[PPT] - Microstructure, mutually exciting processes Delattre, A. Iuga, PowerPoint Presentation

SLIDE 1

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Microstructure, mutually exciting processes and market impact

E. Bacry, S. Delattre, A. Iuga, M.H., J.F. Muzy

CMAP-X, Paris 7, CREST and Paris-Est, Universit´ e de Cort´ e

Paris, March 1, 2012

SLIDE 2

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Outline

1 Introduction: inference across scales 2 The Hawkes processes approach 3 Scaling limits and (financial) interpretations 4 Market impact and mutually exciting processes

SLIDE 3

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Modelling across scales

Suppose we have discrete (financial) data X0, X∆, X2∆, . . . , over [0, T]

f a continuous time process Xt with t ∈ [0, T].

Depending on the relative sizes of ∆ and T as n = ⌊T/∆⌋ → ∞ (whenever we conduct an asymptotic study) we are led to model the data differently.

SLIDE 4

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Financial data modelling

Price processes behave differently at different scales:

Coarse scales (daily, hourly): continuous or jump diffusions,
Fine scales (tick data): marked point processes.

Several HF stylised facts:

Microstructure effect known as “mean-reversion” in d = 1,
Covariation instability: Epps effect for d ≥ 2.

SLIDE 5

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Modelling macroscopic data

Time Bund 500 1000 1500 105 110 115 120 125

Figure: German 10Y Bund (FGBL) sampled with ∆ = 1 day (traded price), 04 Avr. 1999 to 06 Dec. 2005. The candidate for X may be a continuous Ito semimartingale dXt = btdt + σtdBt that we observe at times i∆.

SLIDE 6

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Ruling out discretely observed diffusions on fine scales

If Xt is observed over [0, T] at times 0, ∆, 2∆, . . ., define the signature plot as ∆ V∆

X
t :=
i∆≤t
Xi∆ − X(i−1)∆

2. If X is an Itˆ

continuous semimartingale with

dXt = btdt + σtdBt we have V∆{X}t

P

→ t σ2

s ds

as ∆ → 0 with accuracy √ ∆. This suggests to pick ∆ as small as possible... but

SLIDE 7

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Signature plot

50 100 150 200 0.02 0.025 0.03 0.035 0.04 0.045

V

t

(ticks) t (seconds)

Figure: ∆ V∆ for FGBL (43 days, 9-11 AM) on Last Traded Ask.

SLIDE 8

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Coarse-to-fine modelling

time value 5000 10000 15000 20000 25000 30000 115.40 115.45 115.50 115.55 115.60 115.65

Figure: FGBL, 06 Feb 2007, 08:30-17:00 (UTC) sampled with ∆ = 1

second. The candidate for the underlying process X is rather a

marked point process that we observe at times i∆.

SLIDE 9

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Coarse-to-fine modelling (cont.)

time value 500 1000 1500 2000 2500 3000 3500 115.46 115.48 115.50 115.52 115.54 115.56

Figure: FGBL, 06 Feb 2007, 09:00–10:00 (UTC) 1 data every second. The point process approach suggestion is even more pronounced here. The underlying process looks more complex than a simple CTRW.

SLIDE 10

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Toward more realistic price models

We look for a “simple” multivariate price model Defined in continuous time with discrete values on a microscopic scale. That may incorporate microstructure effects. That diffuses on a macroscopic scale. That enables to tackle other HF issues such as Price Impact.

SLIDE 11

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Alternative approaches

Latent price approach In statistics: Gloter and Jacod (2001), Munk and Schmiedt-Hieber (2009), Reiß (2010) In financial econometrics: Ait-Sahalia, Mykland and Zhang (2003 to 2006). And many more... Podolkii, Vetter, Jacod, Mykland, Zhang, Bandi, Russell, Diebold, Strasser, Barndorff-Nielsen, Hansen, Lund, Shepard, Other approaches for modelling microstructure: Engle Russell (2002), Hautsch (2006), Robert and Rosenbaum (2009) Econophysics literature Order book oriented modelling...

SLIDE 12

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Point process approach

Price process = marked point process.

