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Exogenous versus Endogenous Dynamics in the Price Discovery Process
Market microstructure: Confronting many viewpoints. Paris, France. December 8, 2014
Vladimir Filimonov Chair of Entrepreneurial Risks, ETH Zurich
with Didier Sornette, Spencer Wheatley (ETH Zurich); David Bicchetti, Nicolas Maystre (UNCTAD)
Two views on the price discovery mechanism
Efficient Markets (exogenous dynamics)
Prices are just reflecting news: the market fully and instantaneously absorbs the flow of information and faithfully reflects it in asset prices.
In particular, financial crashes are the signature of exogenous negative news of large impact.
News Prices News Prices
“Reflexivity” of markets(endogenous dynamics)
Markets are subjected to internal feedback loops (e.g. created by collective behavior such as herding or informational cascades).
Prices do influence the fundamentals and this newly-influenced set of fundamentals then proceed to change expectations, thus influencing prices.
2
“Anomalies”
Anomalies § January effect § Monday Effect § Size Effect § Price-Earnings Ratio effect § “Weather effects” § Speculative Bubbles § Herding of portfolio managers !
Puzzles § Excess volatility puzzle § Price moves on news releases
3
“Stylized facts” § Long memory in volatility § Heavy tails of returns § Intermittency and volatility clustering § Multifractal properties § Leverage effect § Gain/loss asymmetry § Volume/volatility correlation § Extreme drawdowns
Microscopic observations § Long memory in signs of orders § Market impact § “Latent” liquidity
Sources of endogeneity in financial markets
■ Imperfection and cognitive biases of trading agents; ■ Imitation and informational cascades leading to herding; ■ Hedging strategies (also cross-excite markets); ■ Replication of index with ETFs and pricing of structured products
(also contribute to cross-excitation between markets) ■ Model-induced feedback loops (e.g. BS in 1987, CDO in 2008); ■ Mark-to-market accounting rules; ■ Margin/leverage trading and margin-calls; ■ Methods of optimal portfolio execution and order splitting; ■ Speculation, based on technical analysis, including algorithmic
trading; ■ High frequency trading (HFT) as a subset of algorithmic trading; ■ Stop-loss orders and etc.
4
Algorithmic and High-Frequency Trading
5
Adoption of electronic trading by asset classes
0%10%20%30%40%50%60%70%80%90%
100%
2001
2002
2003
2004
2005
2006
2007
2008
2009
e201
0
e201
1
e201
2
EquitiesFuturesOptionsFX
Source: Aite Group (2010)
Fraction of HFT activity in equity trading volume
0%
10%
20%
30%
40%
50%
60%
70%
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
*
353739404139
29
951
4949515556
61
52
35
2621
USEurope
Source: TABB Group (2014)
■ How does adoption of algorithmic (including HFT) trading affect price discovery mechanisms?
■ Is it possible to quantify the interplay between exogeneity (external impact) and endogeneity (internal self-excitation) in price formation?
■ How efficient are financial markets?
6
Modeling feedback: Autoregressive modelsDiscrete-time processes § ARMA/ARIMA models family
§ GARCH models family
§ SEMF (Self-excited multifractal) process
Point processes § ACD (Autoregressive conditional duration) models family
§ ACI (Autoregressive conditional intensity) model
7
Xt +pX
i=1
aiXt�i = c+ ⇠t +qX
j=1
bj⇠t�i
rt = �t⇠t, �2t = ! +
qX
i=1
↵i⇠2t�i +
pX
j=1
�j�2t�i
rt = �t⇠t, �t = � exp
� 1
�
t�1X
i=1
riht�i�1
!, ht =
(h0 exp(��t),
h0t�'�1/2
⌧i = �i✏i, �i = ! +qX
j=1
↵i⌧i�j +pX
k=1
�j�i�k
�(t) = �0(t) exp��N(t)
�s(t), �N(t) =
pX
i=1
↵i
1�
Z TN(t)�i+1
TN(t)�i
�(t)dt
!+
qX
j=1
�j�N(t)�j
Modeling feedback: Self-excited Hawkes process
Applications of the Hawkes model: ■ High-frequency price dynamics ■ Order book construction ■ Critical events and estimation of VaR ■ Default times in a portfolio of companies
Self-excited Hawkes process is the point process whose intensity is conditional on its history:
0 100 200 300 400 500 600 700 800 900 10000
20
40
60
80
100
120
Time
Intensity
Exogenous activity
Endogenous feedback
Background intensity Self-excitation part
■ Triggered seismicity (earthquakes) ■ Sequence of genes in DNA ■ Epileptic seizures of brain ■ Crime and violence propagation 8
�(t|Ht�) = µ+ nX
ti<t
�(t� ti)
Branching structure of Hawkes process
Crucial parameter of the branching process is the “branching ratio” (n), which is defined as an average number of “daughters” per one “mother”
For n < 1 system is subcritical (stationary evolution)For n = 1 system is critical (tipping point)For n > 1 system is supercritical (with prob.>0 will explode to infinity)
In subcritical regime, the branching ratio (n) is equal to the fraction of endogenously generated events among the whole population.
