Microstructure, mutually exciting processes and market impact E. Bacry, S. Microstructure, mutually exciting processes Delattre, A. Iuga, M.H., and market impact J.F. Muzy Introduction: inference across scales E. Bacry, S. Delattre, A. Iuga, M.H., J.F. Muzy The Hawkes processes approach CMAP-X, Paris 7, CREST and Paris-Est, Universit´ e de Cort´ e Scaling limits and (financial) interpretations Paris, March 1, 2012 Market impact and mutually exciting processes
Outline Microstructure, mutually exciting processes and market impact 1 Introduction: inference across scales E. Bacry, S. Delattre, A. Iuga, M.H., J.F. Muzy 2 The Hawkes processes approach Introduction: inference across scales The Hawkes 3 Scaling limits and (financial) interpretations processes approach Scaling limits and (financial) 4 Market impact and mutually exciting processes interpretations Market impact and mutually exciting processes
Modelling across scales Microstructure, mutually exciting processes and market impact Suppose we have discrete (financial) data E. Bacry, S. Delattre, A. X 0 , X ∆ , X 2∆ , . . . , over [0 , T ] Iuga, M.H., J.F. Muzy of a continuous time process X t with t ∈ [0 , T ]. Introduction: inference across scales Depending on the relative sizes of The Hawkes processes ∆ and T as n = ⌊ T / ∆ ⌋ → ∞ approach Scaling limits and (financial) (whenever we conduct an asymptotic study) we are led to interpretations model the data differently. Market impact and mutually exciting processes
Financial data modelling Microstructure, mutually exciting processes and market impact E. Bacry, S. Delattre, A. Price processes behave differently at different scales: Iuga, M.H., J.F. Muzy - Coarse scales (daily, hourly): continuous or jump diffusions, Introduction: - Fine scales (tick data): marked point processes. inference across scales Several HF stylised facts: The Hawkes - Microstructure effect known as “mean-reversion” in d = 1, processes - Covariation instability: Epps effect for d ≥ 2. approach Scaling limits and (financial) interpretations Market impact and mutually exciting processes
Modelling macroscopic data Microstructure, mutually exciting processes and market impact 125 E. Bacry, S. 120 Delattre, A. Iuga, M.H., J.F. Muzy 115 Bund Introduction: 110 inference across scales 105 The Hawkes processes approach 0 500 1000 1500 Time Scaling limits and (financial) interpretations Market impact Figure: German 10Y Bund (FGBL) sampled with ∆ = 1 day (traded and mutually exciting price), 04 Avr. 1999 to 06 Dec. 2005. The candidate for X may be a processes continuous Ito semimartingale dX t = b t dt + σ t dB t that we observe at times i ∆.
Ruling out discretely observed diffusions on fine scales Microstructure, mutually If X t is observed over [0 , T ] at times 0 , ∆ , 2∆ , . . . , define exciting processes and the signature plot as market impact E. Bacry, S. Delattre, A. � � � � � 2 . ∆ � V ∆ X t := X i ∆ − X ( i − 1)∆ Iuga, M.H., J.F. Muzy i ∆ ≤ t Introduction: inference If X is an Itˆ o continuous semimartingale with across scales dX t = b t dt + σ t dB t we have The Hawkes processes approach � t P Scaling limits σ 2 V ∆ { X } t → s ds and (financial) interpretations 0 √ Market impact and mutually as ∆ → 0 with accuracy ∆. exciting processes This suggests to pick ∆ as small as possible... but
Signature plot Microstructure, mutually exciting processes and market impact 0.045 E. Bacry, S. Delattre, A. Iuga, M.H., 0.04 J.F. Muzy Introduction: 0.035 inference V � t across scales (ticks) The Hawkes 0.03 processes approach Scaling limits 0.025 and (financial) interpretations Market impact 0.02 50 100 150 200 � t (seconds) and mutually exciting processes Figure: ∆ � V ∆ for FGBL (43 days, 9-11 AM) on Last Traded Ask.
Coarse-to-fine modelling Microstructure, mutually exciting processes and 115.65 market impact 115.60 E. Bacry, S. Delattre, A. Iuga, M.H., 115.55 J.F. Muzy value 115.50 Introduction: inference 115.45 across scales The Hawkes 115.40 processes approach 0 5000 10000 15000 20000 25000 30000 Scaling limits time and (financial) interpretations Figure: FGBL, 06 Feb 2007, 08:30-17:00 (UTC) sampled with ∆ = 1 Market impact and mutually second. The candidate for the underlying process X is rather a exciting processes marked point process that we observe at times i ∆.
Coarse-to-fine modelling (cont.) Microstructure, mutually exciting processes and market impact 115.56 E. Bacry, S. Delattre, A. 115.54 Iuga, M.H., J.F. Muzy 115.52 value Introduction: 115.50 inference across scales 115.48 The Hawkes processes 115.46 approach 0 500 1000 1500 2000 2500 3000 3500 Scaling limits time and (financial) interpretations Figure: FGBL, 06 Feb 2007, 09:00–10:00 (UTC) 1 data every second. Market impact and mutually The point process approach suggestion is even more pronounced here. exciting processes The underlying process looks more complex than a simple CTRW.
Toward more realistic price models Microstructure, mutually exciting processes and market impact E. Bacry, S. We look for a “simple” multivariate price model Delattre, A. Iuga, M.H., Defined in continuous time with discrete values on a J.F. Muzy microscopic scale. Introduction: inference That may incorporate microstructure effects. across scales That diffuses on a macroscopic scale. The Hawkes processes approach That enables to tackle other HF issues such as Price Scaling limits Impact. and (financial) interpretations Market impact and mutually exciting processes
Alternative approaches Microstructure, mutually exciting processes and Latent price approach market impact In statistics: Gloter and Jacod (2001), Munk and E. Bacry, S. Delattre, A. Schmiedt-Hieber (2009), Reiß (2010) Iuga, M.H., J.F. Muzy In financial econometrics: Ait-Sahalia, Mykland and Zhang (2003 to 2006). Introduction: inference And many more... Podolkii, Vetter, Jacod, Mykland, across scales Zhang, Bandi, Russell, Diebold, Strasser, The Hawkes Barndorff-Nielsen, Hansen, Lund, Shepard, processes approach Other approaches for modelling microstructure: Scaling limits and (financial) Engle Russell (2002), Hautsch (2006), Robert and interpretations Rosenbaum (2009) Market impact and mutually Econophysics literature Order book oriented modelling... exciting processes
Point process approach Microstructure, mutually exciting processes and market impact Price process = marked point process. E. Bacry, S. Delattre, A. Marks : jumps up/down by 1 tick, Iuga, M.H., J.F. Muzy Jump times: time stamps of price changes. The price process is the result (sum) of a “upward change Introduction: inference or price” and a “downward change of price” (two counting across scales processes). The Hawkes processes approach By coupling random intensities of the counting processes, Scaling limits we create local oscillations that reproduce empirical and (financial) interpretations microstructure effects. Market impact and mutually exciting processes
Hawkes processes Microstructure, mutually exciting processes and market impact A (linear) Hawkes process is a counting process N t E. Bacry, S. constructed via its stochastic intensity Delattre, A. Iuga, M.H., J.F. Muzy � t λ ( t ) = µ + φ ( t − s ) dN s Introduction: inference 0 across scales The Hawkes Mathematically tractable choice: φ ( x ) = α e − β x with processes approach simple interpretation. Scaling limits � t 0 φ ( t − s ) dN s = � and (financial) (One has T n < t φ ( t − T n ).) interpretations Non-explosion constraint: � φ � L 1 < 1. Market impact and mutually exciting processes
Price model in dimension 1 Microstructure, Continuous-time price model living on a tick-grid: mutually exciting processes and market impact X t = N + t − N − t E. Bacry, S. Delattre, A. with N ± Iuga, M.H., t Hawkes processes with respective stochastic intensities J.F. Muzy � λ + ( t ) [0 , t ) e − β ( t − s ) dN − Introduction: = µ + + α s inference across scales � The Hawkes λ − ( t ) [0 , t ) e − β ( t − s ) dN + = µ − + α processes s approach Scaling limits and (financial) µ ± : exogeneous intensity. interpretations Market impact α et β : mutually exciting intensities generating a and mutually exciting “mean-reverting effect” for S t . processes α e − β x � φ ( x ) with � φ � L 1 < 1 in the sequel.
Scaling limits Microstructure, mutually exciting processes and market impact E. Bacry, S. Delattre, A. First step: Iuga, M.H., J.F. Muzy 1 closed-form formulas for the mean “signature plot” when Φ( x ) = α e − β x (through the explicit computation of the Introduction: inference Bartlett spectrum, case with stationary increments) in across scales dimension 1 and 2. The Hawkes 2 Statistical fits and discussion of further data filtering. processes approach Second step: scaling (diffusive) limit for arbitrary φ . Scaling limits and (financial) interpretations Market impact and mutually exciting processes
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