Marks : jumps up/down by 1 tick, Jump times: time stamps of price changes.

The price process is the result (sum) of a “upward change

r price” and a “downward change of price” (two counting

processes). By coupling random intensities of the counting processes, we create local oscillations that reproduce empirical microstructure effects.

SLIDE 13

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Hawkes processes

A (linear) Hawkes process is a counting process Nt constructed via its stochastic intensity λ(t) = µ + t φ(t − s)dNs Mathematically tractable choice: φ(x) = αe−βx with simple interpretation. (One has t

0 φ(t − s)dNs = Tn<t φ(t − Tn).)

Non-explosion constraint: φL1 < 1.

SLIDE 14

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Price model in dimension 1

Continuous-time price model living on a tick-grid: Xt = N+

t − N− t

with N±

t Hawkes processes with respective stochastic intensities

     λ+(t) = µ+ + α

[0,t) e−β(t−s)dN−

s

λ−(t) = µ− + α

[0,t) e−β(t−s)dN+

s

µ±: exogeneous intensity. α et β: mutually exciting intensities generating a “mean-reverting effect” for St. αe−βx φ(x) with φL1 < 1 in the sequel.

SLIDE 15

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Scaling limits

First step:

1 closed-form formulas for the mean “signature plot” when

Φ(x) = αe−βx (through the explicit computation of the Bartlett spectrum, case with stationary increments) in dimension 1 and 2.

2 Statistical fits and discussion of further data filtering.

Second step: scaling (diffusive) limit for arbitrary φ.

SLIDE 16

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Mean “signature plot” and scaling limits

Time-space renormalisation X (T)(t) := T −1/2X(tT), t ∈ [0, 1] Realised volatility V∆

X (T)

:=

∆−1

i=1
X (T)(i∆) − X (T)

(i − 1)∆ 2 ≈ 1 ∆T E

X(∆T) − X(0)

2 Mean signature plot V(t) := 1 t E

X(t) − X(0)

2 Interpretation V(∆T) ≈ V∆

X (T)

.

SLIDE 17

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Mean signature plot

If µ+ = µ− = µ we have a closed-form expression for V(t) = 2µ 1 − α/β

1

(1 + α/β)2 + +(1 − 1 (1 + α/β)2 )1 − exp

− (α + β)t
(α + β)t

SLIDE 18

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical mean signature plot on real data

Bund 10Y : 21 days, 9-11 AM - Last Traded Ask Mean square regression fit

50 100 150 200 0.02 0.025 0.03 0.035 0.04 0.045 50 100 150 200

V!t

(ticks)

!t (seconds)

SLIDE 19

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 20

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 21

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 22

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 23

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 24

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 25

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 26

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 27

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 28

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Empirical and mean signature plot over 2 hours, 1 single day

SLIDE 29

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Scaling limit , arbitrary φ, µ± = µ

Step 1 : price decomposition introducing a martingale X (T)(t) = T −1/2 N+

tT − N− tT

= M(T)

t

+ B(T)

t

, with M(T)

t

= T −1/2 N+

tT − N− tT

− B(T)

t

martingale and B(T)

t

= T −1/2 tT

λ+(s) − λ−(s)
ds predictable

SLIDE 30

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Scaling limit in dimension 1 (cont.)

Step 2: Convergence of the compensator B(T)

t

= T −1/2 tT ds s φ(s − u)d(N−

u − N+ u )

= −

[0,t)

dX (T)(u) t−u Tφ(sT)ds = −

[0,t)

X (T)(u) φT(t − u)

Dirac mass ×φL1

du ≈ − φL1X (∞)(t). In the limit X (∞)(t) = −φL1X (∞)(t) + M(∞)

t

SLIDE 31

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Scaling limit in dimension 1 (cont.)

Step 3: Convergence of the martingale part

M(T)

t = T −1

tT

λ+(s) + λ−(s)
ds

= 2µt + T −1 tT ds s φ(s − u)d

N+

u + N− u

= 2µt +

t

M(T)

u φT(t − u)du

≈ 2µt + t M(T)u φT(t − u)du ≈ 2µt + φL1M(T)t Conclusion

M(T)

t P

− → 2µ 1 − φL1 t

SLIDE 32

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Scaling limit in dimension 1 (cont.)

