Statistical Inference for Ergodic Point Processes and Applications to Limit Order Book

Transcription

1 Statistical Inference for Ergodic Point Processes and Applications to Limit Order Book Simon Clinet PhD student under the supervision of Pr. Nakahiro Yoshida Graduate School of Mathematical Sciences, University of Tokyo CREST, Japan Science and Technology Agency Toyama Symposium October 2, Introduction Most financial transactions take place nowadays in electronic markets. Participating to continuous-time double auctions, agents can freely send buying or selling orders at different prices that are automatically matched according precise rules. As this matching process is rather complex and the orders sent by market participants are asynchronous, they are centralized in an Electronic Limit Order Book (also denoted LOB), waiting to be executed according to their price and time priority. A LOB is thus a multidimensional queuing system, each dimension representing a price level, and each queue containing the waiting orders that have not been executed yet, sorted by their arrival time. Agents can then interact with this dynamical system via three elementary mechanisms. They may submit a buying (resp. selling) limit order that will increase the size of one queue on the bid (resp. ask ) side of the LOB. They also may send a buying (resp. selling) market order that will immediately consume the corresponding liquidity at the best available price. Finally they can submit cancellations orders to remove one of their latent limit order in the LOB. Driven by these simple events, the characteristics of the Order Book, such as its mid price, its shape, or the number of orders submitted in a given time window are subject to random fluctuations as time passes. As the macroscopic price movements of an asset are determined by the evolution of its LOB through time, understanding the stochastic structure of this object is a fundamental issue. It is also a way to describe in a high-dimensional context the microstructure phenomena related to the stock price movements. Indeed, phenomena such as the Epps effect, that are traditionally represented by a noise process in many studies that are only interested in the modelling of the price process itself, are captured when the whole limit order book is described. As the availability of high-frequency financial data has made in-depth analysis of LOB dynamics accessible, it is now possible to use statistical tools in order to describe either empirical facts about their shapes and their evolution 14, 4], or to design and select stochastic models that are able to reproduce some of those This work was in part supported by CREST Japan Science and Technology Agency; Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No (Scientific Research), No (Challenging Exploratory Research); NS Solutions Corporation; and by a Cooperative Research Program of the Institute of Statistical Mathematics. 1

2 stylized facts. For the latter, many studies have been conducted in order to model the stochastic behavior of limit order books, the simplest approaches being zero-intelligence models, in which interarrival times are exponentially distributed. The seminal work about this basic representation 2] has been followed by many extensions, see for example 1, 6, 8, 7, 15]. Lately, more complex dynamics have been studied, including Markovian models 11], Hawkes process driven bid-ask prices 3, 22], or even mixture of those approaches 2]. Since a LOB is mechanically driven by the orders that are submitted through time, many authors choose to see a LOB through the stochastic structure of the interarrival times between two events. In other words, a Limit Order Book is often described as a high-dimensional point process, whose components are integer measures counting the waiting times between two orders of the same type, and at the same price level. In a parametric context, estimating the parameter θ based on the observations is thus a very general and crucial issue that may take place in two distinct asymptotics. As, at least for liquid stocks, a tremendously large number of events happen during short periods of times, the heavy traffic limit (very large number of events on a finite period of time) seems to be a good way to construct consistent estimators. In 17], a sequence of multivariate point processes is thus assumed to be observable on a finite time window. Under suitable assumptions on the sequence of stochastic intensities itself, it is shown that even in this non-ergodic context it is possible to conduct the quasi likelihood analysis procedure (QLA for short). In particular, both quasi maximum likelihood estimator (QMLE) and quasi bayesian estimator (QBE) are consistent and asymptotically mixed normal. On the other hand, in this work, we are interested in the long run characteristics of the LOB seen as a point process. as the time parameter T tends to infinity, assuming that the LOB satisfies suitable ergodicity assumptions, we aim at taking advantage of this regularity in order to derive the asymptotic properties of the QMLE and the QBE. This problematic is of course not new since the consistency and the asymptotic normality of the maximum likelihood estimator for ergodic stationary point processes was shown a few decades ago in 16] and 18]. Furthermore, maximum likelihood estimations have also been empirically conducted for the abovementioned models, but the fact that the point process is ergodic is sometimes unclear. In this article we thus give general ergodicity and regularity assumptions for point processes under which all the results from the general QLA can be derived. In particular, we do not necessarily require the stationarity of the point process. We then give some applications and examples that are picked from the literature, and that verify those general conditions. 2 Multivariate point process Let B = (Ω, F, F, P), F = (F t ) t R+, be a stochastic basis, and assume that we are given a multidimensional point process N t = (N α t ) α I, I = {1,..d}. For the sake of simplicity N is supposed to be defined on R +, N = (N t ) t R+, N =, and its components N α are supposed to have no common jumps. For some finite dimensional compact space Θ R n, we consider the family of increasing processes Λ t (θ) = t λ(s, θ)ds, and we assume that there exists θ Θ such that λ(t, θ ) is the F t -intensity of N t. The process Ñ t = N t Λ t (θ ) is thus a local martingale. All along the paper, if x designates a real number, a vector or a matrix, x = i x i. If X is a random variable, X p = E X p ] 1 p. For a measured space (E, E), C b (E, R) is the set of continuous, bounded functions from (E, E) to (R, B(R)). In order to derive the asymptotic properties of the quasi maximum likelihood estimator (QMLE), we consider the following assumptions : A1] The mapping λ : Ω R + Θ R + is F B(R + ) B(Θ)-measurable. Moreover, almost surely : (i) for any θ Θ, s λ(s, θ) is left continuous. 2

