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1 Semi-Markov model for market microstructure and HF exchanging LPMA, College Paris Diderot and JVN Institute, VNU, Ho-Chi-Minh City NUS-UTokyo Workshop on Quantitative Finance Singapore, 26-27 september 2013 Joint use Pietro FODRA EXQIM and LPMA, College Paris Diderot

2 Financial data modelling Continuous time cost process (P t ) t over [, T ] observed at P , P τ. P nτ Different modelling of P based on scales τ and T. Macroscopic scale (hourly, daily observation data): Itô semimartingale Microscopic scale (tick data) High frequency

3 Euribor contract, 2010, for several observation scales EURIBOR [ T: 1 Y / freq: one hour ] Value 99.1 99.2 99.3 2010/01/04 10:00:00 2010/05/03 09:00:00 2010/09/01 09:00:00 2010/12/31 17:00:00 Date EURIBOR [ T: 1 d / freq: 1 sec ] Value 99.13 99.14 99.15 99.16 99.17 99.18 99.19 2010/05/05 09:00:03 2010/05/05 11:30:00 2010/05/05 14:00:00 2010/05/05 16:30:00 Date

4 Figure. Eurostoxx contract, 2010 may Semi-Markov 5, 9h-9h15, model for market tick frequency microstructure and HF exchanging Stylized info on HF data Microstructure effects Discreteness of costs: jump occasions and expenses variations (tick data) Mean-reversion: negative autocorrelation of consecutive variation prices Irregular spacing of jump occasions: clustering of exchanging activity EUROSTOXX [ T: 15m / freq: tick by tick ] 2655 2660 2665 2670 May 05 09:00:00 May 05 09:04:05 May 05 09:08:00 May 05 09:12:04

5 Limit order book Almost all modern equities exchanges organized utilizing a mechanism of Limit Order Book (LOB): 500 Best Bid 375 Best Ask Volume 250 BID/ASK spread 125 48,455 48,46 48,465 48,47 48,475 48,48 48,485 48,49 48,495 48,5 48,505 48,51 48,515 48,52 48,525 48,53 Cost Figure.

Immediate picture in the LOB

6 High frequency finance Two primary streams in literature: Types of intra-day asset cost Latent process approach: Gloter and Jacod (01), Ait Sahalia, Mykland and Zhang (05), Robert and Rosenbaum (11), etc Point process approach: Bauwens and Hautsch (06), Cont and de Larrard (10), Bacry et al. (11), Abergel, Jedidi (11), Sophisticated models made to reproduce microstructure effects, frequently for cause of volatility estimation High frequency exchanging problems Liquidation and market making within the LOB: Almgren, Cris (03), Alfonsi and Schied (10, 11), Avellaneda and Stoikov (08), etc Stochastic control means of optimal exchanging strategies according to classical types of asset cost (arithmetic or geometric Brownian motion, diffusion models)

7 Objective Create a bridge between both of these streams of literature: Create a simple model for asset cost in Limit Order Book (LOB) realistic: captures primary stylized info on microstructure Diffuses round the macroscopic scale Simple to estimate and simulate tractable (easy to evaluate and implement) for dynamic optimization overuse injury in high frequency exchanging Markov renewal and semi-markov model approach

8 References P. Fodra and H. Pham (2013a): Semi-Markov model for market microstructure, preprint on arxiv or ssrn P. Fodra and H. Pham (2013b): High frequency exchanging within the Markov renewal model.

9 Model-free description of asset mid-cost (constant bid-ask spread) Marked point process Evolution within the univariate mid cost process (P t ) based on: The timestamps (T k ) k from the jump occasions N t counting process: N t = inf: modeling of volatility clustering, i.e. info on spikes in power market activity The marks (J k ) k valued in Z , representing (modulo the tick size) the cost increment at T k. modeling within the microstructure noise via mean-reversion of cost increments P J J J T T T t

10 Semi-Markov model approach Markov Renewal Process (MRP) to explain (T k, J k ) k. Largely present in reliability Independent paper by d Amico and Petroni (13) using also semi Markov model for asset prices

11 Jump side modeling For simplicity, we assume J k = 1 (on data, this really is 99,9% within the occasions). J k valued in : side within the jump (upwards or downwards) J k = J k 1 B k (1) (B k ) k i.i.d. with law: P[B k = ±1] = 1±α 2 with α [ 1, 1). (J k ) k irreducible Markov chain with symmetric transition matrix: ) Q α = ( 1+α 2 1 α 2 1 α 2 1+α 2 Remark: arbitrary random jump size can be viewed as by presenting an i.i.d. multiplication take into account (1).

