Statistical and mathematical properties of limit order books Fr ed - - PowerPoint PPT Presentation

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

Statistical and mathematical properties of limit

• rder books

Fr´ ed´ eric Abergel

Chair de finance quantitative Laboratoire MAS ´ Ecole Centrale Paris

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

Plan

1

What is a limit order book ?

2

Empirical properties Competitive liquidity Predictability

3

Mathematical modelling Zero-intelligence models Interplay between liquidity providing and taking

4

Conclusion

5

References

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

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(1) initial state

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(2) liquidity is taken

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(3) wide spread

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(4) liquidity returns

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(5) liquidity returns

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(6) final state Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Plan

1

What is a limit order book ?

2

Empirical properties Competitive liquidity Predictability

3

Mathematical modelling Zero-intelligence models Interplay between liquidity providing and taking

4

Conclusion

5

References

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Statistical properties of the LOB

Basic questions asked since the first LOB studies, see [Chakraborti et al., 2011] for a comprehensive survey:

1 When will the next event take place ? 2 What type of event will it be ? 3 Where will it take place ?

In a more recent past, conditional statistics have been extensively studied, see [Muni Toke, 2009], [Eisler et al., 2011].

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Market making

Two examples [Muni Toke, 2009] of such ”conditional” statistics...

Figure: Evidence of liquidity providing

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Market taking

... leading to more sophisticated models.

Figure: Evidence of liquidity taking

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Market imperfections

Other practically relevant questions can be asked. For instance, what can the LOB tell us about

1 the sign of the next trade ? 2 the size of the price change ?

Recent results [Zheng et al., 2011] shed some light on these aspects.

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Trade sign prediction

Of the utmost interest is the prediction of the sign of the next trade. The LOB provides us with helpful information.

5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0

Conditional probability of TradeSign, BNPP.PA

Bid−ask volume ratio Conditional probability Depth=1

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Trade-through prediction

Some events are more informational than others. [Pomponio, Abergel, 2011] is a detailed study of multiple-limit trades, or trade-throughs. Trade-throughs occur when the liquidity dries out, and this can be read on the LOB. Question: what are the best predictors for a trade-through ?

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Upward jump

Prediction of an upward jump, i.e., a multiple-limit trade on the ask side.

−10 −8 −6 −4 −2 −12 −10 −8 −6 −4 −2 2

TECF.PA, upward jump, Sep 2009

Coefficients

37 31 19 5 AskSize1 AskGap1 BidGap1 BidSize1 BidSize2 BidGap2 AskGap2

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Competitive liquidity Predictability

Downward jump

Similar results hold for downward jumps.

−10 −8 −6 −4 −2 −10 −5

TECF.PA, downward jump, Sep 2009

Log Lambda Coefficients

36 36 22 6 BidSize1 BidGap1 AskGap1 BidGap2 AskSize1 AskGap2 AskSize3

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Plan

1

What is a limit order book ?

2

Empirical properties Competitive liquidity Predictability

3

Mathematical modelling Zero-intelligence models Interplay between liquidity providing and taking

4

Conclusion

5

References

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Mathematical modelling

The LOB is described as a point process. Several types of events (orders) can happen. Two events cannot occur simultaneously (simple process). Main questions to be addressed:

1 Stationarity 2 Price dynamics 3 Scaling Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Zero-intelligence

In [Smith et al., 2003], the authors describe and analyze an elementary LOB model, with dLi±

t

λi±

L

∆P τ dMt λ±

M

1 τ dC i±

t

λi+

C

ai τ , λi−

C

bi τ τ is the lot size, ∆P, the tick size, and the ai, bi’s are the available liquidity i ticks away from the best opposite limit.

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Model description

The dynamics can be decomposed into two basic types of events:

1 a change in one of the ai’s or bi’s. 2 a renumbering (shift) after a change of one of the best

available limits. It is a Markovian LOB model with Posson arrivals and proportional cancellation rate.

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Diffusive behaviour

Such a simple model already captures some empirically observed properties of the price dynamics: Long time diffusive behaviour...

