Systemic risk at high frequency: price cojumps and Hawkes factor models

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1 Systemic risk at high frequency: price cojumps and Hawkes factor models Fabrizio Lillo Scuola Normale Superiore di Pisa, University of Palermo (Italy) and Santa Fe Institute (USA) FisMat Milan, September 9, / 31

2 In collaboration with This project has been developed in QuantLab: G. Bormetti L.M. Calcagnile F. Corsi S. Marmi M. Treccani QUANT QuantLab is a joint Research Lab between SNS & List Spa. See 2 / 31

3 Motivations Financial markets are intrinsically unstable and display large price fluctuations Often these fluctuations occur on very short time scales, and sometimes revert quickly. It is commonly believed that in recent year there has been an increasing frequency of these events Neither idiosyncratic news nor market wide news can explain the frequency and amplitude of price jumps (Joulin et al. 2008) Nanex research suggested a sharp increase of price jump frequency and related it to market regulation when rebalancing their positions, High Frequency Traders may compete for liquidity and amplify price volatility (Kirilenko et al. 2010) Flash crash was not only about E-Mini S&P 500 futures (where everything started), but propagated in a very short time to ETFs, stocks, options, etc 3 / 31

4 Price jumps Price jumps are discontinuities in the price process. Mathematically it is typically described by a compound Poisson noise term in the price equation X t = t 0 µ sds + t 0 N t σ sdw s + l=1 J t 4 / 31

5 Price jumps are very common Figure: Left. Intraday jump of the DAX futures. Right. Twitter flash crash of April 23, / 31

6 May 6, 2010 Flash Crash On May 6, 2010 markets dropped 1% per minute, reaching a low of more than 10% 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$ There is growing concern about the possible role of High Frequency Trader as responsible of large price jumps when rebalancing their positions, High Frequency Traders may compete for liquidity and amplify price volatility (Kirilenko et al. 2010) 6 / 31

7 Stock price jumps and systemic risk Flash crash was not only about E-Mini S&P 500 futures (where everything started) Because of arbitration, when the e-mini changes price with high volume, Many ETFs are repriced (quotes updated, trades executed). The component stocks of ETFs are also repriced, along with many indexes. All the option chains for the ETFs, their components and indexes are also repriced. Trades being executed at irrational prices as low as one penny or as high as $ 100,000 Flash crash: a 20 millesecond cascade (Nanex) HFTs as means of contagion (Gerig 2012) Motivation for our study: how frequent are systemic events? How can we model them? 7 / 31

8 Questions Dynamics of jumps: (Bormetti et al., Modelling systemic price cojumps with Hawkes factor models, arxiv: ) Is there a dynamics of jumps? Does a jump change the probability of another jump in the near future? Time clustering of jumps? Is there a contagion between jumps in different assets? Cross-asset excitation of jumps? How frequent are systemic co-jumps? Have market become more unstable in recent years? Have markets become more jumpy in recent years? Is the contagion of jumps across assets faster today? Are market more exposed to systemic events? 8 / 31

9 Data description Data Sample Tick by Tick Data of FTSE 40 Italy, traded at Milan stock exchange 20 high liquidity stocks 88 Days of Executions (March-June 2012) One minute time scale for log returns Jump detection methods are obviously very sensitive to outliers, data anomalies, etc. Outlier removal: Brownlees - Gallo (2006) algorithm Merging / Splitting and Volatility auctions are automatically subtracted Removal of intraday volatility pattern We need a statistically robust identification method for jumps 9 / 31

10 Jumps identification Most of the jump identification methods are essentially estimating the local volatility and then testing that the ratio between absolute return and volatility is above a given threshold This approach implicitly requires the definition of a time scale used to compute returns (in our case, one minute) We define Jump: r σ > θ (we choose θ = 4 as in Bouchaud et al., 2008) The volatility estimation is the most critical step and differentiates methods We used the realized bipower variation (Barndorff-Nielsen and Shephard (2003, 2004) ˆσ 2 bv,t = µ 2 1 r i r i+1 = µ 2 1 α i>0 (1 α) i 1 r t i r t i 1, 2 with µ 1 = and α = (gives 50% of weight to the π closest 22 observations). 10 / 31

