carnets d’ordres pilotés par des processus de...

Carnets d’ordres pilotés par des processus deHawkes

workshop sur les Mathématiques des marchés financiers en hautefréquence

Frédéric AbergelChaire de finance quantitative

fiquant.mas.ecp.fr/limit-order-books

14 Avril 2015

Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 1/43

1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity

2 Mathematical modelling of limit order booksZero-intelligence models

Stability and long-time dynamicsLarge-scale limit of the price process

Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics

3 Simulation with Hawkes processes

What is a limit order book ?

The limit order book is the list, at a given time, of all buy and selllimit orders, with their corresponding prices and volumes

The order book evolves over time according to the arrival of neworders

3 main types of orders:limit order: specify a price at which one is willing to buy (sell) a certainnumber of sharesmarket order: immediately buy (sell) a certain number of shares at thebest available opposite quote(s)cancellation order: cancel an existing limit order

Price dynamics

The price dynamics becomes a by-product of the order book dynamics

The order book evolves over time according to the arrival of neworders3 main types of orders:

limit order: specify a price at which one is willing to buy (sell) a certainnumber of sharesmarket order: immediately buy (sell) a certain number of shares at thebest available opposite quote(s)cancellation order: cancel an existing limit order

Price dynamics

The order book evolves over time according to the arrival of neworders3 main types of orders:

limit order: specify a price at which one is willing to buy (sell) a certainnumber of sharesmarket order: immediately buy (sell) a certain number of shares at thebest available opposite quote(s)cancellation order: cancel an existing limit order

Price dynamics

Limit order book events

Timestamp Side Level Price Quantity33480.158 B 1 121.1 48033480.476 B 2 121.05 163633481.517 B 5 120.9 131833483.218 B 1 121.1 42033484.254 B 1 121.1 55633486.832 A 1 121.15 18733489.014 B 2 121.05 139733490.473 B 1 121.1 34233490.473 B 1 121.1 30433490.473 B 1 121.1 25633490.473 A 1 121.15 237

Table: Tick by tick data file sample

Trades

Timestamp Last Last quantity33483.097 121.1 6033490.380 121.1 21433490.380 121.1 3833490.380 121.1 48

Table: Trades data file sample.

These data are made available (for free) to the scientific communityhttp://fiquant.mas.ecp.fr/liquidity-watch

Trades and order book data processing

For each stock and each trading day:1 Parse the tick by tick data file to compute order book state variations;2 Parse the trades file and for each trade:

1 Compare the trade price and volume to likely market orders whosetimestamps are in [tTr −∆t , tTr + ∆t], where tTr is the trade timestampand ∆t is a predefined time window;

2 Match the trade to the first likely market order with the same price andvolume and label the corresponding event as a market order;

3 Remaining negative variations are labeled as cancellations.

Doing so, we have an average matching rate of around 85% for CAC 40stocks. As a byproduct, one gets the sign of each matched trade.

Arrival time of orders

0 2 4 6 8 10 12 14

Interarrival time

BNPP.PALognormal

ExponentialWeibull

0.0001

1 10 100 1000 10000

Lifetime for cancelled limit orders

Power law ∝ x-1.4

BNPP.PAFTE.PA

RENA.PASOGN.PA

Figure: Distribution of estimated lifetime of cancelled limit orders.

0.0001

1 10 100 1000 10000

Lifetime for executed limit orders

Power law ∝ x-1.5

BNPP.PAFTE.PA

RENA.PASOGN.PA

Figure: Distribution of estimated lifetime of executed limit orders.

30000 35000 40000 45000 50000 55000 60000 65000

in ∆

Time of day (seconds)

BNPP.PABNPP.PA quadratic fit

FTE.PAFTE.PA quadratic fit

Figure: Normalized average number of market orders in a 5-minute interval.

30000 35000 40000 45000 50000 55000 60000 65000

in ∆

Time of day (seconds)

BNPP.PABNPP.PA quadratic fit

FTE.PAFTE.PA quadratic fit

Figure: Normalized average number of limit orders in a 5-minute interval.

Competitive liquidity

Muni Toke (2011)

Figure: Evidence of liquidity providing

Competitive liquidity

Muni Toke (2011)Anane et al. (2015)

Figure: Evidence of liquidity taking

Mathematical modelling

A set of reasonable assumptions:

the limit order book is described as a point process;

several types of events can happen;

two events cannot occur simultaneously (simple point process).