Time0 0 01 1 1 1 1 1 1 12 2 2 22 22 2 23 3 3 34
n = 0.88
9
Calibration of the modelParametric estimation § Maximum Likelihood
§ Ozaki T. (1979). Maximum likelihood estimation of Hawkes' self-exciting point processes. Annals of the Institute of Statistical Mathematics, 31(1), 145–155.
§ Method of moments § da Fonseca J., Zaatour R. (2014). Hawkes Process: Fast Calibration, Application to Trade Clustering and Diffusive
Limit. Journal of Futures Markets, 34(6), 548–579.
Non-parametric estimation § Expectation-Maximization (EM) algorithm
§ Marsan D., Lengline O. (2008). Extending Earthquakes' Reach Through Cascading. Science, 319(5866), 1076–1079.
§ Spectral method based on second order statistics § Bacry E., Dayri K., Muzy J.-F. (2012). Non-parametric kernel estimation for symmetric Hawkes processes. Application
to high frequency financial data. EPJB, 85(5), 157.
§ Integral equation method § Bacry E., Muzy J.-F. (2014). Second order statistics characterization of Hawkes processes and non-parametric
estimation. arXiv:1401.0903.
Goodness-of-fit § Residual analysis
§ Ogata Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9–27.
10
Calibration issues
§ General § Choice of the kernel:
§ Robustness of estimation § Specific shape of the HF-part of the kernel
§ Multiple minima of optimization problem § Renormalization of the estimates in case
of time-varying background intensity
§ Specific to HF data § Quality of timestamps § RTH and overnight trading
§ Non-stationarity § Scheduled macroeconomic announcements § Intraday trends § Day-to-day variations of trading activity
11• Filimonov V., Sornette D. (2014) Swiss Finance Institute Research Paper No. 13-60. arXiv:1308.6756
0
250
500
2002
0
200
400
0
200
400
2004
0
150
300
0
200
400
2006
0
200
400
0
1000
2000
2008
0
750
1500
0 500 1000 1500 20000
750
1500
2009
0 200 400 600 800 1000 1200 14000
500
1000
0
10000
20000
2010
0
1000
2000
0
7500
15000
2011
0
700
1400
0 20 40 60 80 1000
7500
15000
Time between FAST/FIX packages, msec
2012
0 5 10 15 20 250
300
600
Processing time, msec
0
250
500
2002
0
200
400
0
200
400
2004
0
150
300
0
200
400
2006
0
200
400
0
1000
2000
2008
0
750
1500
0 500 1000 1500 20000
750
1500
2009
0 200 400 600 800 1000 1200 14000
500
1000
0
10000
20000
2010
0
1000
2000
0
7500
15000
2011
0
700
1400
0 20 40 60 80 1000
7500
15000
Time between FAST/FIX packages, msec
2012
0 5 10 15 20 250
300
600
Processing time, msec
0
0.5
1
Sep
04
Raw data
0
0.4
0.8After "detrending"
0
0.4
0.8
Sep
170
0.4
0.8
0
3
6Se
p 18
0
2.5
5
0
0.4
0.8
Sep
19
0
0.4
0.8
0
0.75
1.5
Oct
11
0
0.7
1.4
10:00 12:00 14:00 16:000
0.4
0.8
Time (EST)
Oct
29
10:00 12:00 14:00 16:000
0.25
0.5
Time (EST)
2009
Proxy: mid-quote price dynamics
Time
Price Last transaction priceBest bid priceBest ask priceMid-quote priceTransactionMid-quote price change
Limit orders to sell
Limit orders to buy
Bid
Ask
Sell market order
Buy market order
Price
12
Methodology
61.8
62
62.2
62.4
62.6
Pric
e
March 23, 2007
09:00 10:00 11:00 12:00 13:00 14:000
50
100
150
200
250
Time
Inte
nsity
10:30 10:32 10:34 10:36 10:38 10:4062.2
62.25
62.3
Price
n = 0.43
■ We split the entire interval of the analysis into 10 minutes intervals, rolling them with a step of 1 minute within the RTH
■ In each of these windows we have calibrated the Hawkes model with the short-term exponential kernel on the timestamps of mid-quote price changes
■ Each calibration resulted in a single estimation of the branching ration (n)
■ We have performed residual analysis and rejected “bad” fits (using KS-test)
■ Collecting all estimates for each month (~6000-7000 estimates) we have averaged them to construct the “endogeneity index” for the given month
13
Mechanisms of self-reflexivitymilliseconds seconds minutes hours days weeks months years
High-frequency trading
Stop-loss orders
Algorithmic trading
Optimal execution
Margin calls
Imitation and long-term herding
Hedging in pricing models
14
Short-term endogeneity in E-mini S&P 500
0
50M
100M
Volu
me
0
1M
2M
3M
4M
Num
ber o
f eve
nts
Monthly volumeNumber of events per month
0
0.05
0.1
Vola
tility
500
1000
1500
Pric
e
Daily volatilityDaily closing price
Back
grou
nd a
ctiv
ity
0
0.5
1
Year
Bran
chin
g ra
tio
1998 2000 2002 2004 2006 2008 2010 20120.3
0.4
0.5
0.6
0.7
0.8
Trading activity proxied by volume and
number of mid-price changes
Dynamics of price and volatility
Rate of exogenous events (triggered by idiosyncratic
“news”)
Branching ratio that quantifies endogeneity of the system
(fraction of endogenous events in the system)
• Filimonov V., Sornette D. (2012) Physical Review E 85(5), 056108 • Filimonov V., Bicchetti D., Maystre N., Sornette D. (2014) J. of Int. Money and Finance, 42, 174-192 15
Short-term endogeneity (E-Mini S&P 500)
16
0
50M
100M
150M
Volu
me
0
2M
4M
6M
Num
ber o
f eve
nts
a Two months’ volumeNumber of events per 2 months
0
0.05
0.1
Vola
tility
500
1000
1500
Pric
e
b Daily volatilityDaily closing price
Back
grou
nd a
ctiv
ity c
0
0.2
0.4
0.6
0.8
Year
Bran
chin
g ra
tio
d
1998 2000 2002 2004 2006 2008 2010 20120.3
0.4
0.5
0.6
0.7
0.8
Figure 4: Dynamics of (a) volume and activity measured in number of mid-quote pricechanges, (b) daily closing price and daily volatility, (c) estimated background intensity (µ̂,see text) and (d) branching ratio (n̂, see text) for the E-mini S&P 500 futures over theperiod 1998–2012. Each point in panels (c) and (d) represents averaged estimates overtwo months interval prior to the point in time windows of 10 minutes for ∆ = 100 msec(squares), ∆ = 200 msec (crosses with black line), ∆ = 300 msec (circles) and ∆ = 1 sec(dots with blue line). The shaded area gives the 25%–75% quantile range obtained withthe same two months estimates for ∆ = 200 msec. In the analysis we have considered onlyestimates performed within hours of active trading (see table 1).
56
• Filimonov V., Bicchetti D., Maystre N., Sornette D. (2014) J. of Int. Money and Finance, 42, 174-192
Short-term endogeneity (Crude Oil: Brent and WTI)
0
0.05
0.1
Vola
tility
0
50
100
150
Pric
e
Daily volatilityDaily closing price
0
1M
2M
3M
4M
5M
Volu
me
Year
Bran
chin
g ra
tio
2005 2006 2007 2008 2009 2010 2011 2012
0.4
0.5
0.6
0.7
0.8
0
0.05
0.1
Vola
tility
0
50
100
150
Pric
e
Daily volatilityDaily closing price
0
2M
4M
6M
8M
10M
Volu
me
Year
Bran
chin
g ra
tio
2005 2006 2007 2008 2009 2010 2011 2012
0.5
0.6
0.7
0.8
• Filimonov V., Bicchetti D., Maystre N., Sornette D. (2014) J. of Int. Money and Finance, 42, 174-192
Brent Crude (ICE Europe) WTI (NYMEX)
17
4
1998 2000 2002 2004 2006 2008 2010 20120
0.5
1
1.5
2
Bra
nchi
ng ra
tio, n
n = 1
1998 2000 2002 2004 2006 2008 2010 20120.001
0.01
0.1
1
10
Hig
h Fr
eque
ncy
Cut
-off
~ τ
0
Moore’s law
1998 2000 2002 2004 2006 2008 2010 20120
0.1
0.2
0.3
0.4Ex
pone
nt - ε
ε = 0.15
1998 2000 2002 2004 2006 2008 2010 20120
0.050.1
0.150.2
0.250.3
0.350.4
0.450.5
Bas
e In
tens
ity µ Base Intensity
Event rate
FIG. 1: Results for the MLE fitting of the Hawkes process with power-law kernel to mid-point changes in the E-mini S&PFutures contract. The fit is performed over two-month periods between Jan. 1998 and Dec. 2011. Top-Left: The branchingratio n appears to change little and fluctuates around the critical value n = 1 for the 14-year period studied. Top-Right: Thehigh frequency cut-off τ0, on the other hand, decreases exponentially over time as a result of the increasing frequency at whichevents are correlated, this trend is not too far out of line with Moore’s law, the oft-cited observation that the processing speedof computers doubles every 18 months. Bottom-Left: The base intensity µ (in events per second) that has remained roughlyconstant over time, apart from spikes during crisis periods. Bottom-Right: The exponent ϵ entering the power-law kernel φ(τ ),that has also remained roughly constant around 0.15 – see below.
