HEC Lausanne

High-frequency data modelling using Hawkes processes

Valerie Chavez-Demoulin1

joint work J.A McGill

1Faculty of Business and Economics,University of Lausanne, Switzerland

Boulder, April 2016

Boulder, April 2016 1 / 25

HEC LausanneHigh frequency financial data

The availability of high frequency data on transactions, quotes andorder flow has revolutionized data processing and has led to statisticalmodeling challenges

At the same time, the frequency of submission of orders has increasedand the time to execution of market orders has dropped from morethan 25 milliseconds in 2000 to less than a millisecond in 2014.

Average number Number of priceof orders in 10s changes (June 26, 2008)

Citigroup 4469 12499General Electric 2356 7862General Motors 1275 9016

Table: Taken from R.Cont, 2011, http://ssrn.com/abstract=1748022

Boulder, April 2016 2 / 25

HEC LausanneTrade database structure

SYMBOL DATE TIME PRICE SIZE G127 CORR CONDNVDA 20060901 9:30:04 28.67 264600 0 0 -NVDA 20060901 9:30:04 28.67 264509 0 2 -NVDA 20060901 9:30:05 28.65 100 0 0 -NVDA 20060901 9:30:05 28.65 200 0 0 -

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36.5

37.0

37.5

38.0

38.5

Pric

e

09:30:03 10:15:19 11:35:30 13:54:00 15:02:55 15:45:54

Presence of erroneous trades: moving average filteringIrregularly spaced in time: linear interpolation

Boulder, April 2016 3 / 25

HEC LausanneNVDA negative log-returns from 2006 to 2008(15-minutes linearly interpolated)

−0.

10.

00.

10.

20.

3

Time(15min intervals)

Neg

ativ

e lo

g re

turn

s

2006−01−03 2007−01−03 2008−01−02 2008−12−31

NVDA, July to December 2008 negative returns exceedances :

0.02

0.04

0.06

0.08

Time(15min intervals)

Neg

ativ

e lo

g re

turn

s

2008−07−24 2008−09−03 2008−10−13 2008−11−20 2008−12−31

Boulder, April 2016 4 / 25

HEC LausanneExtreme Value Theory:Classical Peaks-Over-Threshold (iid case)

0 20 40 60 80 100

2040

6080

Time

x

The number of exceedances over u has a Poisson distribution withmean λ

Conditional on n exceedances, their sizes Wj = Zj − u are a randomsample of size n from the generalized Pareto distribution (GPD)

Gξ,σ(w) =

{1− (1 + ξw/σ)

−1/ξ+ , ξ 6= 0,

1− exp(−w/σ), ξ = 0.

Boulder, April 2016 5 / 25

HEC LausanneNVDA log-returns from 2006 to 2008 (15-minutesinterpolated) time differences between exceedances overhigh threshold

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 20 40 60 80 100 120

010

020

030

0

Exponential

Diff

eren

ce in

tim

e

does not seem exponentially distributedhigher than expected frequencies for short distance between extremes

Boulder, April 2016 6 / 25

HEC LausanneHigh-frequency financial data features

Returns not iid

Absolute returns highly correlated

Volatility appears to change randomly with time

Returns are leptokurtic or heavy–tailed

Extremes appear in clusters

Boulder, April 2016 7 / 25

HEC LausanneDeclustering methods exist:An example for daily returns

0.04

0.08

0.12

Original dataR

etur

ns in

%

1975 1980 1985 1990 1995

0.04

0.08

0.12

Declustered data

Ret

urns

in %

1975 1980 1985 1990 1995Boulder, April 2016 8 / 25

HEC LausanneIdea

the classical Peaks-Over-Thresholds (POT) model of EVT assumes anhomogeneous Poisson process

the inter-event times would be independent exponential randomvariables

the Figures above contradict the classical model

Aim: Treats the negative log-returns above a high threshold u as arealization of a marked point process:

Using a first model that combines a self-exciting process for thethreshold exceedance times with Generalized Pareto model for thethreshold excess sizes (mark sizes)

Using a second model that considers a self-exciting POT model

Boulder, April 2016 9 / 25

HEC LausanneOverall loglikelihood (POT method)

Since number and mark size for the threshold exceedances are assumedindependent

l(λ, σ, ξ) = log

P(Nu = n)n∏

j=1

gξ,σ(wj)

.

= n log λ− λ− n log σ − (1 + 1/ξ)n∑

j=1

log(1 + ξwj/σ)+

so inference can be performed separately for the frequency of exceedancesand their sizes.