3 (ii) for any s R +, θ λ(s, θ) is in C 2 (Θ). A2] The intensity processes and their first derivatives are non degenerate in the following sense : 2 (i) for any p > 1, sup t R+ i= supθ Θ i θ λ(t, θ) p < + (ii) for any p > 1, sup t R+ sup θ Θ λ(t, θ) 1 1 {λ(t,θ) } p < + (iii) For any θ Θ, λ(t, θ) = iff λ(t, θ ) = A3] Ergodicity. There exists a mapping π : C b (R 4, R) Θ R such that for any (ψ, θ) C b (R 4, R) Θ the following convergence holds : 1 T ψ(λ(s, θ ), λ(s, θ), θ λ(s, θ), T θλ(s, 2 θ))ds P π(ψ, θ) (1) In section 3 we define the likelihood function and the QMLE. We then state its consistency and its asymptotic normality. In section 3.4 we finally derive the convergence of moments of both the QMLE and the QBE under stronger versions of A1]-A4]. 3 Statistical inference 3.1 Definitions and asymptotic properties of the QMLE For any θ Θ, the log-likelihood function can be expressed as (see 9], proposition 7.2.III) : l T (θ) = α I log(λ α (s, θ))dn α s α I λ α (s, θ)ds (2) up to an addition by a term independent of θ. A quasi maximum likelihood estimator ˆθ T asymptotically maximizes the rescaled likelihood function in the following sense: Put, l T (ˆθ T ) T l T (θ) max o P (1) (3) θ Θ T The key lemma to show the consistency of the QMLE is the following Lemma 3.1. There exists Y(θ) such that Y T (θ) = 1 T (l T (θ) l T (θ )) (4) sup Y T (θ) Y(θ) θ Θ The proof of the above lemma makes extensive use of the ergodicity assumption. Indeed, Y has actually the following form : Y(θ) = lim T + 1 T α I ] λ α (s, θ) λ α (s, θ ) log λα (s, θ) λ α (s, θ ) λα (s, θ ) ds (5) To conclude, we need to assume the following non-degeneracy condition on Y : 3