In fitting the model we only consider market events oc-curring during Regular Trading Hours (09:30 to 16:15).Furthermore, we take account of the ‘U-shape’ intra-dayactivity profile. Without appropriate detrending, thefitting procedure would interpret the slow variation inevent rate during the day as arising from the shape ofthe Hawkes kernel. To this end, we consider a detrendedHawkes process model
λ(t|θ) =1
w(t)
!
µ+
" t
0w(s)φ(t − s|θ)dN(s)
#
(15)
where w(t) is a periodic function defined over the trad-ing day that dictates the weight to be appropriated toevents occurring at different hours. The aim is to giveless weight to events occurring during the morning andlate afternoon when high activity is to be expected. Theweights are chosen to be the reciprocal of the empiri-cal average daily event rate at that moment of the dayw(t) = 1
R(t) normalised such that E[R(t)] = E[w(t)] = 1.The model parameters as a function of time are cal-
culated between 1998 and 2011 on non overlapping two-month periods using an estimate of the function R(t)made for each year. All regular trading hours in each
two-month period are concatenated to form a single con-tinuous point-process for which the log-likelihood of Eq.(14) and Eq. (15) is maximised with respect to the fourparameters θ = {µ, n, ϵ, τ0}. The results are shown inFig. 1.
Our key observation is that the branching ratio ap-pears to change little over the course of the fourteen yearsstudied but fluctuates about the critical value nc = 1 cor-responding to the onset of stationarity. This means, thatwithin the Hawkes framework, the majority of events ob-served in the market can be attributed to the long-rangeinfluence kernel, both now and in 1998. However, wesee that the short lag cut-off of the kernel has decreasedexponentially over time. This can be attributed to thesteady emergence of higher frequency trading despite lit-tle change in the event rate. It may surprise the readerthat the event rate has not increased significantly in the14-year period (the event rate, for example in 2009 iscomparable with that in 1999) despite the growth of theelectronic futures market, but understand that the eventsstudied are mid-point changes, not trades or order-bookevents and as such it is a better proxy for volatility thantransaction quantity or volume.
Long-term endogeneity
18• Hardiman S., Bercot N., Bouchaud J.-P. (2013). EPJB, 86(10), 442.