We obtain a) M(T)(t)

d

− →

2µ

1−φL1 Wt, where W is a

Wiener process and b) the representation X (∞)(t) = −φL1X (∞)(t) + M(∞)

t

. a) + b) yield the final result X (T)(t)

d

− → X (∞)

t

= 1 1 + φL1

2µ

1 − φL1 Wt

SLIDE 33

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Discussion

Microscopic variance (assuming a stationary regime) E

λ+(t) + λ−(t)
=

2µ 1 − φ1 Macroscopic variance σ2 = 2µ 1 − φ1 1 (1 + φ1)2

SLIDE 34

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Macroscopic trace of microstructure

Macroscopic variance σ2 = 2µ 1 − φ1 1 (1 + φ1)2 The influence of φ does not disappear on large scales : it can be quantified by looking at φ1 = x ∈ [0, 1) 1 1 − x 1 (1 + x)2 which is negative as long as x ≤ 0.61 and minimum at x = 1

3.

SLIDE 35

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Influence of φ1 on the diffusive variance

Histogram of φ1 (fitted on the signature plot) 140 days - 9 : 11am - 12am: 2pm - 2 : 4pm Mean ≈ 0.29 (Bund) and ≈ 0.36 (Bobl)

SLIDE 36

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

In dimension 2

115.2 115.3 115.4 115.5 115.6

Bund and Bobl

Time Price 13:00 13:48 14:36 15:24 16:12 17:00 17:48 Bund Bobl 109.0 109.1 109.2 109.3 109.4

Figure: FGBL/FGBM

SLIDE 37

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

In dimension 2: Epps effect

In the same Itˆ

semimartingale setting, we have

convergence of the quadratic covariation CV∆

S(1), S(2)

t :=

i∆≤t
S(1)

i∆ − S(1) (i−1)∆

S(2)

i∆ − S(2) (i−1)∆

P

→ S(1), S(2)t Same prescription as for the realized volatility: pick ∆ as small as possible... but

SLIDE 38

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Epps effect

50 100 150 200 0. 2 0. 4 0. 6 0. 8

Figure: ∆ CV∆

S(1), S(2)

(normalized) with S(1) = FGBL, S(2) = FGBM, 40 days, 9-11AM.

SLIDE 39

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Price in dimension 2

1 Start from two processes X and Y constructed as before. 2 Introduce a supplementary coupling on the intensities of

the two processes and create a dependence structure UpwardX-UpwardY and DownwardX-DownwardY .

3 (We ignore further possible coupling UpwardX-DownwardY and

DownwardX-UpwardY between X and Y .)

SLIDE 40

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Representation of X and Y

Set X(t) = N+

X (t) − N− X (t) and Y (t) = N+ Y (t) − N− Y (t)

with λ±

X(t) = µ± X+

[0,t)

ΦX,X(t−s)dN∓

X (s)+

[0,t)

ΦX,Y (t−s)dN±

Y (s)

and λ±

Y (t) = µ± Y +

[0,t)

ΦY ,Y (t−s)dN±

X (s)+

[0,t)

ΦY ,X(t−s)dN±

X (s)

SLIDE 41

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Simulation over 1000 seconds

200 400 600 800 5 10 20 30 s y1 200 400 600 800 10 20 30 s y2

Figure: Sample simulation in dimension 2

SLIDE 42

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

In dimension 2

The same asymptotic analysis can be conducted. Scaling limit: 2-dimensional Brownian motion with explicit macroscopic covariance. We can fit the Epps curve analogously.

SLIDE 43

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Market impact

A (meta)-order is executed at a certain rate ˙ vs (traded volume per unit of time) during [t0, t0 + T]. What is the effect of this meta-order on the baseline price process? MI(t) := E

Xt − Xt0
, t ≥ t0

given the information of the meta order execution.

