Modelling systemic price cojumps with Hawkes factor models

Modelling systemic price cojumps with Hawkes factor models Michele Treccani Joint work with G. Bormetti, L.M. Calcagnile, F. Corsi, S. Marmi and F. Lillo XV Workshop on Quantitative Finance Florence, January 23-24, 2014 1 / 21

What is QuantLab QuantLab is a joint research laboratory between SNS & List S.p.A. QuantLab is a quantitative finance research laboratory aimed at providing innovative solutions to the challenges faced by traders, regulators and exchanges in the evolving area of nowadays electronic trading. G. Bormetti F. Lillo S. Marmi L. M. Calcagnile M. Treccani QUANT 2 / 21

Stock price jumps Financial markets are intrinsically unstable and display large price fluctuations. Often these fluctuations: occur on very short time scales they are systemic: simultaneously on different stocks and instruments revert quickly during the day. Systemic cojumps archetype: The Flash Crash On May 6, 2010 markets dropped 1% per minute, reaching a low of more than 10%, the biggest one-day point decline. During the flash crash 8 stocks in the S&P500 were traded at 1 cent (e.g. Accenture). Others (e.g. Apple and Hewlett-Packard) were traded at 100,000$. Questions Is the Flash Crash an isolated event or the tip of the iceberg? Is there an underlying dynamics of these cojumps? Is the contagion of jumps across assets faster today? Are markets more exposed to systemic events than before the advent of HFT? 3 / 21

Data description Data Sample Tick by Tick Data of FTSE MIB 40 Italy 1 Min log returns of 20 high liquidity stocks, 88 Days of Executions (March-June 2012) Outlier removal(brownlees - Gallo (2006)), Merging / Splitting and Volatility auctions are automatically subtracted Removal of intraday volatility pattern Jump r σ > θ Jump Definition with σ2 t = µ 2 1 α i>0 (1 α) i 1 r t i r t i 1, 2 with µ 1 = π and α = 0.032. We find an average 1.2 jumps per stock per day, ranging from 0.7 (Mediaset) to 2.1 (Enel) per day. 4 / 21

Multiple cojumps in the Italian stock market Time series of the number of stocks n co-jumping simultaneously co jumps day Mar Apr May Jun Jul n = 1 2 <= n <= 3 4 <= n <= 8 9 <= n <= 20 9 10 11 12 13 14 15 16 17 hour A large number of cojumps involving a sizable number of assets! 5 / 21

Comparing with independent Poisson processes random co jumps day Mar Apr May Jun Jul 9 10 11 12 13 14 15 16 17 hour n = 1 2 <= n <= 3 4 <= n <= 8 9 <= n <= 20 log 10(counts) 3 2 1 0 0 10 20 Number of cojumping stocks Figure : Left. Simulated independent multivariate Poisson process with intensities as in real data. Right. Histogram of the number of stocks jumping simultaneously in one minute. Real data (filled circles) vs. independent Poisson model (empty circles). Multiple jump (MJ): the event of at least two jumps of the stock price occurring inside a time window w. Nw ˆp MJ w = i=1 1 si 2 N Assicurazioni Generali w ˆp MJ w 0.06 0.04 0.02 Empirical Poisson mean Poisson 99% c.l. Poisson 95% c.l. 0 0 50 100 w (minute) In our data sample the null is rejected for 18 stocks out of 20. Strong evidence of time clustering of jumps and violation of the univariate Poisson model. 6 / 21

Dynamic Intensity Models: Hawkes Processes Hawkes Processes (Hawkes, 1971) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Intensity Events 0 5 10 15 20 Time 4 0 0 201 3 3 3 3 22 2 22 2 2 2 11 1 1 1 1 1 Time Figure : Left. Example of a simulated univariate Hawkes process. A blue triangle signals the occurrence of a count. A single exponential kernel was employed with λ = 1.2, α = 0.5, β = 0.9. Right. Branching structure of the Hawkes process (top) and events on the time axis (bottom). This picture corresponds to a branching ratio equal to n = 0.88. (from Filimonov and Sornette, PRE 2012). A univariate point process N t is called a Hawkes process if it is a linear self-exciting process. For our purposes, we focus on a particular realization of Hawkes process with the following parametrization: I (t) = λ + t i <t αe β(t t i ). where λ is called the base intensity, α 0 is a scale parameter, β > 0 controls the strength of decay. 7 / 21

