What is a limit order book ? Empirical properties Mathematical modelling Conclusion References Statistical and mathematical properties of limit order 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 What is a limit order book ? 1 Empirical properties 2 Competitive liquidity Predictability Mathematical modelling 3 Zero-intelligence models Interplay between liquidity providing and taking Conclusion 4 References 5 Fr´ ed´ eric Abergel Order book modelling
What is a limit order book ? Empirical properties Mathematical modelling Conclusion References (1) initial state (2) liquidity is taken (3) wide spread 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 -10 -10 -10 -20 -20 -20 -30 -30 -30 -40 -40 -40 95 100 105 110 95 100 105 110 95 100 105 110 (4) liquidity returns (5) liquidity returns (6) final state 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 -10 -10 -10 -20 -20 -20 -30 -30 -30 -40 -40 -40 Fr´ ed´ eric Abergel Order book modelling 95 100 105 110 95 100 105 110 95 100 105 110
What is a limit order book ? Empirical properties Competitive liquidity Mathematical modelling Predictability Conclusion References Plan What is a limit order book ? 1 Empirical properties 2 Competitive liquidity Predictability Mathematical modelling 3 Zero-intelligence models Interplay between liquidity providing and taking Conclusion 4 References 5 Fr´ ed´ eric Abergel Order book modelling
What is a limit order book ? Empirical properties Competitive liquidity Mathematical modelling Predictability Conclusion References 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 Competitive liquidity Mathematical modelling Predictability Conclusion References 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 Competitive liquidity Mathematical modelling Predictability Conclusion References 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 Competitive liquidity Mathematical modelling Predictability Conclusion References 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 Competitive liquidity Mathematical modelling Predictability Conclusion References Trade sign prediction Of the utmost interest is the prediction of the sign of the next trade. The LOB provides us with helpful information. Conditional probability of TradeSign, BNPP.PA 1.0 Depth=1 0.8 Conditional probability 0.6 0.4 0.2 0.0 0 5 10 15 20 Bid−ask volume ratio Fr´ ed´ eric Abergel Order book modelling
What is a limit order book ? Empirical properties Competitive liquidity Mathematical modelling Predictability Conclusion References 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 Competitive liquidity Mathematical modelling Predictability Conclusion References Upward jump Prediction of an upward jump, i.e. , a multiple-limit trade on the ask side. TECF.PA, upward jump, Sep 2009 37 31 19 5 0 2 BidSize1 AskGap1 AskGap2 BidSize2 0 BidGap2 BidGap1 −2 Coefficients AskSize1 −4 −6 −8 −10 −12 Fr´ ed´ eric Abergel Order book modelling −10 −8 −6 −4 −2
What is a limit order book ? Empirical properties Competitive liquidity Mathematical modelling Predictability Conclusion References Downward jump Similar results hold for downward jumps. TECF.PA, downward jump, Sep 2009 36 36 22 6 0 BidGap1 BidGap2 AskSize1 AskSize3 0 AskGap2 AskGap1 Coefficients BidSize1 −5 −10 −10 −8 −6 −4 −2 Fr´ ed´ eric Abergel Order book modelling Log Lambda
What is a limit order book ? Empirical properties Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References Plan What is a limit order book ? 1 Empirical properties 2 Competitive liquidity Predictability Mathematical modelling 3 Zero-intelligence models Interplay between liquidity providing and taking Conclusion 4 References 5 Fr´ ed´ eric Abergel Order book modelling
What is a limit order book ? Empirical properties Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References 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 Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References Zero-intelligence In [Smith et al. , 2003], the authors describe and analyze an elementary LOB model, with ∆ P dL i ± λ i ± t L τ 1 λ ± dM t M τ a i b i λ i + dC i ± τ , λ i − t C C τ τ is the lot size, ∆ P , the tick size, and the a i , b i ’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 Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References Model description The dynamics can be decomposed into two basic types of events: 1 a change in one of the a i ’s or b i ’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 Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References Diffusive behaviour Such a simple model already captures some empirically observed properties of the price dynamics: Long time diffusive behaviour... x 10 -4 2 Var[P(t + lag) - P(t)] 1 0 50 100 150 200 250 300 Time lag (sec) 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 Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References Signature plot ... and microstructural effects, the signature plot. -6 1.25 x 10 FTE.PA mide price (March 2011) Estimated model 1.2 1.15 1.1 Var[P(t + lag) - P(t)] / lag 1.05 1 0.95 0.9 0.85 0.8 5 10 15 20 25 30 35 40 45 50 55 60 Time lag (sec) Fr´ ed´ eric Abergel Order book modelling
What is a limit order book ? Empirical properties Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References Mathematical results Main mathematical properties 1 Theorem 6.1 in [Abergel, Jedidi 2011] There exists a Lyapunov function V = � | a i | + � | b i | 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 Zero-intelligence models Mathematical modelling Interplay between liquidity providing and taking Conclusion References 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. -6 2.6 x 10 FTE.PA mide price (March 2011) Estimated model 2.4 2.2 2 Var[P(t + lag) - P(t)] / lag 1.8 1.6 1.4 1.2 1 Fr´ ed´ eric Abergel Order book modelling 0.8 5 10 15 20 25 30 35 40 45 50 55 60
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