SLIDE 44

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Market impact

Figure: Source: Moro, Vicente, Moyano, Gerig, Farmer, Vaglica, Lillo, Mantegna, 2009

SLIDE 45

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Market impact: stylised facts

Concave impact while trading - Square-root law Relaxation after trading (power-law?), Bouchaud et al. Is market impact permanent?

SLIDE 46

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Towards a market impact model

Market price baseline modelling: Xt = N+

t − N− t where

N±

t specified by their random intensities

λ±(t) = µ + t φ(t − s)dN∓

s .

Exogeneous single buy order of size v at time s0

Impact on λ+ by an additive term ˜ λ+(t) = λ(t) + g +

v (t − s0)

Likewise, impact on λ−(t) by an additive term g −

v (t − s0)

Instantaneous impact function g±

v (t − s0) ≈ f (v)δs0(dt)

SLIDE 47

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

A market impact model

Impact model for an exogeneous buy order at time s0 and volume v    ˜ λ+(t) = λ+(t) + g+

v (t − s0)

˜ λ−(t) = λ−(t) + g−

v (t − s0)

Impact model for an exogeneous strategy ˙ vt starting at time t0      ˜ λ+(t) = λ+(t) + t

t0 ds f (˙

vs)g+(t − s) ˜ λ−(t) = λ−(t) + t

t0 ds f (˙

vs)g−(t − s)

SLIDE 48

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Market impact computation

Impact model for a single buy order of size v and time t = s0 with g±

v (t − s0) = f (v)g±(t − s0)

MI(t) = f (v) [G(t − s0) − ψ ⋆ G(t − s0)] with G(t) = t

s0

ds (g+(s) − g−(s)), ψ =

+∞

n=1

φ⋆n. Impact model for an exogeneous strategy ˙ vt starting at time t = t0 MI(t) = t

t0

ds f (˙ vs) [G(t − s) − ψ ⋆ G(t − s)]

SLIDE 49

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Permanent versus non permanent market impact

For a single buy order of size v and time t0 : MI(t) = f (v) [G(t − t0) − ψ ⋆ G(t − t0)] , where G(t) = t

g+(s) − g−(s)
ds

Permanent impact: g+1 = g−1 Non permanent impact: G(t) → 0 when t → ∞, that is g+1 = g−1.

SLIDE 50

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

A simple non-permanent impact model

We need g+1 = g−1. Impact on ˜ λ+(t) at s0 for a single trade: take g+(t − s0) = C1δs0(dt). Assume that a buy order on ˜ λ−(t) is digested in the same way “as if an upward jump had occurred”: ˜ λ−(t) = µ + t φ(t − s)

dN+(s) + C2δs0(ds)
which implies g−(t) = φ(t − s0).

Having non permanent impact imposes C1f (v) = C2φ1. Finally, take a constant continuous strategy ˙ vt ≡ cte for t ∈ [t0, t0 + T] and integrate g± in s0 over [t0, t0 + T].

SLIDE 51

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

A simple non-permanent impact model

5 10 15 20 25 0.5 1 1.5 2 2.5 t (seconde) MI(g) α=0.2, β= 1.5, δ = 0.25 α=0.1213, β= 1.2, δ = 0.25

φ(t) = α(t + γ)−β.

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Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

To be compared with the empirical results of Farmer et al.

SLIDE 53

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Square root law

Market impact model with mutually exciting processes MI(t) = t

t0

ds f (˙ vs) [G(t − s) − ψ ⋆ G(t − s)] , On average, we retrieve the impact model of Gatheral (2009) X(t) = X(t0) + t

t0

ds f (˙ vs)K(t − s) + centred noise

SLIDE 54

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes

Square root law

For strategies with constant rate ˙ vt, assuming the square root law for the impact, the no-arbitrage argument of Gatheral theory implies that if φ(t) ∼ tβ then necessarily β = 3/2. Empirical findings on FGBL (Al Dayri, Bacry and Muzy, 2011) suggest β ≈ 1.1.

SLIDE 55

Microstructure, mutually exciting processes and market impact

E. Bacry, S.

Delattre, A. Iuga, M.H., J.F. Muzy Introduction: inference across scales The Hawkes processes approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes