Carnets d ordres pilotés par des processus de Hawkes

Transcription

1 Carnets d ordres pilotés par des processus de Hawkes workshop sur les Mathématiques des marchés financiers en haute fréquence Frédéric Abergel Chaire 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

2 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 2/43

3 What is a limit order book? The limit order book is the list, at a given time, of all buy and sell limit orders, with their corresponding prices and volumes The order book evolves over time according to the arrival of new orders Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 3/43

4 What is a limit order book? The limit order book is the list, at a given time, of all buy and sell limit orders, with their corresponding prices and volumes The order book evolves over time according to the arrival of new orders 3 main types of orders: limit order: specify a price at which one is willing to buy (sell) a certain number of shares market order: immediately buy (sell) a certain number of shares at the best available opposite quote(s) cancellation order: cancel an existing limit order Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 3/43

5 What is a limit order book? The limit order book is the list, at a given time, of all buy and sell limit orders, with their corresponding prices and volumes The order book evolves over time according to the arrival of new orders 3 main types of orders: Price dynamics limit order: specify a price at which one is willing to buy (sell) a certain number of shares market order: immediately buy (sell) a certain number of shares at the best available opposite quote(s) cancellation order: cancel an existing limit order The price dynamics becomes a by-product of the order book dynamics Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 3/43

6 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 4/43

7 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 5/43

8 Limit order book events Timestamp Side Level Price Quantity B B B B B A B B B B A Table: Tick by tick data file sample Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 6/43

9 Trades Timestamp Last Last quantity Table: Trades data file sample. These data are made available (for free) to the scientific community Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 7/43

10 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 whose timestamps are in [t Tr t, t Tr + t], where t Tr is the trade timestamp and t is a predefined time window; 2 Match the trade to the first likely market order with the same price and volume 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 40 stocks. As a byproduct, one gets the sign of each matched trade. Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 8/43

11 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 9/43

12 Arrival time of orders Empirical density BNPP.PA Lognormal Exponential Weibull Interarrival time Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 10/43

13 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 11/43

14 1 0.1 Power law x -1.4 BNPP.PA FTE.PA RENA.PA SOGN.PA 0.01 Probability functions e-05 1e-06 1e Lifetime for cancelled limit orders Figure: Distribution of estimated lifetime of cancelled limit orders. 1 Institute for Pure and Applied Mathematics Hawkes-process-driven LOBPower law x /43

15 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 13/43

16 3 Number of market orders submitted in t=5 minutes BNPP.PA BNPP.PA quadratic fit FTE.PA FTE.PA quadratic fit Time of day (seconds) Figure: Normalized average number of market orders in a 5-minute interval. 3.5 Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 14/43

17 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 15/43

18 Competitive liquidity Muni Toke (2011) Figure: Evidence of liquidity providing Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 16/43

19 Competitive liquidity Muni Toke (2011)Anane et al. (2015) Figure: Evidence of liquidity taking Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 17/43

20 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 18/43

21 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... Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 19/43

22 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 Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 19/43

23 Some notations The order book is represented by a finite-size vector of quantities X(t) := (a(t); b(t)) := (a 1 (t),..., a K (t); b 1 (t),..., b K (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 +P B 2 : mid-price A(p), B(p): cumulative number of sell (buy) orders up to price level p Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 20/43

24 Notations a 7 B P S A P a 8 a a 6 a 9 a1 a 2 a3 a4 a 5 b 6 b 4 b 3 b 2 b 1 b 5 b 9 b b 7 b 8 Figure: Order book notations Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 21/43

25 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 22/43

26 Zero-intelligence model Inspired by Smith et al. (2003), Abergel and Jedidi (2013) analyzes an elementary LOB model where the events affecting the order book are described by independent Poisson processes: M ± : arrival of new market order, with intensity λm± q ; L ± : arrival of a limit order at level i, with intensity λl ± P i ; i q C ± a : cancellation of a limit order at level i, with intensity λ C+ i i i q and b λ C i i q Cancellation rate is proportional to the outstanding quantity at each level. Under these assumptions, (X(t)) t 0 is a Markov process with state space S Z 2K. Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 23/43

27 Order book dynamics i 1 da i (t) = q a k dm + (t) + qdl + (t) qdc + (t) i i k=1 + K + (J M (a) a) i dm (t) + (J L i (a) a) i dl i (t) + K i=1 (J C i (a) a) i dc i (t), db i (t) = similar expression, where J M±, J L ± i, and J C± i are shift operators corresponding to the effect of order arrivals on the reference frame i=1 Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 24/43