11 Number of detected jumps ISIN jumps jumps up jumps down single jumps cojumps IT (47%) 55 (53%) 53 (51%) 50 (49%) IT (46%) 34 (54%) 38 (60%) 25 (40%) IT (50%) 61 (50%) 97 (80%) 24 (20%) IT (49%) 47 (51%) 56 (60%) 37 (40%) IT (53%) 60 (47%) 55 (43%) 72 (57%) IT (47%) 31 (53%) 44 (75%) 15 (25%) IT (41%) 105 (59%) 150 (84%) 28 (16%) IT (50%) 62 (50%) 76 (62%) 47 (38%) IT (43%) 107 (57%) 107 (57%) 81 (43%) IT (43%) 89 (57%) 95 (61%) 60 (39%) IT (40%) 42 (60%) 41 (59%) 29 (41%) IT (57%) 55 (43%) 79 (61%) 50 (39%) IT (53%) 45 (47%) 74 (78%) 21 (22%) IT (55%) 33 (45%) 46 (62%) 28 (38%) IT (49%) 53 (51%) 72 (70%) 31 (30%) IT (41%) 68 (59%) 85 (74%) 30 (26%) IT (51%) 49 (49%) 65 (65%) 35 (35%) IT (42%) 69 (58%) 57 (48%) 61 (52%) LU (46%) 32 (54%) 32 (54%) 27 (46%) NL (45%) 47 (55%) 51 (59%) 35 (41%) total average (47%) 57.2 (53%) 11 / 31

12 Multiple cojumps in the Italian stock market Time series of the number of stocks n co-jumping simultaneously Jul n = 1 2 n 3 4 n 8 9 n 20 Jun Day May Apr Mar Hour A large number of cojumps involving a sizable number of assets! 12 / 31

13 Comparing with independent Poisson processes Jul 3 Day Jun May Apr Mar Hour log 10 (counts) Number of cojumping stocks 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). There is a large number of jumps per stock: 1.2 jumps per stock per day At the daily scale, the number of jumps is Poisson distributed There is a large number of simultaneous jumps of groups of stocks, not explained by Poisson Can we identify evidence of time clustering of jumps on the same or on different assets? 13 / 31

14 A multi-scale statistical test based on multiple jumps and cross jumps detection A multiple jump (MJ) of a stock is the event of at least two jumps of the stock s price occurring inside a time window. A cross-jump (CJ) between two stocks is the event of at least one jump in the series of each stocks occurring inside a given time window. s i the number of jumps in window i of length w estimator for multiple jump probability in a window of length w over a sampling period of length N ˆp MJ w = N w i=1 1 si 2 N w, estimator for cross-jump probability of stock l and k in a window of length w between stock l and k w N ˆp w CJ i=1 1 s l = i 1 1 s i k 1 N. w For each window length w we estimate empirically these quantities and we compare with 99% and 95% confidence bands of the tested null model We correct for multiple hypothesis testing using Bonferroni correction 14 / 31

15 Testing multiple jumps for individual stocks under the Poisson null Empirical Poisson mean Poisson 99% c.l. Poisson 95% c.l. ˆp MJ w w (minute) Figure: MJ probability test under Poisson null for the Italian asset Assicurazioni Generali. 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. 15 / 31

16 Dynamic Intensity Models: Hawkes Processes Hawkes Processes (Hawkes, 1971) A univariate point process N t is called a Hawkes process if it is a linear self-exciting process, defined by the intensity I (t) = λ(t) + t 0 t ν(t u)dn(u), where λ is a deterministic function called the base intensity and ν is a positive decreasing weight function. The most common parametrization of ν is given by ν(t) = P j=1 α je β j t, for t > 0, where α j 0 are scale parameters, β j > 0 controls the strength of decay, and P N is the order of the process. For single exponential kernel and λ(t) constant, the expected number of jumps in a time interval of length T is [ t0 +T ] λ E dn t = 1 α/β T. 16 / 31

17 Hawkes processes Intensity Events Time 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 = (from Filimonov and Sornette, PRE 2012). Applied to earthquakes (Omori law), epidemiology, genomics, crime, finance, etc. The parameter n = ν(s)ds plays the role of a criticality parameter and 0 the process becomes non stationary at n > / 31

18 Testing multiple jumps for individual stocks under Null Hawkes ˆp MJ w Empirical Monte Carlo mean Monte Carlo 99% c.l. Monte Carlo 95% c.l w (minute) w (minute) Figure: MJ probability test under Hawkes null for the assets Assicurazioni Generali (left) and Intesa Sanpaolo (right). Univariate Hawkes processes are able to capture the dynamics and time clustering of jumps of real data 18 / 31

19 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 (t) + 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? 19 / 31

20 Failure of the bivariate Hawkes models Empirical Monte Carlo mean Monte Carlo 99% c.l. Monte Carlo 95% c.l. ˆp CJ w w (minute) Figure: CJ probability test under independent Hawkes null for the pair of Italian assets Generali - Intesa Sanpaolo. Multivariate Hawkes processes fail to describe both the single stock and the cross stock time clustering of jumps observed in real data When calibrated on real data, Hawkes processes strongly underestimate the simultaneous jumps of stocks. 20 / 31