4 A4] For any θ Θ {θ }, Y(θ). Theorem 3.2. Under A4], a QMLE estimator ˆθ T is consistent. ˆθ T P θ The asymptotic normality of the QMLE can be then obtained following the standard procedure. Writing : θ lt L (θ ) = α I and λ α (s, θ ) 1 θ λ α (s, θ )1 {λ α (s,θ ) }dñ α s (6) 2 θl L T (θ) = α I α I + α I It is easy to deduce the existence of the Fisher information : θ ( λ α (s, θ) 1 θ λ α (s, θ) ) 1 {λ α (s,θ ) }dñ α t (7) ( θ λ α ) 2 (s, θ)λ α (s, θ) 2 λ α (s, θ )1 {λ α (s,θ ) }ds (8) 2 θλ α (s, θ)λ α (s, θ) 1 (λ α (s, θ) λ α (s, θ ))1 {λα (s,θ ) }ds (9) Lemma 3.3. There exists Γ R n n such that : Under A1],A2] and A3], if V T is a ball centered on θ shrinking to {θ }, then : sup T 1 θl 2 T (θ) + Γ P θ V T (1) Once again the ergodicity assumption is crucial here since we have the following representation for Γ : 1 Γ = lim T + T α I Finally, thanks to the following lemma Lemma 3.4. we have : ( ) 1 θ l ut (θ ) d Γ 1 2 (Wu ) u,1] ut u,1] ( θ λ α ) 2 (s, θ )λ α (s, θ ) 1 1 {λ α (s,θ ) }ds (11) where W is a standard Brownian motion, and d designs the convergence in distribution of the whole process. This reduces to show a functional central limit theorem. From the previous lemmas we easily deduce the following convergence. Theorem 3.5. Put θ T a QMLE estimator. we have : T (θt θ ) d Γ 1 2 N Where N is a standard normal distribution, and provided that Γ is invertible. 4

5 3.2 QLA and convergence of moments In this section we slightly strengthen the previous assumptions about ergodicity and derivability of the intensity process in order to apply the quasi likelihood analysis and get the convergence of moments of the QMLE as well as the convergence of moments for the QBE. We write C (R 4, R) the set of functions ψ : (x, y, z, t) ψ(x, y, z, t) from R 4 to R of class C 1 such that ψ and its gradient are of polynomial growth in (x, y, z, t, 1 x, 1 y ). A1 ] The mapping λ : Ω R + Θ R + is F B(R + ) B(Θ)-measurable. Moreover, almost surely : (i) for any θ Θ, s λ(s, θ) is left continuous. (ii) for any s R +, θ λ(s, θ) is in C 3 (Θ). A2 ] The intensity processes and their first derivatives are non degenerate in the following sense : 3 (i) for any p > 1, sup t R+ i= supθ Θ θ iλ(t, θ) p < + (ii) for any p > 1, sup t R+ supθ Θ λ(t, θ) 1 1 {λ(t,θ) } p < + (iii) For any θ Θ, λ(t, θ) = iff λ(t, θ ) = A3 ] Ergodicity. There exists a mapping π : C (R 4, R) Θ R and there exists < γ < 1 2 such that for any (ψ, θ) C (R 4, R) Θ the following convergence holds : sup T γ 1 T ψ(λ(s, θ ), λ(s, θ), θ λ(s, θ), 2 θ Θ T θλ(s, θ))ds π(ψ, θ) (12) p A4 ] Define χ as: We assume that χ = inf Y(θ) θ Θ\{θ } θ θ 2 χ > For a given prior density p on the space Θ, we define the quasi Bayesian estimator as follows: ] 1 θ T = exp(l T (θ))p(θ)dθ θexp(l T (θ))p(θ)dθ (13) Θ Θ We can then apply results from 21] about quasi-likelihood analysis based on polynomial type large deviations. To derive our main result about convergence of moments we represent the likelihood process as follows. For any u U T = {u R k θ + T 1/2 u Θ}, we write θ u = θ + T 1/2 u and define the likelihood field : The main result is the following : Z T (u) = exp{l T (θ u ) l T (θ )} Theorem 3.6. Under A1 ]-A4 ], the two following results hold: Polynomial type large deviation inequality for every L >, there exists C L such that : ] P sup Z T (u) e r C L u U T, u >r r L Convergence of moments If ˆθ T is the QMLE and θ T the QBE, we have : E f( ] T (ˆθ T θ )) Ef(Γ 1 2 N )] E f( ] T ( θ T θ )) Ef(Γ 1 2 N )] for any continuous f with polynomial growth. 5