Testing under Null Hawkes 0.015 Empirical Monte Carlo mean Monte Carlo 99% c.l. 0.01 Monte Carlo 95% c.l. 0.5 ˆp MJ w 0.005 0.25 0 0 10 20 30 w (minute) 0 0 250 500 w (minute) Univariate Hawkes processes are able to capture the dynamics and time clustering of jumps of single asset real data (Generali, Intesa Sanpaolo) Cross-jump: the event of at least one jump in the series of each stocks l,k occurring inside a given time window w. ˆp CJ w = N w i=1 1 s l i 1 1 s k i 1 N w Generali Intesa ˆp CJ w 0.015 0.01 0.005 0 Empirical Monte Carlo mean Monte Carlo 99% c.l. Monte Carlo 95% c.l. 0 10 20 30 w (minute) Univariate Hawkes fail to describe the cross stock time clustering of empirical jumps 8 / 21

Ansatz: 1 Factor Model + Idiosyncratic with 20 stocks Stock X can jump: responding with Prob. P x to a jump of the (unobservable) Factor, following its own Idiosyncratic Process. 1 Factor model calibration N F Time Series of at least J synchronous jumps (J = 4, 5, 6...) λ F T = #N F P 1λ F T = #N 1F... = P 1,..., P 20, λ F P 20λ F T = #N 20F 1 Calibrate (λ H, α, β) on N F time series 2 Perturb only λ H = λ F /(1 α β ) Idiosyncratic contribution For each stock (only Hawkes scenario): Define the complement set: N i F N i \ N F Calibrate (λ H,i, α i, β i ) on the i th N i F time series λ i T = #N i F Perturb only λ H,i = λ i /(1 α β ) 9 / 21

1 Factor Hawkes + Idiosyncratic Hawkes: results 0.01 Empirical Monte Carlo mean Monte Carlo 99% c.l. Monte Carlo 95% c.l. 0.01 ˆp MJ w 0.005 ˆp CJ w 0.005 0 0 0 10 20 30 0 10 20 30 0.02 0.01 ˆp CJ w 0.01 0.005 0 0 10 20 30 w (minute) 0 0 10 20 30 w (minute) Figure : From the top left clockwise: MJ probability test under N factor model null for the asset Generali; CJ probability test for the pairs Generali-Mediobanca, Generali-Banca Popolare Milano, and Generali-Intesa Sanpaolo. Easy to calibrate P 1,..., P 20, λ F Parsimonious formulation: can be extended to arbitrary number N of stocks (O(N) parameters) 10 / 21

Russell 3000 data for a historical perspective Data 1-minute data of the Russell 3000 components 13 years (2000 2012) 140 most liquid stocks for each year high-frequency cleaning procedures as before (outliers, stock splits, intraday volatility pattern) How markets instability has changed: 2000 vs 2012 2000: Many jumps involving few stocks 2012: Few jumps involving many stocks 5 11 / 21

Evolution of systemic cojumps total jump minutes cojump multiplicity 10 #(jump minutes) 0 20000 40000 60000 relative #(cojump minutes) 0.000 0.010 0.020 0.030 cojump multiplicity 30 cojump multiplicity 60 relative #(cojump minutes) 0.000 0.004 0.008 relative #(cojump minutes) 0.000 0.001 0.002 0.003 www.quantlab.it Total number of jumps and the number of single asset jumps has actually declined in recent years The number of multiple asset co-jumps has significantly increased This effect is stronger for systemic cojumps, i.e. cojumps involving a large number of assets 12 / 21