�(t|Ht�) = µ+ nX
ti<t
�(t� ti), �(t� ti) =
8>>><
>>>:
✓c✓
(t+ c)✓+1
1
Z
"X 1
⇠✓+1i
e�t/⇠i � Se�t/⇠�1
#, ⇠i = ⌧0m
i
Susceptibility at a coarse-grained scale
19• Filimonov V., Sornette, D. (2014). Swiss Finance Institute Research Paper No 14-48. arXiv:1407.5037
Dynamics of number of extreme intraday drawdowns (drawdowns with duration >1min and the size at the power-law tail of the distribution) per day for futures on major world's indices
1998 2000 2002 2004 2006 2008 2010 20121 msec
10 msec
100 msec
1 sec
Year
Brent CrudeWTIE−mini S&P 500
Acceleration of market makers
• Filimonov V., Sornette D. (2014) La vie èconomique, 5, 20-22 20
Med
ian
mar
ket m
aker
s’ re
actio
n tim
e
Distribution of mid-price changes inter-event times
21
Tâtonnement process and price discovery
22
“Acceleration” versus “clustering”
23
4
1998 2000 2002 2004 2006 2008 2010 20120
0.5
1
1.5
2
Bra
nchi
ng ra
tio, n
n = 1
1998 2000 2002 2004 2006 2008 2010 20120.001
0.01
0.1
1
10
Hig
h Fr
eque
ncy
Cut
-off
~ τ
0
Moore’s law
1998 2000 2002 2004 2006 2008 2010 20120
0.1
0.2
0.3
0.4
Expo
nent
- ε
ε = 0.15
1998 2000 2002 2004 2006 2008 2010 20120
0.050.1
0.150.2
0.250.3
0.350.4
0.450.5
Bas
e In
tens
ity µ Base Intensity
Event rate
FIG. 1: Results for the MLE fitting of the Hawkes process with power-law kernel to mid-point changes in the E-mini S&PFutures contract. The fit is performed over two-month periods between Jan. 1998 and Dec. 2011. Top-Left: The branchingratio n appears to change little and fluctuates around the critical value n = 1 for the 14-year period studied. Top-Right: Thehigh frequency cut-off τ0, on the other hand, decreases exponentially over time as a result of the increasing frequency at whichevents are correlated, this trend is not too far out of line with Moore’s law, the oft-cited observation that the processing speedof computers doubles every 18 months. Bottom-Left: The base intensity µ (in events per second) that has remained roughlyconstant over time, apart from spikes during crisis periods. Bottom-Right: The exponent ϵ entering the power-law kernel φ(τ ),that has also remained roughly constant around 0.15 – see below.
In fitting the model we only consider market events oc-curring during Regular Trading Hours (09:30 to 16:15).Furthermore, we take account of the ‘U-shape’ intra-dayactivity profile. Without appropriate detrending, thefitting procedure would interpret the slow variation inevent rate during the day as arising from the shape ofthe Hawkes kernel. To this end, we consider a detrendedHawkes process model
λ(t|θ) =1
w(t)
!
µ+
" t
0w(s)φ(t − s|θ)dN(s)
#
(15)
where w(t) is a periodic function defined over the trad-ing day that dictates the weight to be appropriated toevents occurring at different hours. The aim is to giveless weight to events occurring during the morning andlate afternoon when high activity is to be expected. Theweights are chosen to be the reciprocal of the empiri-cal average daily event rate at that moment of the dayw(t) = 1
R(t) normalised such that E[R(t)] = E[w(t)] = 1.The model parameters as a function of time are cal-
culated between 1998 and 2011 on non overlapping two-month periods using an estimate of the function R(t)made for each year. All regular trading hours in each
two-month period are concatenated to form a single con-tinuous point-process for which the log-likelihood of Eq.(14) and Eq. (15) is maximised with respect to the fourparameters θ = {µ, n, ϵ, τ0}. The results are shown inFig. 1.
Our key observation is that the branching ratio ap-pears to change little over the course of the fourteen yearsstudied but fluctuates about the critical value nc = 1 cor-responding to the onset of stationarity. This means, thatwithin the Hawkes framework, the majority of events ob-served in the market can be attributed to the long-rangeinfluence kernel, both now and in 1998. However, wesee that the short lag cut-off of the kernel has decreasedexponentially over time. This can be attributed to thesteady emergence of higher frequency trading despite lit-tle change in the event rate. It may surprise the readerthat the event rate has not increased significantly in the14-year period (the event rate, for example in 2009 iscomparable with that in 1999) despite the growth of theelectronic futures market, but understand that the eventsstudied are mid-point changes, not trades or order-bookevents and as such it is a better proxy for volatility thantransaction quantity or volume.
�(t) =✓c✓
t1+✓�(t� c)
�(t) =✓c✓
(t+ c)1+✓�(t)
�(t) =1
Z
"M�1X
i=0
1
⇠✓+1i
e�t/⇠i � Se�t/⇠�1
#, ⇠i = ⌧0m
i
High-frequency part of long-term endogeneity
24
4
1998 2000 2002 2004 2006 2008 2010 20120
0.5
1
1.5
2
Bra
nchi
ng ra
tio, n
n = 1
1998 2000 2002 2004 2006 2008 2010 20120.001
0.01
0.1
1
10
Hig
h Fr
eque
ncy
Cut
-off
~ τ
0
Moore’s law
1998 2000 2002 2004 2006 2008 2010 20120
0.1
0.2
0.3
0.4
Expo
nent
- ε
ε = 0.15
1998 2000 2002 2004 2006 2008 2010 20120
0.050.1
0.150.2
0.250.3
0.350.4
0.450.5
Bas
e In
tens
ity µ Base Intensity
Event rate
FIG. 1: Results for the MLE fitting of the Hawkes process with power-law kernel to mid-point changes in the E-mini S&PFutures contract. The fit is performed over two-month periods between Jan. 1998 and Dec. 2011. Top-Left: The branchingratio n appears to change little and fluctuates around the critical value n = 1 for the 14-year period studied. Top-Right: Thehigh frequency cut-off τ0, on the other hand, decreases exponentially over time as a result of the increasing frequency at whichevents are correlated, this trend is not too far out of line with Moore’s law, the oft-cited observation that the processing speedof computers doubles every 18 months. Bottom-Left: The base intensity µ (in events per second) that has remained roughlyconstant over time, apart from spikes during crisis periods. Bottom-Right: The exponent ϵ entering the power-law kernel φ(τ ),that has also remained roughly constant around 0.15 – see below.
In fitting the model we only consider market events oc-curring during Regular Trading Hours (09:30 to 16:15).Furthermore, we take account of the ‘U-shape’ intra-dayactivity profile. Without appropriate detrending, thefitting procedure would interpret the slow variation inevent rate during the day as arising from the shape ofthe Hawkes kernel. To this end, we consider a detrendedHawkes process model
λ(t|θ) =1
w(t)
!
µ+
" t
0w(s)φ(t − s|θ)dN(s)
#
(15)
where w(t) is a periodic function defined over the trad-ing day that dictates the weight to be appropriated toevents occurring at different hours. The aim is to giveless weight to events occurring during the morning andlate afternoon when high activity is to be expected. Theweights are chosen to be the reciprocal of the empiri-cal average daily event rate at that moment of the dayw(t) = 1
R(t) normalised such that E[R(t)] = E[w(t)] = 1.The model parameters as a function of time are cal-
culated between 1998 and 2011 on non overlapping two-month periods using an estimate of the function R(t)made for each year. All regular trading hours in each
two-month period are concatenated to form a single con-tinuous point-process for which the log-likelihood of Eq.(14) and Eq. (15) is maximised with respect to the fourparameters θ = {µ, n, ϵ, τ0}. The results are shown inFig. 1.
Our key observation is that the branching ratio ap-pears to change little over the course of the fourteen yearsstudied but fluctuates about the critical value nc = 1 cor-responding to the onset of stationarity. This means, thatwithin the Hawkes framework, the majority of events ob-served in the market can be attributed to the long-rangeinfluence kernel, both now and in 1998. However, wesee that the short lag cut-off of the kernel has decreasedexponentially over time. This can be attributed to thesteady emergence of higher frequency trading despite lit-tle change in the event rate. It may surprise the readerthat the event rate has not increased significantly in the14-year period (the event rate, for example in 2009 iscomparable with that in 1999) despite the growth of theelectronic futures market, but understand that the eventsstudied are mid-point changes, not trades or order-bookevents and as such it is a better proxy for volatility thantransaction quantity or volume.
4
1998 2000 2002 2004 2006 2008 2010 20120
0.5
1
1.5
2
Bra
nchi
ng ra
tio, n
n = 1
1998 2000 2002 2004 2006 2008 2010 20120.001
0.01
0.1
1
10
Hig
h Fr
eque
ncy
Cut
-off
~ τ
0
Moore’s law
1998 2000 2002 2004 2006 2008 2010 20120
0.1
0.2
0.3
0.4
Expo
nent
- ε
ε = 0.15
1998 2000 2002 2004 2006 2008 2010 20120
0.050.1
0.150.2
0.250.3
0.350.4
0.450.5
Bas
e In
tens
ity µ Base Intensity
Event rate
FIG. 1: Results for the MLE fitting of the Hawkes process with power-law kernel to mid-point changes in the E-mini S&PFutures contract. The fit is performed over two-month periods between Jan. 1998 and Dec. 2011. Top-Left: The branchingratio n appears to change little and fluctuates around the critical value n = 1 for the 14-year period studied. Top-Right: Thehigh frequency cut-off τ0, on the other hand, decreases exponentially over time as a result of the increasing frequency at whichevents are correlated, this trend is not too far out of line with Moore’s law, the oft-cited observation that the processing speedof computers doubles every 18 months. Bottom-Left: The base intensity µ (in events per second) that has remained roughlyconstant over time, apart from spikes during crisis periods. Bottom-Right: The exponent ϵ entering the power-law kernel φ(τ ),that has also remained roughly constant around 0.15 – see below.