6 4 Applications to Limit Order Book As explained in the introduction, one can always associate to a Limit Order Book process the point process counting that counts the different events that occur through time. If the limit order book is constituted of m N queues stored in the process X t Z m, we can associate m one dimensional counting processes for the limit orders, m for the cancellations, and finally 2 more for the market orders, thus a 2m + 2-dimensional point process (C t, L t, M t ) t. We thus consider a few examples for which the ergodicity condition is satisfied. 4.1 When intensities depend on a Markovian underlying process Some models assume the existence of an underlying process (Y t ) t adapted to the filtration F = (F t ) t R+ that the parametrized intensity processes write as : such λ(t, θ) = h(y t, θ) (14) Example 4.1. In 1], Abergel and Jedidi define a model in which the cancellation intensities are linear functions of the size of the queues of the LOB itself, and other intensities remain constant. In other words, any cancellation intensity at a given level α is of the form : λ C,α (t) = λ C,α X α (t) (15) This feedback enables the authors to show that the vector process containing the sizes of the queues is thus a Markovian V-geometric process (see 13, 12] for a deep insight of this notion). The ergodicity condition A3 ] is thus automatically verified for this simple case. Example 4.2. Another example comes from 19]. In this article, the authors consider that the price process of an asset S t is driven by a Brownian motion. They also assume that the point processes counting the limit orders on the best bid and best ask are Cox processes whose stochastic intensities are functions of the fractional part of S t, Y t := {S t } = S t mod 1. They construct then a non-parametric estimator of the function h using the ergodicity properties of the Markov process Y. It is immediate to see that a parametrized model {h(., θ) θ Θ} would satisfy our ergodicity condition. Example 4.3. Finally, in 11], the model described in 4.1 is generalized. X is defined as a general pure jump Markov process. This, in turn, implies that the intensity vector is a pure function of the LOB state : λ(t, θ) = h(x t, θ) (16) If the intensities of cancellation are sufficiently large when the size of the limits becomes too big, it is shown in 11] that once again the markov Process X is V-geometrically ergodic. 4.2 When intensities follow an exponential Hawkes dynamic In this section we are interested in the special case of multivariate Hawkes process with exponential kernel. Let N t = (Nt α ) α I, I = {1,..d}, N =, be a multidimensional point process and write F N = (Ft N ) t R+, where Ft N = σ{n s s t} is the canonical filtration of N. We say that N is a linear Hawkes process or Hawkes self-exciting process starting from if there exist h : R + R d d + and ν (R +) d such that the Ft N -intensity λ(t) of N writes : 6

