except it removes observations corresponding to 3 minutes around 1:00, 3 minutes around 2:00 (start of the European regime), 3+30 minutes around 8:30 (start of Regime 3 and the 9:00 announcements), and 16 minutes at the end of trading. Andersen, Bondarenko, Kyle, and Obizhaeva Intraday Invariance 18/29
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