In fitting the model we only consider market events oc-curring during Regular Trading Hours (09:30 to 16:15).Furthermore, we take account of the ‘U-shape’ intra-dayactivity profile. Without appropriate detrending, thefitting procedure would interpret the slow variation inevent rate during the day as arising from the shape ofthe Hawkes kernel. To this end, we consider a detrendedHawkes process model
λ(t|θ) =1
w(t)
!
µ+
" t
0w(s)φ(t − s|θ)dN(s)
#
(15)
where w(t) is a periodic function defined over the trad-ing day that dictates the weight to be appropriated toevents occurring at different hours. The aim is to giveless weight to events occurring during the morning andlate afternoon when high activity is to be expected. Theweights are chosen to be the reciprocal of the empiri-cal average daily event rate at that moment of the dayw(t) = 1
R(t) normalised such that E[R(t)] = E[w(t)] = 1.The model parameters as a function of time are cal-
culated between 1998 and 2011 on non overlapping two-month periods using an estimate of the function R(t)made for each year. All regular trading hours in each
two-month period are concatenated to form a single con-tinuous point-process for which the log-likelihood of Eq.(14) and Eq. (15) is maximised with respect to the fourparameters θ = {µ, n, ϵ, τ0}. The results are shown inFig. 1.
Our key observation is that the branching ratio ap-pears to change little over the course of the fourteen yearsstudied but fluctuates about the critical value nc = 1 cor-responding to the onset of stationarity. This means, thatwithin the Hawkes framework, the majority of events ob-served in the market can be attributed to the long-rangeinfluence kernel, both now and in 1998. However, wesee that the short lag cut-off of the kernel has decreasedexponentially over time. This can be attributed to thesteady emergence of higher frequency trading despite lit-tle change in the event rate. It may surprise the readerthat the event rate has not increased significantly in the14-year period (the event rate, for example in 2009 iscomparable with that in 1999) despite the growth of theelectronic futures market, but understand that the eventsstudied are mid-point changes, not trades or order-bookevents and as such it is a better proxy for volatility thantransaction quantity or volume.
Dynamics of high-frequency part of a response (memory) function for critical Hawkes model with
power-law kernel and decreasing cut-off time
“Criticality” versus “non-stationarity”
25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.5
0.6
0.7
0.8
0.9
1
1.1
µ2
Bran
chin
g ra
tio, n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5
0.6
0.7
0.8
0.9
1
1.1
n2
Bran
chin
g ra
tio, n
• Filimonov V., Sornette D. (2014) Swiss Finance Institute Research Paper No. 13-60. arXiv:1308.6756
Estimated branching ratio (n) in case of regime switch in background intensity
(2 independent samples with µ1=1 and µ2, n=0.5)
Estimated branching ratio (n) in case of regime switch in branching ratio intensity
(2 independent samples with n1=0.5 and n2)
“Criticality” versus “non-stationarity” (cont.)
26• Filimonov V., Sornette D. (2014) Swiss Finance Institute Research Paper No. 13-60. arXiv:1308.6756
0
1000
2000
3000
4000
5000
6000
7000
8000
2002 0
5000
10000
15000
20000
2007 0
10000
20000
30000
40000
50000
2011
0
1000
2000
3000
4000
5000
6000
7000
8000
2002 0
5000
10000
15000
20000
2007 0
10000
20000
30000
40000
50000
2011
Daily numbers of mid-quote price changes counted over RTH for the Front Month
Contract of the E-mini S&P 500 Futures (time period of February 1 to April 1)
Branching ratio estimated on synthetic time series concatenated from independent Poisson processes with daily
intensities taken from the real data (red curve). The black curve corresponds to the case where days with
trading durations smaller than 5 hours are ignored.
Year1998 2000 2002 2004 2006 2008 2010 2012
Estim
ated
bra
nchi
ng ra
tio, n
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Beyond standard Hawkes model: Kernel
27
10^−
710^−
510^−
310^−
110^1
10^−5 10^−2 10^1 10^4
f(τ)
τ
Estimation of the kernel shape for exponential kernel (black), approximated power law with a cut-off (grey) and non-parametric estimation using EM-algorithm (blue).
Solid lines present median for sample sizes of 1000 (~5-10 min), 2000 (~10-20 min) and 4000 (~20-60 min) mid-quote price changes. Dashed lines represent quartiles.