The main questions of interest are

the stationarity and ergodicity of the order book;

the dynamics of the induced price...

... particularly, its behaviour at larger time scales.

⇒ Linking microstructure modelling and continuous-time finance

Mathematical modelling

A set of reasonable assumptions:

the limit order book is described as a point process;

several types of events can happen;

two events cannot occur simultaneously (simple point process).

The main questions of interest are

the stationarity and ergodicity of the order book;

the dynamics of the induced price...

... particularly, its behaviour at larger time scales.

⇒ Linking microstructure modelling and continuous-time finance

Some notations

The order book is represented by a finite-size vector of quantities

X(t) := (a(t); b(t)) := (a1(t), . . . , aK (t); b1(t), . . . , bK (t));

a(t): ask side of the order book

b(t): bid side of the order book

∆P: tick size

q: unit volume

P = PA+PB

2 : mid-price

A(p),B(p): cumulative number of sell (buy) orders up to price levelp

Notations

1a 3a 4a2a5a

1b2b3b4b

APBP S

Figure: Order book notations

Zero-intelligence model

Inspired by Smith et al. (2003), Abergel and Jedidi (2013) analyzes anelementary LOB model where the events affecting the order book aredescribed by independent Poisson processes:

M±: arrival of new market order, with intensityλM±

L±i : arrival of a limit order at level i, with intensityλL±

i ∆P

C±i : cancellation of a limit order at level i, with intensity λC+

λC−i|bi |

q⇒ Cancellation rate is proportional to the outstanding quantity ateach level.

Under these assumptions, (X(t))t≥0 is a Markov process with statespace S ⊂ Z2K .

Order book dynamics

dai(t) = −

q − i−1∑k=1

dM+(t) + qdL+i (t) − qdC+

+ (JM−(a) − a)idM−(t) +K∑

(JL−i (a) − a)idL−i (t)

(JC−i (a) − a)idC−i (t),

dbi(t) = similar expression,

where JM± , JL±i , and JC±i are shift operators corresponding to the effect oforder arrivals on the reference frame

The shift operator

For instance, the shift operator corresponding to the arrival of a sellmarket order is

JM−(a) =

0, 0, . . . , 0︸︷︷︸k times

, a1, a2, . . . , aK−k

k := inf{p :

p∑j=1

|bj | > q} − inf{p : |bp | > 0}.

Infinitesimal generator

Lf (a; b) = λM+

(f([ai − (q − A(i − 1))+]+; JM+

(b))− f)

i (f(ai + q; JL+

i (b))− f)

i ai(f(ai − q; JC+

i (b))− f)

+ λM−(f(JM−(a); [bi + (q − B(i − 1))+]−

)− f

K∑i=1

λL−i (f

(JL−i (a); bi − q

)− f)

λC−i |bi |(f

(JC−i (a); bi + q

)− f)

Stability of the order book

Stationary order book distribution

Abergel and Jedidi (2013) If λC = min1≤i≤N{λC±i } > 0, then

(X(t))t≥0 = (a(t); b(t))t≥0 is an ergodic Markov process. In particular(X(t)) has a unique stationary distribution Π. Moreover, the rate ofconvergence of the order book to its stationary state is exponential.

The proof relies on the use of a Lyapunov functionThe proportional cancellation rate helps a lot... and so do the boundaryconditions!

A general approach to study price asymptotics

Combining ergodic theory and martingale convergence

The ergodicity of the order book allows for a direct study of theasymptotic behaviour of the price, based on Foster-Lyapunov-typecriteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and theconvergence of martingales Ethier and Kurtz (2005):

the evolution of the price is dPt =∑K ′

i=1 Fi(Xt )dNit ;

the rescaled, centered price is P̃nt ≡

Pnt−∫ nt

∑K ′i=1 Fi(Xs)λids√

its predictable quadratic variation is < P̃n, P̃n >t =∫ nt

∑K ′i=1(Fi(Xs))

2λidsn

ergodicity ensures the convergence of∫ nt

∑K ′i=1(Fi(Xs))

2λidsnt as n → ∞

This approach is easily extended to state-dependent intensitiesA similar approach - in a somewhat different modelling context - is usedin recent papers by Horst and co-authors Horst and Paulsen (2015).

A general approach to study price asymptotics

Combining ergodic theory and martingale convergence

The ergodicity of the order book allows for a direct study of theasymptotic behaviour of the price, based on Foster-Lyapunov-typecriteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and theconvergence of martingales Ethier and Kurtz (2005):

the evolution of the price is dPt =∑K ′

i=1 Fi(Xt )dNit ;

the rescaled, centered price is P̃nt ≡

Pnt−∫ nt

∑K ′i=1 Fi(Xs)λids√

its predictable quadratic variation is < P̃n, P̃n >t =∫ nt

∑K ′i=1(Fi(Xs))

2λidsn

ergodicity ensures the convergence of∫ nt

∑K ′i=1(Fi(Xs))

2λidsnt as n → ∞

This approach is easily extended to state-dependent intensitiesA similar approach - in a somewhat different modelling context - is usedin recent papers by Horst and co-authors Horst and Paulsen (2015).

A general approach made precise I

Write as above

Pt = P0 +∑

0Fi(Xs)dNi

and its compensator

Qt =∑

0λiFi(Xs)ds.

Defineh =

λiFi(X)

and let

α =a.s.lim

t→+∞

0λi(Fi(Xs))ds =

∫h(X)Π(dX).

Finally, introduce the solution g to the Poisson equation

Lg = h − α

A general approach made precise II

and the associated martingale

Zt = g(Xt ) − g(X0) −

0Lg(Xs)ds ≡ g(Xt ) − g(X0) − Qt + αt .

Then, the deterministically centered, rescaled price

P̄nt ≡

Pnt − αnt√

converges in distribution to a Wiener process σ̄W . The asymptoticvolatility σ̄ satisfies the identity

σ̄2 = limt→+∞

0λi((Fi−∆i(g))(Xs))2ds ≡

∫λi((Fi−∆i(g))(X))2Π(dx).

where ∆i(g)(X) denotes the jump of the process g(X) when the processNi jumps and the limit order book is in the state X .

Hawkes processes

Zero-intelligence models fail to capture the dependencies betweenvarious types of orders:

clustering of market orders;

interplay between liquidity taking and providing;

leverage effect,

as has been demonstrated in many contributions, see e.g. Muni Toke(2011)Muni Toke and Pomponio (2012).

Hawkes processes

Hawkes processes provide an ad hoc tool to describe the mutualexcitations of the arrivals of different types of orders.In D dimensions, the process Nj

s has a stochastic intensity λjt such that

λjt = λ

D∑p=1

0φjp(t − s)dNp

s . (2.1)

A simplifying choice is the exponential kernel

φjp(s) = αjp exp(−βjps) (2.2)

leading to markovian processes. A classical result states that the processis stationary iff the spectral radius of the matrix

βjp] (2.3)

is < 1, see Brémaud and Massoulié (1996).

Introducing dependence in the order flow

As an example, Muni Toke (2011) study variations of the model below

A better order book model

λM(t) = λM0 +

0αMMe−βMM(t−s)dNM(s),

λL (t) = λL0 +

0αLL e−βLL (t−s)dNL (s)

0αML e−βML (t−s)dNM(s).

MM and LL effect for clustering of orders

ML effect as observed on data

No global LM effect observed on data

Infinitesimal generator

LF(−→a ;−→b ;−→µ ) = λM+

(F([ai − (q − A(i − 1))+]+; JM+

(−→b );−→µ + ∆M+

(−→µ ))− F)

i (F(ai + q; JL+

i (−→b );−→µ + ∆L+

i (−→µ ))− F)

i ai(F(ai − q; JC+

i (−→b )

)− F)

+ λM−(F

(JM−(−→a ); [bi + (q − B(i − 1))+]−;−→µ + ∆M−(−→µ )

)− F

K∑i=1

λL−i (F

(JL−i (−→a ); bi − q;−→µ + ∆L−i (−→µ )

)− F)

λC−i |bi |(f

(JC−i (−→a ); bi + q

)− f)

D∑i,j=1

βijµij∂F∂µij

. (2.4)

Large-time behaviour

Results similar to those obtained in the zero-intelligence case holdAbergel and Jedidi (2015)

Large-time behaviour for Hawkes-process driven LOB

Under the usual stationarity conditions for the intensities, thereexists a Lyapunov function V =

∑|ai |+

∑|bi |+

∑Ukλk and the LOB

converges exponentially to its stationary distribution Π

The rescaled, (deterministically) centered price converges to aWiener process

The proofs rely on the same decomposition as in the Poisson arrivalcase, thanks to the fact that the proportional cancellation rate remainsbounded away from zeroSome extra care is required to prove that the solution to the Poissonequation is in L2(Π(dx))

Hawkes model

In Muni Toke (2011), the flow of limit and market orders are modelled byHawkes processes NL and NM with intensities λ and µ defined as follows:

µM(t) = µM0 +

0αMMe−βMM(t−s)dNM

λL (t) = λL0 +

0αLMe−βLM(t−s)dNM

0αLL e−βLL (t−s)dNL

Numerical results on the order book: fitting

Fitting and simulation

Model µ0 αMM βMM λ0 αLM βLM αLL βLL

HP 0.22 - - 1.69 - - - -LM 0.22 - - 0.79 5.8 1.8 - -MM 0.09 1.7 6.0 1.69 - - - -

MM LL 0.09 1.7 6.0 0.60 - - 1.7 6.0MM LM 0.12 1.7 6.0 0.82 5.8 1.8 - -

MM LL LM 0.12 1.7 5.8 0.02 5.8 1.8 1.7 6.0Common parameters: mP

1 = 2.7, νP1 = 2.0, sP

1 = 0.9V1 = 275,mV

2 = 380λC = 1.35, δ = 0.015

Table: Estimated values of parameters used for simulations.

Impact on arrival times

Figure: Empirical density function of the distribution of the interval times between a marketorder and the following limit order for three simulations, namely HP, MM+LL, MM+LL+LM,compared to empirical measures. In inset, same data using a semi-log scale

Impact on the bid-ask spread

0 0.05 0.1 0.15 0.2 0.25 0.3

Empirical BNPP.PAHomogeneous Poisson

Hawkes MMHawkes MM+LM

0 0.05 0.1 0.15 0.2

Figure: Empirical density function of the distribution of the bid-ask spread forthree simulations, namely HP, MM, MM+LM, compared to empirical measures. Ininset, same data using a semi-log scale. X-axis is scaled in euro (1 tick is 0.01euro).

0 0.05 0.1 0.15 0.2

Empirical BNPP.PAHomogeneous Poisson

Hawkes MM+LLHawkes MM+LL+LM

0 0.05 0.1 0.15 0.2

Figure: Empirical density function of the distribution of the bid-ask spread threesimulations, namely HP, MM, MM+LM, compared to empirical measures. In inset,same data using a semi-log scale. X-axis is scaled in euro (1 tick is 0.01 euro).

References I

Abergel, F. and Jedidi, A. (2013). A mathematical approach to order bookmodelling. International Journal of Theoretical and Applied Finance,16(5).

Abergel, F. and Jedidi, A. (2015). Long time behaviour of a Hawkesprocess-based limit order book.

Anane, M., Abergel, F., and Lu, X. F. (2015). Mathematical modeling ofthe order book: New approach of predicting single-stock returns.

Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkesprocesses. Ann. Probab., 24:1563–1588.

Ethier, S. N. and Kurtz, T. G. (2005). Markov processes characterizationand convergence. Wiley.

Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions ofthe Poisson equation. Annals of Probability, 24(2):916–931.

Horst, U. and Paulsen, M. (2015). A law of large numbers for limit orderbooks.

References II

Meyn, S. and Tweedie, R. (1993). Stability of markovian processes III:Foster-lyapunov criteria for continuous-time processes. Advances inApplied Probability, 25:518–548.

Muni Toke, I. (2011). Market Making in an order book model and itsimpact on the spread. In Abergel, F., Chakrabarti, B. K., Chakraborti,A., and Mitra, M., editors, Econophysics of Order-driven Markets,pages 49–64. Springer.

Muni Toke, I. and Pomponio, F. (2012). Modelling trade-throughs in alimited order book using Hawkes processes. Economics: TheOpen-Access, Open-Assessment E-Journal, 6.

Smith, E., Farmer, J. D., Gillemot, L., and Krishnamurthy, S. (2003).Statistical theory of the continuous double auction. QuantitativeFinance, 3(6):481–514.

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