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Markovian Order Book ModellingStability and Scaling Limits
Aymen Jedidi
Ecole Centrale Paris, [email protected]
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
Aymen Jedidi Markovian Order Book Modelling 2/25
Objectives
1 Analyse stability of stochastic order book models.2 Nature of the price process at large time scales.
Aymen Jedidi Markovian Order Book Modelling 3/25
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.
Aymen Jedidi Markovian Order Book Modelling 4/25
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.
The price dynamics is the result of the interplay betweenthe order book and the order flow.
Aymen Jedidi Markovian Order Book Modelling 4/25
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.
The price dynamics is the result of the interplay betweenthe order book and the order flow.
Aymen Jedidi Markovian Order Book Modelling 4/25
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.
Aymen Jedidi Markovian Order Book Modelling 5/25
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.
Aymen Jedidi Markovian Order Book Modelling 5/25
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
λi+C
|X ti |
τ.
(Xt) is a Markov process with state space X ⊂ Z2K .
Aymen Jedidi Markovian Order Book Modelling 6/25
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).
Aymen Jedidi Markovian Order Book Modelling 7/25
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).
Aymen Jedidi Markovian Order Book Modelling 7/25
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(1) initial state
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(2) liquidity is taken
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(3) wide spread
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(4) liquidity returns
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(5) liquidity returns
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(6) final state
Figure 1: Illustration.
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Order Book Dynamics
da it = −
τ − i−1∑k=1
ak
+
dM+t + τdL i+
t − τdC i+t
+ (J−M(a) − a)idM−t +K∑
i=1
(J i−L (a) − a)idL i−
t
+K∑
i=1
(J i−C (a) − a)idC i−
t ,
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.
Aymen Jedidi Markovian Order Book Modelling 9/25
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)
t.
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)
+K∑
i=1
λi+C
ai
τ(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
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)
t.
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)
+K∑
i=1
λi+C
ai
τ(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.
Aymen Jedidi Markovian Order Book Modelling 11/25
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.
Aymen Jedidi Markovian Order Book Modelling 11/25
Stability of the Order Book
Proof. Let V(a; b) = τ +∑K
i=1 a i +∑K
i=1 |bi |.
LV(a; b) ≤K∑
i=1
(λi+L + λi−
L )τ − (λ+M + λ−M)τ −K∑
i=1
(λi+C a i + λi−
C |bi |)
+K∑
i=1
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.
Aymen Jedidi Markovian Order Book Modelling 12/25
Stability of the Order Book
Corollary 1 [AJ11]
The spread process St := QAt − QB
t = Ψ(Xt), isergodic—expected as (St) is bounded by K + 1.
Aymen Jedidi Markovian Order Book Modelling 13/25
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 .
Aymen Jedidi Markovian Order Book Modelling 14/25
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 .
Aymen Jedidi Markovian Order Book Modelling 14/25
Rate of Convergence to the Stationary State
Proof. A by-product of the proof of theorem 1.
Aymen Jedidi Markovian Order Book Modelling 15/25
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:
uij =
= −qij
qii=
qij∑k,i qik
if i , j
0 otherwise,
where qij are the transition rates of (Xt).
Aymen Jedidi Markovian Order Book Modelling 16/25
Large-scale Limit of the Price Process
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 .
Aymen Jedidi Markovian Order Book Modelling 17/25
Large-scale Limit of the Price Process
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 .
Aymen Jedidi Markovian Order Book Modelling 17/25
Large-scale Limit of the Price Process
Theorem 3 [AJ11]The rescaled centered price process
P(t) = limn→∞
P(bntc) − E [P(bntc)]√
n
is a Brownian motion in the limit of n going to infinity.
Aymen Jedidi Markovian Order Book Modelling 18/25
Large-scale Limit of the Price Process
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.
Aymen Jedidi Markovian Order Book Modelling 20/25
Thank you for your attention.
Aymen Jedidi Markovian Order Book Modelling 21/25
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.
Aymen Jedidi Markovian Order Book Modelling 22/25
-20 -15 -10 -5 0 5 10 15 20
-6
-4
-2
0
2
4
6
Distance from best opposite quote (ticks)
Qua
ntity
Figure 2: Average depth profile.
Aymen Jedidi Markovian Order Book Modelling 23/25
0 50 100 150-4
-3
-2
-1
0
1
2
3
4
Event time
p(t)
-p(0
) (t
icks
)
Bid priceAsk priceMid priceMicro priceTrade price
Figure 3: Price sample path (∼ 100 events).
Aymen Jedidi Markovian Order Book Modelling 24/25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
-30
-20
-10
0
10
20
30
40
50
60
Event time
p(t)
-p(0
) (t
icks
)
Figure 4: Price sample path (∼ 10, 000 events).
Aymen Jedidi Markovian Order Book Modelling 25/25