12 Mean reversion Underneath the stationary possibility of (J k ) k, we’ve: α = correlation(j k, J k 1 ) Estimation of α: ˆα n = 1 n n J k J k 1 k=1 α 87, 5%, ( Euribor3m, 2010, 10h-14h) Strong mean reversion of cost returns

13 Timestamp modeling Conditionally on , the succession of inter-arrival jump occasions is i.i.d. with distribution function F ± and density f ±. F ± (t) = P [ S k t J k J k 1 = ±1 ]. Remarks The succession (S k ) k is (unconditionally) i.i.d with distribution: f + F = 1 + α 2 F + + 1 α F. 2 h + = 1+α 2 1 F may be the intensity reason behind cost begin exactly the same direction, h = 1 α f 2 1 F may be the intensity reason behind cost begin the selection direction

14 Non parametric estimation of jump intensity 5 10 15 f h[ ] h[+] .05 .-15 .-25 .-35 .4 .45 .5 .55 .6 .65 .7 .75 .8 .85 .9 .95 1 s (quantile) Figure. Estimation of h ± as reason behind the renewal quantile

15 Simulated cost Cost simulation 10 9 8 7 6 5 4 3 2 7 P 47 111 176 349 447 533 599 688 755 818 886 968 1039 1121 1215 1295 1406 1493 1623 1700 1785 t(s) Figure. half an hour simulation Cost simulation 62 56 50 44 38 32 26 20 14 9 4 P 804 1919 3148 4384 5623 6851 8079 9311 10650 12117 13670 15129 16650 18105 19558 21010 22463 23943 25491 26951 28424 t(s) Figure. eventually simulation

16 Diffusive behavior at macroscopic scale Scaling: P (T ) t = P tt T, t [, 1]. Theorem lim T P(T ) (d) = σ W, where W could be a Brownian motion, and σ 2 may be the macroscopic variance: ( 1 + α ) σ 2 = λ. 1 α with λ 1 = tdf (t).

17 Mean signature plot (recognized volatility) We think about the problem of delayed renewal process: S n F, n 1, with finite mean 1/λ, and S 1 density λ(1 F ) Cost process P has stationary increments Proposition V (τ) := 1 ( 2α ) 1 τ E[ (P τ P ) 2 ] = σ 2 + Gα (τ) 1 α (1 α)τ, where G α (t) = E[α Nt ] is clearly given via its Laplace-Stieltjes transform α in relation to F (s) := e st df (t). V ( ) = σ 2, and V ( + ) = λ. Remark: Similar expression much like Robert and Rosenbaum (09) or Bacry et al. (11).

18 Signature Plot C(tau) .0005 .0010 .0015 .0020 .0025 Data Simulation Gamma Poisson 500 1000 1500 2000 2500 3000 3500 tau(s) Figure. Mean signature plot for α

19 Markov embedding of cost process Define the the best cost jump direction: I t = J Nt, t , valued in along with the passed time because the last jump: S t = t sup T k, t . T k t Your cost process (P t ) valued by fifty percentδZ is part of a Markov process with three observable condition variables (P t, I t, S t ) with generator: L(p, i, s) = + h + (s) [ (p + 2δi, i, ) (p, i, s) ] s + h (s) [ (p 2δi, i, ) (p, i, s) ],

20 Exchanging issue Problem in the agent (market maker) who submits limit orders on sides within the LOB: limit buy order inside the best bid cost and limit sell order inside the best ask cost, for your exact purpose to achieve multiplication. We have to model industry order flow, i.e. the counterpart exchange the limit order

21 Market trades A company order flow is modelled getting reasonable point process (θ k, Z k ) k. θ k. arrival time period of this marketplace order M t counting process Z k valued in < 1, +1>: side within the trade. Z k = 1: exchange the most effective BID cost (market sell order) Z k = +1: exchange the most effective ASK cost (market buy order) index n θ k best ask best bid traded cost Z k 1 9:00:01.123 98.47 98.46 98.47 +1 2 9:00:02.517 98.47 98.46 98.46-1 3 9:00:02.985 98.48 98.47 98.47-1 Dependence modeling between market order flow and cost in LOB: Cox marked point process

22 Trade timestamp modeling The counting process (M t ) on the market order timestamps (θ k ) k could be a Cox process with conditional intensity λ M (S t ). Types of parametric forms reproducing intensity decay when s is large: λ exp M (s) = λ + λ 1 s r e ks λ power M (s) = λ + λ 1s r 1 + s k. with positive parameters λ , λ 1, r, k, believed by MLE.

