Markovian Order Book Modelling - polytechnique · 2 Market order: Immediately buy (sell) a certain...

Markovian Order Book ModellingStability and Scaling Limits

Aymen Jedidi

Ecole Centrale Paris, Franceaymen.jedidi@ecp.fr

Modelling and Managing Financial Risks ConferenceParis, January 2011

Aymen Jedidi Markovian Order Book Modelling 1/25

Outline

1 Introduction

2 Model Setup

3 Infinitesimal Generator

4 Stability of the Order Book

5 Large-scale Limit of the Price Process

6 Summary

Objectives

1 Analyse stability of stochastic order book models.2 Nature of the price process at large time scales.

Definitions

The Order Book is the list of all buy and sell limit orders,with their corresponding prices and volumes, at a giveninstant of time.

There are 3 types of orders:1 Limit order: Specify a price at which one is willing to buy

(sell) a certain number of shares;2 Market order: Immediately buy (sell) a certain number of

shares at the best available opposite quote;3 Cancellation order: Cancel an existing limit order.

The price dynamics is the result of the interplay betweenthe order book and the order flow.

Definitions

The Order Book is the list of all buy and sell limit orders,with their corresponding prices and volumes, at a giveninstant of time.There are 3 types of orders:

1 Limit order: Specify a price at which one is willing to buy(sell) a certain number of shares;

2 Market order: Immediately buy (sell) a certain number ofshares at the best available opposite quote;

3 Cancellation order: Cancel an existing limit order.

Definitions

The Order Book is the list of all buy and sell limit orders,with their corresponding prices and volumes, at a giveninstant of time.There are 3 types of orders:

1 Limit order: Specify a price at which one is willing to buy(sell) a certain number of shares;

2 Market order: Immediately buy (sell) a certain number ofshares at the best available opposite quote;

3 Cancellation order: Cancel an existing limit order.

Notations

The order book is represented by a finite-size vector ofquantities:

Xt := (at ; bt) := (a1t , . . . , a

Kt ; b1

t , . . . , bKt );

at : ask side of the order book;bt : bid side of the order book;

∆p: tick size;τ: unit volume;P = PAsk+PBid

2 : mid-price.

Notations

The order book is represented by a finite-size vector ofquantities:

Xt := (at ; bt) := (a1t , . . . , a

Kt ; b1

t , . . . , bKt );

at : ask side of the order book;bt : bid side of the order book;∆p: tick size;τ: unit volume;P = PAsk+PBid

2 : mid-price.

Model Setup

The events affecting the order book are described byindependent Poisson processes:

M±t : arrival of new market order ↪→ arrival rate λ±M ;L±i

t , i ∈ {1, . . . ,K } : arrival of a limit order i ticks awayfrom the best opposite quote ↪→arrival rate λi±

L ;C±i

t , i ∈ {1, . . . ,K }: cancellation of a limit order i ticksaway from the best opposite quote ↪→ arrival rate

|X ti |

(Xt) is a Markov process with state space X ⊂ Z2K .

Reference Frame and Boundary Conditions

Finite moving reference frame of size 2K : Ask sideranges from 1 to K ticks away from the best availableopposite quote. Idem for bid side of the order book.

Constant boundary conditions outside the movingframe: Every time the moving frame leaves a price level,the number of shares at that level is set to a∞ (or b∞,depending on the side of the order book).

Reference Frame and Boundary Conditions

Finite moving reference frame of size 2K : Ask sideranges from 1 to K ticks away from the best availableopposite quote. Idem for bid side of the order book.Constant boundary conditions outside the movingframe: Every time the moving frame leaves a price level,the number of shares at that level is set to a∞ (or b∞,depending on the side of the order book).

95 100 105 110

(1) initial state

95 100 105 110

(2) liquidity is taken

95 100 105 110

(3) wide spread

95 100 105 110

(4) liquidity returns

95 100 105 110

(5) liquidity returns

95 100 105 110

(6) final state

Figure 1: Illustration.

Order Book Dynamics

da it = −

τ − i−1∑k=1

dM+t + τdL i+

t − τdC i+t

+ (J−M(a) − a)idM−t +K∑

(J i−L (a) − a)idL i−

(J i−C (a) − a)idC i−

db it = similar expression,

where J±M, J i±L , and J i±

C are shift operators corresponding tothe effect of order arrivals on the reference frame.

Infinitesimal Generator

The infinitesimal generator of (Xt)t≥0 is the operator L ,defined to act on sufficiently regular functionsf : Rn → R, by

Lf(x) = limt↓0

E[f(Xt)|X0 = x] − f(x)

We have:

Lf (a; b) = λ+M

(f((a i − (τ − A i−1)+)+; J+

M (b))− f

K∑i=1

λi+L (f

(a i + τ; J i+

L (b))− f)

(a i − τ; J i+

C (b))− f)

+ similar terms for the events affecting the bid side.

Infinitesimal Generator

The infinitesimal generator of (Xt)t≥0 is the operator L ,defined to act on sufficiently regular functionsf : Rn → R, by

Lf(x) = limt↓0

E[f(Xt)|X0 = x] − f(x)

We have:

Lf (a; b) = λ+M

(f((a i − (τ − A i−1)+)+; J+

M (b))− f

K∑i=1

λi+L (f

(a i + τ; J i+

L (b))− f)

(a i − τ; J i+

C (b))− f)

+ similar terms for the events affecting the bid side.Aymen Jedidi Markovian Order Book Modelling 10/25

Stability of the Order Book

A Markov Process (Xt)t≥0 is ergodic if an invariantprobability distribution π exists and

limt→∞||Q t(x, .) − π|| = 0,∀x ∈ X,

where ||.|| is the total variation norm defined by

||µ|| := supA∈B(X)

|µ(A)| − infA∈B(X)

|µ(A)|.

and (Q t) are the transition functions of of the Markovprocess.

Theorem 1 [CST10], [AJ11]

If λC = min1≤i≤K {λ±iC } > 0, then (Xt) = (at ; bt) is an ergodic

Markov process. In particular (Xt) has a stationarydistribution π.

A Markov Process (Xt)t≥0 is ergodic if an invariantprobability distribution π exists and

limt→∞||Q t(x, .) − π|| = 0,∀x ∈ X,

where ||.|| is the total variation norm defined by

||µ|| := supA∈B(X)

|µ(A)| − infA∈B(X)

|µ(A)|.

and (Q t) are the transition functions of of the Markovprocess.

Theorem 1 [CST10], [AJ11]

If λC = min1≤i≤K {λ±iC } > 0, then (Xt) = (at ; bt) is an ergodic

Markov process. In particular (Xt) has a stationarydistribution π.

Proof. Let V(a; b) = τ +∑K

i=1 a i +∑K

i=1 |bi |.

LV(a; b) ≤K∑

(λi+L + λi−

L )τ − (λ+M + λ−M)τ −K∑

(λi+C a i + λi−

C |bi |)

λi−L (iS − i)+a∞ +

K∑i=1

λi+L (iS − i)+|b∞|

≤ K(λ+L + λ−L )τ − (λ+M + λ−M)τ − λC f(a; b)

+ K(K + 1)(λ−L a∞ + λ+L |b∞|)

LV(x) ≤ −cV(x) + d, (GDC)

where λ±L = max1≤i≤K {λi±L } and λC = min1≤i≤K {λ

i±C } > 0.

Corollary 1 [AJ11]

The spread process St := QAt − QB

t = Ψ(Xt), isergodic—expected as (St) is bounded by K + 1.

Rate of Convergence to the Stationary State

Actually, (Xt) is V -uniformly ergodic.Denote, as above, ||.|| the total-variation norm:

||µ|| := supA∈B(S)

|µ(A)| − infA∈B(S)

|µ(A)|,

And define the V -norm distance between Q1,Q2 by:

|||Q1 − Q2|||V := supx∈X

||Q1(x, .) − Q2(x, .)||VV(x)

and the outer-product:

[1 ⊗ π](x,A) := π(A), x ∈ X,A ∈ B(X).

Theorem 2 [MT93], [AJ11]There exist β < 1 and B < ∞ such that

|||Q t − 1 ⊗ π|||V ≤ Bβ−t .

Actually, (Xt) is V -uniformly ergodic.Denote, as above, ||.|| the total-variation norm:

||µ|| := supA∈B(S)

|µ(A)| − infA∈B(S)

|µ(A)|,

And define the V -norm distance between Q1,Q2 by:

|||Q1 − Q2|||V := supx∈X

||Q1(x, .) − Q2(x, .)||VV(x)

and the outer-product:

[1 ⊗ π](x,A) := π(A), x ∈ X,A ∈ B(X).

Theorem 2 [MT93], [AJ11]There exist β < 1 and B < ∞ such that

|||Q t − 1 ⊗ π|||V ≤ Bβ−t .

Proof. A by-product of the proof of theorem 1.

Large-scale Limit of the Price Process

In what follows, I will focus on the embedded discrete-timeMarkov chain:

(Xn)n∈N,

defined by its transition probabilities uij:

= −qij

qij∑k,i qik

if i , j

0 otherwise,

where qij are the transition rates of (Xt).

Lemma 1 [MT93]

(Xn) is V−geometrically mixing, that is, there exist B > 0 and0 < β < 1 such that for all g2, h2 ≤ V and l, n ∈ Z :∣∣∣Ex [g(Xn)h(Xn+l)] − Ex [g(Xn)]Ex [h(Xn+l)]

∣∣∣ ≤ Bβl (1 + ρnV(x)) .

Lemma 2 [MT93]

Let h = h − π(h), g = g − π(g). The stationary version of (Xt)satisfies a geometric mixing condition, that is, there existsB ′ > 0 such that for all l, n ∈ Z :

Eπ[g(Xn)h(Xn+l)

]≤ B ′βl .

Lemma 1 [MT93]

(Xn) is V−geometrically mixing, that is, there exist B > 0 and0 < β < 1 such that for all g2, h2 ≤ V and l, n ∈ Z :∣∣∣Ex [g(Xn)h(Xn+l)] − Ex [g(Xn)]Ex [h(Xn+l)]

∣∣∣ ≤ Bβl (1 + ρnV(x)) .

Lemma 2 [MT93]

Let h = h − π(h), g = g − π(g). The stationary version of (Xt)satisfies a geometric mixing condition, that is, there existsB ′ > 0 such that for all l, n ∈ Z :

Eπ[g(Xn)h(Xn+l)

]≤ B ′βl .

Theorem 3 [AJ11]The rescaled centered price process

P(t) = limn→∞

P(bntc) − E [P(bntc)]√

is a Brownian motion in the limit of n going to infinity.

Proof.1 Using Lemma 2 (geometric-mixing), there exists

(δ(l))l∈N such that

supn∈N,|g|,|h|≤1

|E[g(Xn)h(Xn+l)] − E[g(Xn)]E[h(Xn+l)]| ≤ δ(l),

and ∑l

δ(l) < ∞;

2 The price increments1 δPn = Φ(Xn,Xn−1, εn) are weaklydependent and have finite variance;

3 The Functional CLT holds.

1Φ : Z2K × Z2K × {−1, 1} → ±{0,∆, 2∆, . . . ,K∆}.Aymen Jedidi Markovian Order Book Modelling 19/25

Summary

A Markov-chain order book model was described;The order book is “stable” (ergodic);The convergence to the stationary state happensgeometrically fast (under a certain norm);The large-scale limit of the price process is a Brownianmotion.

Thank you for your attention.

References

F. Abergel and A. Jedidi.Forthcoming.2011.

P. Billingsley.Convergence of Probability Measures.Wiley, 1999.

R. Cont, S. Stoikov, and R. Talreja.A stochastic model for order book dynamics.Operations Research, 2010.

S. Meyn and R.L. Tweedie.Markov Chains and Stochastic Stability.Cambridge University Press, 1993.

I. Muni-Toke.Market making behaviour in an order book model and its impact on the spread.Econophysics of Order-driven Markets, Springer-Verlag, 2010.

E. Smith, D. Farmer, L. Guillemot, and S. Krishnamurthy.Statistical theory of the continuous double auction.Quantitative Finance, 2003.

-20 -15 -10 -5 0 5 10 15 20

Distance from best opposite quote (ticks)

Figure 2: Average depth profile.

0 50 100 150-4

Event time

Bid priceAsk priceMid priceMicro priceTrade price

Figure 3: Price sample path (∼ 100 events).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Event time

Figure 4: Price sample path (∼ 10, 000 events).

Markovian Order Book Modelling - polytechnique · 2 Market order: Immediately buy (sell) a certain...

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