7 for any α I. λ α (t) = ν α + β I t h αβ (t s)dn β s Given A = a αβ ] αβ and C = c αβ ] αβ R d d +, we say that N is an exponential Hawkes process if the kernel functions h αβ are of the form : h αβ (s) = c αβ e a αβs Those processes were introduced by Hawkes in 1971, see 1], and were extensively used to model earthquakes and their aftershocks. Lately they have been used in finance to represent the fact that market and limit orders seem to trigger each other, see for example 2]. Let us recall in the following a few results about this family of processes. We define the elementary excitations as : ɛ αβ (t) = t h αβ (t s)dn β s and the matrix E(t) = ] ɛ αβ(t). Finally we define the excitation ratio matrix as Φ = cαβ αβ a αβ ]αβ, and ρ(φ) its spectral radius. We have the following results : Proposition 4.4. Assume that ρ(φ) < 1. Then : (i) There exists a unique stationary point process N defined on the whole real line and on the same probability space as N, such that its F N t -intensity λ verifies : λ α (t) = ν α + β I t h αβ (t s)d N β s (ii) Let S be the shift operator, meaning that for any t R +, S t N = (N s+t ) s R+. Then N is stable in the following sense : S t N D N R+ where D designates the weak convergence associated to the vague topology on integer valued measures. See 5] for deeper explanations about this mode of convergence. Proposition 4.5. Assume that ρ(φ) < 1. Then : (i) The elementary excitation process E is a Markov Process taking values in R d d +. (ii) E is V -geometrically ergodic, and moreover V : R+ d d R + can be chosen as V (ɛ) = e M,ɛ for some M = m αβ ] αβ R d d + and with M, ɛ = α,β I m αβɛ αβ. We now consider an exponential Hawkes process as a model parametrized by the triplet (ν, c, a) R + R d d + R d d +. More precisely we consider a compact parameter state Θ R + R d d + R+ d d such that for any triplet θ = (ν, c, a), for any α, β I, < ν ν α ν < + c c αβ c < + < a a αβ ā < + The following result is a consequence of the previous propositions : 7

8 Theorem 4.6. Assume that ρ(φ) < 1. Then the non-stationary exponential Hawkes process verifies A1 ]-A4 ] and the QLA applies. The main difficulty consists in showing the ergodicity assumption A3 ]. Indeed, given the form of the intensities, the process (λ(s, θ ), λ(s, θ), θ λ(s, θ), θ 2 λ(s, θ)) cannot be written as a pure function of the Markov Process E(t). On the other hand, since the exponential Hawkes intensities are stable and are generated by a fast decreasing kernel, they should be mixing in the following sense : For a given process (X t ) t R+ taking values in some state space E, and a set of functions C from E to R, we say that X is C-mixing if for any φ, ψ C, the quantity ρ u = sup s R + Covφ(X s ), ψ(x s+u )] It turns out that the exponential Hawkes process satisfies the following property, which in turns implies the ergodicity condition A3 ]. B1 ] Mixing (λ(t, θ ), λ(t, θ), θ λ(t, θ), 2 θ λ(t, θ)) t R + is C (R 4, R)-mixing. Moreover the rate ρ verifies : ρ u = o(u ɛ ) for some ɛ > Stability There exists λ such that for any (ψ, θ) C (R 4, R) Θ, 2γ 1 2γ t γ Eψ(λ(t, θ ), λ(t, θ), θ λ(t, θ), 2 θλ(t, θ))] Eψ( λ(θ ), λ(θ), θ λ(θ), 2 θ λ(θ))] In particular, the mapping π writes: π(ψ, θ) = Eψ( λ(θ ), λ(θ), θ λ(θ), 2 θ λ(θ))] A typical example including a Hawkes process is the following : Example 4.7. In 2], the point process associated to the limit and market orders (L t, M t ) t form a stable multivariate exponential Hawkes process. As for 4.1, the cancellation intensities are linear functions of the size of the queues. The ergodicity condition A3 ] is an easy consequence of 4.6. References 1] Abergel, F., Jedidi, A.: A mathematical approach to order book modeling. International Journal of Theoretical and Applied Finance 16(5) (213) 2] Abergel, F., Jedidi, A.: Long time behaviour of a hawkes process-based limit order book. Preprint (215) 3] Bacry, E., Delattre, S., Hoffmann, M.: Modelling microstructure noise with mutually exciting point processes. Quantitative Finance 13(1), (213) 4] Bouchaud, J., Farmer, J., Lillo, F.: How markets slowly digest changes in supply and demand. ArXiv e-prints (28) 5] Bremaud, P., Massoulie, L.: Stability of non-linear hawkes processes. The Annals of Probability 24(3), (1996) 6] Cont, R., de Larrard, A.: Order book dynamics in liquid markets : limit theorems and diffusion approximations. ArXiv e-prints (212) 7] Cont, R., de Larrard, A.: Price dynamics in a markovian limit order market. SIAM Journal on Financial Mathematics (213) 8

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