50 100 150 200 250 300 1 2 x 10

• 4

Time lag (sec) Var[P(t + lag) - P(t)]

Figure: Diffusive behaviour of the price for the low frequencies

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Signature plot

... and microstructural effects, the signature plot.

5 10 15 20 25 30 35 40 45 50 55 60 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 x 10

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Time lag (sec) Var[P(t + lag) - P(t)] / lag

FTE.PA mide price (March 2011) Estimated model

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Mathematical results

Main mathematical properties

1 Theorem 6.1 in [Abergel, Jedidi 2011] There exists a

Lyapunov function V = |ai| + |bi| and the LOB converges to a stationary distribution

2 Theorem 6.4 in [Abergel, Jedidi 2011] The rescaled, centered

price converges to a Brownian motion

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Comparison with other zero-intelligence models

As an alternative, similar models where the cancellation rate is not proportional have been considered. Apparently, their performance is not as good.

5 10 15 20 25 30 35 40 45 50 55 60 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 x 10

• 6

Var[P(t + lag) - P(t)] / lag

FTE.PA mide price (March 2011) Estimated model

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Hawkes processes

Hawkes processes are quite handy in describing the mutual excitation of the arrivals of limit and market orders. Process Nj

s has a stochastic intensity λj t such that

λj

t = λj 0 + D

• p=1

t φjp(t − s)dNp

s .

(1) A typical choice is the exponential kernel φjp(s) = αjp exp(−βjps) (2) leading to Markovian processes.

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Spread distribution

Modelling made better: spread distribution with Hawkes processes for the arrival rates

Figure: Distribution of the B/A spread

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Zero-intelligence models Interplay between liquidity providing and taking

Mathematical properties

Main mathematical properties

1 Under the usual stationarity conditions for the intensities,

there exists a Lyapunov function V = |ai| + |bi| + Ukλk and the LOB converges to a stationary distribution

2 The rescaled, centered price converges to a Brownian motion

The situation is quite similar to the case of Poisson arrivals, because the proportional cancellation rate remains bounded from below.

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

Plan

1

What is a limit order book ?

2

Empirical properties Competitive liquidity Predictability

3

Mathematical modelling Zero-intelligence models Interplay between liquidity providing and taking

4

Conclusion

5

References

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

Conclusion

1 There exists a large body of empirical results 2 Modelling based on simple point processes leads to realistic

behaviour

3 The low frequency price dynamics is consistent with diffusion

models

4 The key to better modelling is to incorporate information on

the behaviour of market participants, see [Challet et al., 2011]

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

Plan

1

What is a limit order book ?

2

Empirical properties Competitive liquidity Predictability

3

Mathematical modelling Zero-intelligence models Interplay between liquidity providing and taking

4

Conclusion

5

References

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

References I

[Muni Toke, 2009] I. Muni-Toke [Abergel, Jedidi 2011] F. Abergel, A. Jedidi, A mathematical approach to order-book modelling [Abergel, Jedidi 2011] F. Abergel, A. Jedidi, in preparation [Smith et al., 2003] Smith, Farmer, Guillemot, Krishnamurthy, Statistical theory of the continuous double auction, [Eisler et al., 2011] Z. Eisler, J.-P. Bouchaud, J. Kockelkoren, Model for the impact of all order book events [Zheng et al., 2011] B. Zheng, E. moulines, F. Abergel, Price jump detection in a limit order book

Fr´ ed´ eric Abergel Order book modelling

What is a limit order book ? Empirical properties Mathematical modelling Conclusion References

References II

[Pomponio, Abergel, 2011] F. Pomponio, F. Abergel, Multiple-limits trades: empirical facts and applications to lead-lag measures [Chakraborti et al., 2011] A. Chakraborti, M. Patriarca, I. Muni Toke, F. Abergel, Econophysics: empirical facts [Challet et al., 2011] D. Challet, D. Morton de la Chappelle, Collective portfolio optimization in brokerage data: the role of transaction cost structure

Fr´ ed´ eric Abergel Order book modelling