Evolution of systemic cojumps, θ dependence total jump minutes cojump multiplicity 10 #(jump minutes) 0 10000 30000 50000 θ = 4 θ = 6 θ = 8 θ = 10 relative #(cojump minutes) 0.000 0.010 0.020 0.030 θ = 4 θ = 6 θ = 8 θ = 10 cojump multiplicity 30 cojump multiplicity 60 relative #(cojump minutes) 0.000 0.004 0.008 0.012 θ = 4 θ = 6 θ = 8 θ = 10 relative #(cojump minutes) 0.000 0.002 0.004 0.006 0.008 0.010 θ = 4 θ = 6 θ = 8 θ = 10 www.quantlab.it Defining jumps as more and more extreme events (θ = 4, 6, 8, 10) does not affect the historical evolution of systemic cojumps. 13 / 21

Factor Hawkes parameter evolution 0.000 0.005 0.010 0.015 lambda 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 0.00 0.02 0.04 0.06 0.08 0.10 alpha 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 0.00 0.05 0.10 0.15 0.20 beta 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 0.0 0.1 0.2 0.3 0.4 0.5 0.6 alpha / beta 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Figure : Historical view of the Hawkes parameters of the factor process (years 2001 2012). Stationarity holds if α β < 1: all years are below the critical threshold β decreases = the typical time scale of self-excitation increases #factor jumps λ 1 α β is slightly heterogeneous across the years 14 / 21

News Impact www.quantlab.it Are systemic events endogenous or do they have an external origin? We adopted the Econoday Database We focus on US expected macro news, likely to affect the systemic events We are able to address from historical point of view (2001-2012) the role of news on our observables What s the fraction of systemic events that can be associated with exogenous events? News ADP Employment Report Beige Book Business Inventories Chairman Press Conference Chicago PMI Construction Spending Consumer Confidence Consumer Price Index Consumer Sentiment Dallas Fed Mfg Survey Durable Goods Orders EIA Petroleum Status Report Empire State Mfg Survey Employment Cost Index Category MMI MMI MMI News Employment Situation Existing Home Sales Factory Orders FOMC Forecasts FOMC Meeting Announcement FOMC Minutes GDP Housing Market Index Housing MMIts Import and Export Prices Industrial Production International Trade ISM Mfg Index ISM Non-Manufacturing Empl. Ind. # News 0 50 100 150 200 250 News occurrence All News "Merit Extra Attention" "Market Moving Indicator" 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Category News MMI Jobless Claims MMI Motor Vehicle Sales New Home Sales MMI Pending Home Sales Index MMI Personal Income and Outlays MMI Philadelphia Fed Survey MMI PMI Manufacturing Index PMI Manufacturing Index Flash MMI Producer Price Index Productivity and Costs MMI Retail Sales MMI S&P Case-Shiller HPI MMI Treasury Budget MMI Treasury International Capital Category MMI MMI MMI MMI MMI MMI MMI 15 / 21

1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Quantifying News Impact 2001 2005 Fraction of cojumps 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 "Merit Extra Attention" News Lag 0 Min Lag 1 Min Lag 2 Min Lag 3 Min Lag 4 Min Lag 5 Min 5 10 50 100 500 5000 50000 Fraction of cojumps 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 "Merit Extra Attention" News Lag 0 Min Lag 1 Min Lag 2 Min Lag 3 Min Lag 4 Min Lag 5 Min 1 10 100 1000 10000 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Cojump Multiplicity, all News 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Cojump Multiplicity, all News 2008 2012 Fraction of cojumps 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 "Merit Extra Attention" News Lag 0 Min Lag 1 Min Lag 2 Min Lag 3 Min Lag 4 Min Lag 5 Min 5 10 50 100 500 5000 Fraction of cojumps 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 "Merit Extra Attention" News Lag 0 Min Lag 1 Min Lag 2 Min Lag 3 Min Lag 4 Min Lag 5 Min 5 10 50 100 500 5000 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Cojump Multiplicity, all News Cojump Multiplicity, all News Black curve: fraction of events with multiplicity m that occur within a 5-minute range from a news release An important fraction of News are (increasingly) irrelevant for cojumps High-multiplicity events are more likely to be news-related than low-multiplicity The digestion time decreases from 2001 to 2012 Even at higher multiplicities, the overall fraction is below 0.4 16 / 21

Endogenous Vs Exogenous Origin Total jump minutes Cojump multiplicity 10 #(jump minutes) 0 10000 30000 50000 Total Jump News Triggered Jumps(5 Mins) relative #(cojump minutes) 0.000 0.010 0.020 0.030 Jump Fraction News Triggered Fraction (5 Mins) Cojump multiplicity 30 Cojump multiplicity 60 relative #(cojump minutes) 0.000 0.002 0.004 0.006 0.008 Jump Fraction News Triggered Fraction (5 Mins) relative #(cojump minutes) 0.000 0.001 0.002 0.003 Jump Fraction News Triggered Fraction (5 Mins) www.quantlab.it The release of US Macro-Economic News explains less and less fraction of Systemic Events, especially at higher multiplicities, its fraction being confined well under 40% 17 / 21

Conclusions A large number of jumps are present in financial markets On individual stocks, jumps are clearly not described by a Poisson process, but display time clustering well described by Hawkes processes We identify a very large number of simultaneous and systemic co-jumps, i.e. sizable sets of stocks simultaneously jumping We propose a Hawkes one factor point process model which is able to describe 1 The time clustering of jumps on individual stocks 2 The time lagged cross excitation between different stocks 3 The large number of simultaneous systemic jumps Individually, stocks have become less jumpy in recent years However the frequency of systemic jumps has considerably increased Systemic cojumps are only marginally related to news 18 / 21

A toy example: 2 Stocks without idiosyncratic terms Without idiosyncratic terms the only variable to specify is the point process describing the factor Poisson factor: λ F (t) = λ Poisson Hawkes factor: λ F (t) = λ Hawkes (t) p 1p 2λ F T = n 12 p 1λ F T = n 1 = P 1, P 2, λ F p 2λ F T = n 2 where n 1 and n 2 are the realized number of jumps of the stock 1 and 2, while n 12 represents the realized number of cojumps between 1 and 2 within the one minute sampling interval. Easy to implement with 2 stocks No unique way to extend to multi stocks 19 / 21

Estimation procedure in a nutshell We start by assuming that all the point processes are independent and we estimate for each stock the probability of observing a jump each time t: the vector of probabilities is p t = ( ) pt 1,... pt N ( = I 1 t t,..., It N t ) We test if the number of jumps that we observe at time t is compatible with the cross independence among the processes. Under the null hypothesis, the probability of observing j jumps at time t is ( ) P (J t = j) = p l 1 t... p l j t 1 pt k. 1 l 1 <...<l j N k {1,...,N}\{l 1,...,l j } Since we repeat the test T times, we adjust the significance level with the Bonferroni s correction. If at time t F we reject the null, we attribute the event to a systemic shock, and we remove it from the set {t s i } We iterate the procedure as many times as required in order to remove all the systemic jumps. The remaining jumps are idiosyncratic. Both types of time series can be fit, for example, with Hawkes processes. 20 / 21

Dynamic Intensity Models: Multivariate Hawkes Processes A K-dimensional Hawkes Processes is a linear self-exciting process, defined by the multivariate intensity I (t) = ( I 1 (t),..., I K (t) ) where the k-type intensity using an exponential kernel with P = 1 is given by I k (t) = λ k + K α km e β km(t ti m ). m=1 t i m <t As an example of bivariate Hawkes, Hewlett 2006 proposed the following model for buy-sell activity λ buy t = µ buy + α buybuye βbuybuy(t u) dnu buy + α buyselle βbuysell(t u) dnu sell (1) u<t u<t λ sell t = µ sell + u<t α sellselle β sellsell(t u) dn sell u + u<t α sellbuye β sellbuy(t u) dn buy u (2) Can a multivariate Hawkes process capture both the multiple- and the cross-jumps? 21 / 21