• Wheatley S., Filimonov V., Sornette D. (2014) Working paper
Beyond standard Hawkes model: Immigration
28• Wheatley S., Filimonov V., Sornette D. (2014) SFI Research Paper No 14-52. arXiv:1407.7118
(0) (0) (0)(1) (2) (1)
(0) (1) (1) (0) (1) (2)
(I)
(II)
Figure 1: Illustration of the conditional intensity functions for (I) the Hawkes process (1)and (II) the Hawkes process with over-dispersed renewal process immigration (3). Thedotted line represents the intensity of immigration, and the solid line corresponds to thetotal intensity. The arrows indicate parenthood and the numbers in parentheses indicatethe generation of offspring.
is the offspring density, a pdf (probability density function) only giving mass to positivesupport. The parameter η is the branching ratio, a non-negative constant determiningthe strength of self-excitation.The Hawkes process can also be considered as a branching process: µ is the immigra-tion intensity, and ηh(t − tN(t)) is the offspring intensity – an inhomogeneous Poissonprocess triggered by each observed point. The branching ratio η is the expected numberof offspring of each point, and the offspring density provides the law for the parent-child inter-event times. Due to the autoregressive nature of the process, it becomesnon-stationary for η > 1 (the case η = 1 is borderline stationary with non-standardscaling properties [Saichev and Sornette, 2014]). The branching process is constructed
as follows: the Poissonian immigration process {T (0)i } introduces immigrant points into
the zeroeth generation; each immigrant generates a subsequent offspring process, whichintroduces first-generation offspring events {T (1)
i }; each offspring may introduce second-
generation offspring {T (2)i } in the same way; and so on over many generations. The
union of the immigrants and offspring of multiple generations defines the Hawkes process({T (0)
i }∪ {T (1)i }∪ . . . ). Further, by summing the mutually independent immigration and
offspring intensities, one recovers the conditional intensity (2). A realization of this con-ditional intensity is in the top panel of Fig. 1.
The standard definition of the Hawkes process (1) assumes constant or time-varyingbut deterministic immigration intensity µ(t), or in other words a Poissonian immigration
process {T (0)i }. Here we extend the definition by introducing the Hawkes Process with
Renewal immigration. For this, we allow the immigration process {T (0)i } to be a general
Renewal process, where waiting times Wi > 0 are i.i.d from some pdf g(w) that is notnecessary exponential. The conditional intensity of this process is then given by:
λ(t|Ht−, I[N(t)]) = µ(t− tI[N(t)]) + Φ(t|Ht−), (3)
4
f(x) = k
x
k�1
�
kexp
�⇣x
�
⌘k�
Hawkes process with renewal immigration (Weibull distribution of inter-event times for background process): !!!Straightforward MLE is not possible.EM-algorithm: § E-step: !
§ M-step:
�(t|Ht�, I(t)) = µ(t� tI(t)) + nX
ti<t
�(t� ti)
✓[m+1]= argmax
✓Q(✓|Ht�,✓
[m])
Q(✓|Ht�,✓[m]) = EZ|Ht�,✓[m] [log L(✓;Ht�,Z)]
References§ Filimonov V., Sornette D. (2011). Self-excited multifractal dynamics. EPL, 94(4), 46003.
http://arxiv.org/abs/1008.1430
§ Filimonov V., Sornette D. (2012) Quantifying reflexivity in financial markets: Toward a prediction of flash crashes. Physical Review E, 85(5), 056108. http://ssrn.com/abstract=1998832
§ Filimonov V., Bicchetti D., Maystre N., Sornette D. (2014) Quantification of the High Level of Endogeneity and of Structural Regime Shifts in Commodity Markets. J. of Int. Money and Finance, 42, 174-192. http://ssrn.com/abstract=2237392
§ Filimonov, V., & Sornette, D. (2014). Mythes, réalités et objectif général des transactions à haute fréquence (HFT: myths, reality and the bigger picture). La Vie Èconomique, 5, 20–22
§ Filimonov V., Sornette D. (2013) Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data. Swiss Finance Institute Research Paper No. 13-60. http://ssrn.com/abstract=2371284
§ Wheatley S., Filimonov V., Sornette D. (2014) Estimation of the Hawkes Process with Renewal Process Immigration using an EM Algorithm. Swiss Finance Institute Research Paper No 14-52. http://ssrn.com/abstract=2477432
§ Filimonov V., Sornette D. (2014) Power law scaling and “Dragon-Kings” in distributions of intraday financial drawdowns. Swiss Finance Institute Research Paper No. 14-48. http://ssrn.com/abstract=2468195
§ Filimonov V., Wheatley S., Sornette D. (2015) Effective measure of endogeneity for the Autoregressive Conditional Duration point processes via mapping to the self-excited Hawkes process. Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 23-37.http://arxiv.org/abs/1306.2245
All papers and preprints are available online at http://www.er.ethz.ch/people/vfilimon/ 29