21 1 Factor Model with 2 Stocks Main idea: A single factor model of jumps of a set of stocks There is one (unobserved) market factor point process describing the jumps When the factor jumps, stock k jumps with probability p k A stock can jump also because of an idiosyncratic point process In equations, I k (t) = v k (t)λ F (t) + λ k (t) (3) where v k (t) B(1, p k ) and λ F (t) and λ k (t) are the intensities of the factor point process and of the idiosyncratic process, respectively. Note that the number of parameters to estimate goes down from O(N 2 ) to O(N). 21 / 31

22 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 22 / 31

23 One Factor Poisson model (without idiosyncratic terms) Self cojumping probability of IT Cross cojumping probability of IT and IT Cojumps frequency o o o o observed MC mean MC 99% c. l. MC 95% c. l. Cojumps frequency o o o o observed MC mean MC 99% c. l. MC 95% c. l Windows length Windows length Figure: Multiple (left) and cross (right) jump probability test against a null model specifically designed for two stocks: single stock jumps are generated by the thinning of a systemic factor driven by a Poisson process. The number of Monte Carlo paths is The factor term explains the cross-cojumps, but the Poissonianity of the factor leads to underestimation of self-cojumps (as expected). 23 / 31

24 1 Factor Model + Idiosyncratic with 20 stocks Consider the whole set of 20 Stocks and define {t s i } for i = 1,..., n s, set of event times for the s-th stock {t F j } for j = 1,..., n F, set of event times when the realized number of cross-cojumps rejects the null hypothesis of cross independence {t s i } = {ts i } \ {t F j } {t s i } for i = 1,..., n s, subset of event times for the s-th stock compatible with the null of cross-independence We introduce a kind of recursive Expectation-Maximization algorithm to estimate The parameters describing the point processes of the factor and of the idiosyncratic terms The systemic (factor) jumps and the idiosyncratic jumps The main technical problem is to distinguish when a multiple jumps is due to a jump of the (unobservable) factor and when it is due to multiple idiosyncratic jumps. 24 / 31

25 Test multiple and cross jumps under a null with One Factor Hawkes + Idiosyncratic Hawkes 0.01 Empirical Monte Carlo mean Monte Carlo 99% c.l. Monte Carlo 95% c.l ˆp MJ w ˆp CJ w ˆp CJ w w (minute) 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. 25 / 31

26 Are financial markets becoming more and more unstable? There is a growing concern that financial markets have become more unstable, i.e. the number of price jumps have increased in recent years HFTs have been blamed to be responsible for the increased instability Our contribution: an historical investigation of the evolution of market instabilities at high frequency Data: The set of the 120 most liquid assets of the Russell 3000 index in the period Need for a careful definition of price jumps Jan 00 Jan 01 Jan 02 Jan 03 Jan 04 Jan 05 Jan 06 Jan 07 Jan 08 Jan 09 Jan 10 Jan 11 1 minute volatility Figure: Left. Number of shocks according to a Credit Suisse report. Right. Volatility of our Russell 3000 data. 26 / 31

27 Systemic cojumps jumps 10 asset cojumps asset cojumps 60 asset cojumps Figure: Evolution of single asset jumps (top left), 10-asset cojumps (top right), 30-asset cojumps (top right), and 60-asset cojumps (top right) Total number of jumps and the number of single asset jumps have actually declined in recent years The number of multiple asset co-jumps have increased significantly This effect is stronger for systemic cojumps, i.e. cojumps involving a large number of assets 27 / 31

28 How markets instability has changed: 2000 vs : Many jumps involving few stocks 2012: Few jumps involving many stocks 5 28 / 31

29 How markets instability has changed: 2000 vs 2012 The Hawkes one factor model reproduces well both the time clustering of jumps and the high synchronization of jumps across assets. Red=2000, Blue=2012 Red=2000, Blue=2012 number of occurrences pdf cojump factor multiplicity p Figure: Left. Histogram of the number of assets involved in a jump in 2000 and Right. Probability density function of the estimated probability p k with which an asset jumps with the Hawkes factor in 2000 and The distribution of multiplicity seems to be described by a power law function. However the tail exponent does not seem to be constant in time The factor component of the jump process has become significantly more relevant in recent years. 29 / 31

30 Systemic cojumps and news arrival Systemic cojumps: Exogenously triggered or endogenously generated? In order to test for the exogenous hypothesis, we consider the news feed mam.econoday.com focusing on macro announcements We compute the fraction of systemic cojumps of N assets which are preceeded by a news in the previous 1-5 minutes Probability of a news trigger cojump factor multiplicity Figure: Fraction of systemic cojumps which are preceeded by a macro news in the previous 1 (black), 2 (cyan), 3 (blue), 4 (green), or 5 (red) minutes as a function of the number of jumping assets. Only approximately 30 40% of the large systemic cojumps can be associated with macro news (for example FOMC announcements) Are the other systemic cojumps endogenously generated? 30 / 31

31 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 31 / 31