Boulder, April 2016 10 / 25

HEC LausanneMarked Point Processes (1)

consider an event process (Ti ,Wi ), i ∈ Zobserved over the period (0; t0] gives data (t1,w1), . . . , (tn,wn)

let Ht , entire history of process up to time t

the joint density of the data seen over (0, t0] can be written

n∏i=1

fTi ,Wi |Hti−1(ti ,wi |Hti−1)P(Tn+1 > t0|Htn)

Boulder, April 2016 11 / 25

HEC LausanneMarked Point Processes (2)

Assuming independence of times Ti and marks Wi the conditional densityfunction is

fTi ,Wi |Hti−1(ti ,wi |Hti−1) = fTi |Hti−1

(ti |Hti−1) · fWi |Hti−1(wi |Hti−1)

leading to the log-likelihood

` =

{n∑

i=1

log fTi |Hi−1(ti ) + logP(Tn+1 > t0|Htn)

}A

+

{n∑

i=1

log fWi |Hti−1(wi |Hti−1)

}B

Boulder, April 2016 12 / 25

HEC LausanneHawkes process

Combining separately a Hawkes process for the counting process of thetimes (part A) and a GPD model for the contributions from the marks(part B), the likelihood becomes

` = log

{n∏

i=1

λ(ti |Hti ) exp

(−∫ t0

0λ(u|Hu)du

)}

−n log σ − (1 + 1/ξ)n∑

i=1

log(1 + ξwi/σ)

with λ(t|Ht) being the conditional intensity

Boulder, April 2016 13 / 25

HEC LausanneSelf-exciting process

We chose the following general form with a constant loading µ0

λ(t|Ht) = µ0 +∑j :tj<t

ω(t − tj ;w j),

where µ0 > 0 and the self-exciting function ω(s) has to be positive whens ≥ 0 and 0 elsewhere. For example, the conditional intensity modelproposed by Ogata (1988) is

λ(t|Ht) = µ0 + ψ∑j :tj<t

eδwj

(t − tj + γ)1+ρETAS

where µ0, ψ, δ, γ, ρ > 0.

Boulder, April 2016 14 / 25

HEC LausanneApplications to high-frequency data:15-minutes linearly interpolated NVDA 2008 negativelog-returns

−0.

100.

05

Time(15min intervals)

Neg

ativ

e lo

g re

turn

s

2008−01−02 2008−04−03 2008−07−02 2008−10−01 2008−12−31

0.00

0.06

0.12

Est

imat

ed in

tens

ity

2008−01−02 2008−04−03 2008−07−02 2008−10−01 2008−12−31

Boulder, April 2016 15 / 25

HEC LausanneEstimated cumulative events numberΛHt (t) =

∫ t0 λ(u|Hu)du and two-sided 95% and 99% confidence limits

based on the Kolmogorov-Smirnov statistic

0 50 100 150 200 250 300 350

050

100

150

200

250

300

350

Transformed time

Cum

ulat

ive

num

ber

of e

xtre

me

loss

es

Boulder, April 2016 16 / 25

HEC LausanneAlternative models (1)

Hawkes process with exponential decay

λ(t|Ht) = µ0 + ψ∑j :tj<t

exp{δwj − γ(t − tj)}.

First order Markov chain for the marks

Wi |Wi−1 = wi ∼ GPD(ξ, σi = exp(a + bwi−1))

whose parameters a, b and ξ can be estimated by maximising

n∏i=2

fWi |Wi−1(wi | wi−1)fw1(w1)

Diagnostic for the marginal distribution based on the residuals

Rj = ξ−1 log (1 + ξwj/σj), j = 1, . . . , n

approximately independent unit exponential variables

Boulder, April 2016 17 / 25

HEC LausanneAlternative models (2):A self-exciting POT modelA model with predictable marks

λ?(t,w) =τ + α1ν(t)

σ + α2ν(t)

(1 + ξ

w − u

σ + α2ν(t)

)−1/ξ−1Where ν(t) =

∑j :tj<t ω(t − tj) uses a self-exciting function ω. This uses

the two-dimensional re-parametrisation of the POT:

Homogeneous Poisson process for the exceedances of the level u with

intensity τ = − lnH(u;µ, β, ξ) = {1 + ξ(u − µ)/β}−1/ξ+ , whereH(y ;µ, β, ξ) is the GEV distribution and σ = β + ξ(u − µ)

Inhomogeneous Poisson process for their sizes with intensity

λ(t,w) = λ(w) =1

σ

(1 + ξ

w − u

σ

)−1/ξ−1Boulder, April 2016 18 / 25

HEC LausanneEstimated intensity of exceeding the threshold fromself-exciting POT model for NVDA with Ogata function

0 1000 2000 3000 4000 5000 6000 7000

24

68

10

Time (15 min intervals)Mar

ks a

mpl

itude

in p

erce

nt

0 1000 2000 3000 4000 5000 6000 7000

0.1

0.3

Time (15 min intervals)

Inte

nsity

0 1000 2000 3000 4000 5000 6000 7000

0.8

1.0

1.2

1.4

Time (15 min intervals)

mea

n of

the

GP

D

Boulder, April 2016 19 / 25

HEC LausanneConditional risk measures

The conditional approach models the intraday clustering of extremesand is used to calculate instantaneous conditional p =95% or 99%VaR over the next period

z t+1p = F−1Zt+1|Ht

(p),

FZt+1|Ht(z) = P(Zt+1 > z |Ht) which is

P(Zt+1 − u > z |Zt+1 > u,Ht)× P(Zt+1 > u|Ht).

P(Zt+1 > u|Ht) represents the conditional probability of an event in(t, t + 1) that is:

1− P{N(t, t + 1) = 0|Ht} = 1− exp

(−∫ t+1

tλ(u|Hu)du

)P(ZT+1 − u > z |Zt+1 > u,Ht) is estimated using the GPD modelwith parameters σ and ξ.

Boulder, April 2016 20 / 25

HEC LausanneConditional VaR and Expected Shortfall

The VaR z t+1p is then the solution of the equation

P(ZT+1 > z t+1p |Ht) = 1− p,

Using the self-exciting POT model the condition VaR is

z t+1p = u +

σ + α2ν(t+)

ξ

{(1− p

τ + α1ν(t+)

)−ξ− 1

}

From the estimated conditional VaR it is straightforward to get anestimation of the conditional Expected Shortfall (ES)

ESt+1p =

zt+1p

1− ξ+σ − ξu1− ξ

.

with parameters replaced by their maximum likelihood estimates

Boulder, April 2016 21 / 25

HEC Lausanne15-minutes 95%VaR (red) and 95%ES (blue)estimates for NVDA negative log-returns using self-excitingPOT models and 95%VaR using classical POT (green)

−0.

10−

0.05

0.00

0.05

Times (15 min intervals)

Neg

ativ

e lo

g re

turn

s

2008−07−24 2008−09−03 2008−10−13 2008−11−20 2008−12−31

Boulder, April 2016 22 / 25

HEC Lausanne15-minutes 95%VaR and 95%ES estimates forNVDA negative log-returns using a competitor model(Chavez-Demoulin et al. (2014))

−0.

10−

0.05

0.00

0.05

Time

Neg

ativ

e lo

g re

turn

s

2008−07−24 2008−09−03 2008−10−13 2008−11−20 2008−12−31

Boulder, April 2016 23 / 25

HEC LausanneBacktesting results

0.95 QuantileExpected 25

POT 59 (0)ETAS ρ = 0 21 (0.24)ETAS ρ 6= 0 22 (0.31)

ETAS? ρ = 0 24 (0.47)ETAS? ρ 6= 0 23 (0.39)

exp? 25 (0.55)NPOT 26 (0.39)

0.99 QuantileExpected 5

POT 14 (0)ETAS ρ = 0 6 (0.39)

exp 6 (0.39)ETAS? ρ = 0 6 (0.39)

exp? 7 (0.24)NPOT 9 (0.11)

0.999 QuantileExpected 5

POT 14 (0)H1 ρ = 0 6 (0.39)H2 ρ 6= 0 6 (0.39)

H3 6 (0.39)H1? 6 (0.39)H2? 6 (0.39)H3? 7 (0.24)

NPOT 9 (0.11)

Boulder, April 2016 24 / 25

HEC LausanneReferences

A.G. Hawkes, 1971, Point spectra of some mutually exciting pointprocesses, Biometrika.

Y. Ogata, 1988, Statistical models for earthquake Occurrences andresiduals analysis for point processes, JASA.

E. Errais, K. Giesecke and L.R. Goldberg, 2010, Affine point processesand portfolio credit risk, SIAM J. Financial Math.

Y. AIt-Sahalia, J. Cacho-Diaz, and R. J. A. Laeven, 2010, Modelingfinancial contagion using mutually exciting jump processes, TheReview of Financial Studies.

V. Chavez-Demoulin, A.C. Davison, A.J. McNeil, 2005, A pointprocess approach to Value-at-Risk estimation, Quantitative Finance.

V. Chavez-Demoulin, P. Embrechts, S.Sardy, 2014 Extreme-quantiletracking for financial time series Journal of Econometrics.

Boulder, April 2016 25 / 25