28 The shift operator For instance, the shift operator corresponding to the arrival of a sell market order is J M (a) = 0, 0,..., 0, a } {{ } 1, a 2,..., a K k, k times with p k := inf{p : b j > q} inf{p : b p > 0}. j=1 Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 25/43

29 Infinitesimal generator Lf (a; b) = λ M+ (f ( [a i (q A(i 1)) + ] + ; J M+ (b) ) f) + + K i=1 K i=1 λ L + i (f ( a i + q; J L + i (b) ) f) λ C+ i a i (f ( a i q; J C+ i (b) ) f) + λ M ( f ( J M (a); [b i + (q B(i 1)) + ] ) f ) + + K i=1 K i=1 λ ( L i (f J L i (a); b i ) q f) λ ( C i b i (f J C i (a); b i + ) q f) Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 26/43

30 Stability of the order book Stationary order book distribution Abergel and Jedidi (2013) If λ C = min 1 i N {λ C± } > 0, then i (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 of convergence of the order book to its stationary state is exponential. The proof relies on the use of a Lyapunov function The proportional cancellation rate helps a lot... and so do the boundary conditions! Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 27/43

31 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 the asymptotic behaviour of the price, based on Foster-Lyapunov-type criteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and the convergence of martingales Ethier and Kurtz (2005): the evolution of the price is dp t = K i=1 F i(x t )dn i t ; the rescaled, centered price is Pn t P nt nt 0 K i=1 F i(x s )λ i ds n its predictable quadratic variation is < Pn, Pn > t = ergodicity ensures the convergence of nt 0 nt 0 K i=1 (F i(x s )) 2 λ i ds n K i=1 (F i(x s )) 2 λ i ds as n nt Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 28/43

32 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 the asymptotic behaviour of the price, based on Foster-Lyapunov-type criteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and the convergence of martingales Ethier and Kurtz (2005): the evolution of the price is dp t = K i=1 F i(x t )dn i t ; the rescaled, centered price is Pn t P nt nt 0 K i=1 F i(x s )λ i ds n its predictable quadratic variation is < Pn, Pn > t = ergodicity ensures the convergence of nt 0 nt 0 K i=1 (F i(x s )) 2 λ i ds n K i=1 (F i(x s )) 2 λ i ds as n nt This approach is easily extended to state-dependent intensities A similar approach - in a somewhat different modelling context - is used in recent papers by Horst and co-authors Horst and Paulsen (2015). Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 28/43

33 A general approach made precise I Write as above and its compensator Define and let α = lim a.s. 1 t + t P t = P 0 + Q t = i i i t 0 t 0 F i (X s )dn i s λ i F i (X s )ds. h = λ i F i (X) t 0 i λ i (F i (X s ))ds = h(x)π(dx). Finally, introduce the solution g to the Poisson equation Lg = h α Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 29/43

34 A general approach made precise II and the associated martingale t Z t = g(x t ) g(x 0 ) Lg(X s )ds g(x t ) g(x 0 ) Q t + αt. 0 Then, the deterministically centered, rescaled price P n t P nt αnt n converges in distribution to a Wiener process σw. The asymptotic volatility σ satisfies the identity σ 2 1 = lim t + t i t 0 λ i ((F i i (g))(x s )) 2 ds λ i ((F i i (g))(x)) 2 Π(dx). where i (g)(x) denotes the jump of the process g(x) when the process N i jumps and the limit order book is in the state X. i Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 30/43

35 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 31/43

36 Hawkes processes Zero-intelligence models fail to capture the dependencies between various 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). Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 32/43

37 Hawkes processes Hawkes processes provide an ad hoc tool to describe the mutual excitations of the arrivals of different types of orders. In D dimensions, the process N j s has a stochastic intensity λ j t such that λ j t = λ j 0 + D p=1 t A simplifying choice is the exponential kernel 0 φ jp (t s)dn p s. (2.1) φ jp (s) = α jp exp( β jp s) (2.2) leading to markovian processes. A classical result states that the process is stationary iff the spectral radius of the matrix is < 1, see Brémaud and Massoulié (1996). [ α jp β jp ] (2.3) Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 33/43

38 Introducing dependence in the order flow As an example, Muni Toke (2011) study variations of the model below A better order book model t λ M (t) = λ M 0 + α MM e βmm(t s) dn M (s), 0 t λ L (t) = λ L 0 + α LL e β LL (t s) dn L (s) + t 0 0 α ML e β ML (t s) dn M (s). MM and LL effect for clustering of orders ML effect as observed on data No global LM effect observed on data Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 34/43

39 Infinitesimal generator LF( a ; b ; µ ) = λ M+ (F + + K i=1 K i=1 ( [a i (q A(i 1)) + ] + ; J M+ ( b ); µ + M+ ( ) µ ) F) ( λ L + i (F a i + q; J L + i ( b ); µ + L + i ( ) µ ) F) ( λ C+ i a i (F a i q; J C+ i ( ) b ) F) + λ M ( F ( J M ( a ); [b i + (q B(i 1)) + ] ; µ + M ( µ ) ) F ) + + K i=1 K i=1 D i,j=1 λ L i ( (F J L i ( a ); b i q; µ + L i ( µ ) ) F) λ ( C i b i (f J C i ( a ); b i + ) q f) β ij µ ij F µ ij. (2.4) Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 35/43

40 Large-time behaviour Results similar to those obtained in the zero-intelligence case hold Abergel and Jedidi (2015) Large-time behaviour for Hawkes-process driven LOB Under the usual stationarity conditions for the intensities, there exists a Lyapunov function V = a i + b i + U k λ k and the LOB converges exponentially to its stationary distribution Π The rescaled, (deterministically) centered price converges to a Wiener process The proofs rely on the same decomposition as in the Poisson arrival case, thanks to the fact that the proportional cancellation rate remains bounded away from zero Some extra care is required to prove that the solution to the Poisson equation is in L 2 (Π(dx)) Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 36/43

41 Plan 1 Stylized facts of limit order books A word on data Arrival times of orders Cancellation of orders Intraday seasonality Competitive liquidity 2 Mathematical modelling of limit order books Zero-intelligence models Stability and long-time dynamics Large-scale limit of the price process Modelling dependencies Hawkes processes Application to order book modelling Stability and long-time dynamics 3 Simulation with Hawkes processes Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 37/43

42 Hawkes model In Muni Toke (2011), the flow of limit and market orders are modelled by Hawkes processes N L and N M with intensities λ and µ defined as follows: t µ M (t) = µ M 0 + λ L (t) = λ L t 0 α MM e β MM(t s) dn M s, α LM e β LM(t s) dn M s + t 0 α LL e β LL (t s) dn L s Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 38/43

43 Numerical results on the order book: fitting Fitting and simulation Model µ 0 α MM β MM λ 0 α LM β LM α LL β LL HP LM MM MM LL MM LM MM LL LM Common parameters: m P = 2.7, 1 νp = 2.0, 1 sp = V 1 mv = λ C = 1.35, δ = Table: Estimated values of parameters used for simulations. Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 39/43

44 Impact on arrival times Figure: Empirical density function of the distribution of the interval times between a market order 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 Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 40/43

45 Impact on the bid-ask spread Empirical BNPP.PA Homogeneous Poisson Hawkes MM Hawkes MM+LM Figure: Empirical density function of the distribution of the bid-ask spread for three simulations, 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). Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 41/43

46 References I Abergel, F. and Jedidi, A. (2013). A mathematical approach to order book modelling. International Journal of Theoretical and Applied Finance, 16(5). Abergel, F. and Jedidi, A. (2015). Long time behaviour of a Hawkes process-based limit order book. Anane, M., Abergel, F., and Lu, X. F. (2015). Mathematical modeling of the order book: New approach of predicting single-stock returns. Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Probab., 24: Ethier, S. N. and Kurtz, T. G. (2005). Markov processes characterization and convergence. Wiley. Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Annals of Probability, 24(2): Horst, U. and Paulsen, M. (2015). A law of large numbers for limit order books. Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 42/43

47 References II Meyn, S. and Tweedie, R. (1993). Stability of markovian processes III: Foster-lyapunov criteria for continuous-time processes. Advances in Applied Probability, 25: Muni Toke, I. (2011). Market Making in an order book model and its impact on the spread. In Abergel, F., Chakrabarti, B. K., Chakraborti, A., and Mitra, M., editors, Econophysics of Order-driven Markets, pages Springer. Muni Toke, I. and Pomponio, F. (2012). Modelling trade-throughs in a limited order book using Hawkes processes. Economics: The Open-Access, Open-Assessment E-Journal, 6. Smith, E., Farmer, J. D., Gillemot, L., and Krishnamurthy, S. (2003). Statistical theory of the continuous double auction. Quantitative Finance, 3(6): Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 43/43