23 First level limit orders First level limit orders Second level limit orders Second level limit orders First level limit orders First level limit orders First level limit orders First level limit orders Second level limit orders Strong and weak side of LOB BEST ASK PRX BEST ASK PRX BEST BID PRX BEST ASK PRX BEST BID PRX BEST BID PRX We call strong side (+) within the LOB, the medial side within the same direction in comparison with last jump, e.g. best ask when cost jumped upwards. We call weak side ( ) within the LOB, the medial side within the alternative way in comparison with last jump, e.g. best bid when cost jumped upwards. We know that trades (market order) arrive mainly around the weak side within the LOB.

24 Trade side modeling The trade sides receive by: Z k = Γ k I θ, k (Γ k ) k i.i.d. valued in with law: for ρ [ 1, 1]. P[Γ k = ±1] = 1 ± ρ 2

25 Interpretation of ρ ρ = corr(z k, I θ ) k ρ = : market order flow arrive individually within the best bid and finest ask (usual assumption within the existing literature) ρ : market orders arrive more frequently within the strong side within the LOB ρ : market orders arrive more frequently within the weak side within the LOB Estimation of ρ: ˆρ n = 1 n n k=1 Z ki θ results in ρ 50%: about k 3 over 4 trades arrive across the weak side. ρ associated with adverse selection

26 Market making strategy Strategy control: foreseeable process (l + t, l t ) t valued in l + t l t = 1: limit order of fixed size L across the strong side: +I t = 1: limit order of fixed size L across the weak side: I t Charges: any transaction is vulnerable to a group cost ε Portfolio process: Cash (X t ) t valued in R, inventory (Y t ) t valued within the set Y of Z

27 Agent execution Execution of limit order occurs when: A company trade can get to θ k across the strong (resp. weak) side if Z k I θ = +1 (resp. 1), by getting an performed quantity k provided with a distribution (cost time priority/prorata) ϑ + L (resp. ) on ϑ L The cost jumps at T k and crosses the limit order cost Remark ϑ ± L cannot be believed on historic data. It should be evaluated getting a backtest obtaining a zero intelligence strategy. Risks: Inventory cost jump Adverse selection in market order trade

28 Market making optimization Value reason for this marketplace making control problem: v(t, s, p, i, x, y) = sup E [ ] PNL T CLOSE(Y T ) η RISK t,t (l +,l ) where η may be the agent risk aversion and: PNL t = X t + Y t P t, (ptf worth the mid cost) CLOSE(y) = (δ + ε) y, (closure market order) RISK t,t = T t Y 2 u d[p] u, (no inventory imbalance)

29 Variable reduction to strong inventory and passed time Theorem The worth function is supplied by: v(t, s, p, i, x, y) = x + yp + ω yi (t, s) where ω q (t, s) = ω(t, s, q) may be the unique viscosity strategy to the integro ODE: [ t + s ] ω + 2δ(h + h )q 4δ 2 η(h + + h )q 2 + max L l + ω + l ,q ll Y in [, T ] R + Y. max L l ω = l ,q+lL Y ω q (T, s) = q (δ + )

30 L l ± = L l ±,M + Ll ±,jump favorable execution of random size L by market order L l ±,M ω := λ ±,M (s) [ω(t, s, q kl) ω(t, s, q) + (+δ ε)kl ] ϑ ± L (dk) unfavorable execution of maximal size L because of cost jump L l ±,jump ω := h ±(s) [ ω(t, , ±q Ll) ω(t, s, q) + ( δ ε)ll ] with λ ±,M (s) := 1 ± ρ 2 λ M (s) (trade intensities)

31 Optimal policy shape: ρ = , execution probability = 10%

32 Optimal policy shape: ρ = , execution probability = 5%

33 Optimal policy shape: ρ = , 33, execution probability = 5%

34 Concluding remarks Markov renewal method of market microstructure + Apparent to find out and simulate + Non parametric estimation according to i.i.d. sample data + dependency between cost return J k and jump time T k + Reproduces well microstructure effects, diffuses on macroscopic scale + Markov embedding with observable condition variables ( Hawkes process approach) + Develop stochastic control formula for HF exchanging – MRP forgets correlation between inter-arrival jump occasions k Extension to multivariate cost model Model with market impact for liquidation problem