Banff, September 23-25, 2016

General Semi-Markov Model for Limit Order Books:Theory, Implementation and Numerics

Katharina Cera & Julia Schmidt

in collaboration with Anatoliy Swishchuk and Tyler Hofmeister

University of Calgaryand

Technical University of Munich

September 25th 2016

Overview

1 Introduction to Limit Order Books and Markets

2 [Cont & Larrard(2013)]: Price Dynamics in a Markovian Limit OrderMarket

3 [Swishchuk & Vadori(2015a)]: A Semi-Markovian Modeling of LimitOrder Markets

4 Our extensions [Swishchuk, Cera, Schmidt & Hofmeister (2016)]

5 References

Katharina Cera & Julia Schmidt Limit Order Books September 25th 2016 2 / 49

Overview





5 References


Introduction to Limit Order Books and Markets

Order Types

Market Order

Buy/ sell certain quantity atbest available price

Usually results in immediateexecution

Limit Order

Buy/ sell at a given price upto certain maximum quantity

Price usually worse, so nodirect execution

The Limit Order Book (LOB) keeps track of incoming and outgoing orders.



Limit Order Book

How a Limit Order Book looks like:



Limit Order Book

The functionality of the Limit Order Book:



Limit Order Book

Incoming Sell Market Order:



Limit Order Book

Incoming Sell Market Order:


Overview





5 References


[Cont & Larrard(2013)]: Price Dynamics in a Markovian Limit Order Market

The Model

[Cont & Larrard(2013)] only models the (best) bid and ask queues:



The Model

qt = (qbt , qat ) is a Markov process, transitions occur when order book

events happen:

Limit Order:arrive at independent, exponential times with rate λ

Market Order:arrive at independent, exponential times with rate µ

Cancellation:arrive at independent, exponential times with rate θ

(Ta/bi , i ≥ 1): duration between two consecutive queue changes at the

ask/ bid

Va/bi : size of associated change in queue size (ask/bid)



The Model

Arriving Buy Limit Order (arrival rate:λ) ⇒Vbi = 1:



The Model

Arriving Buy Limit Order (arrival rate:λ) ⇒Vbi = 1:



The Model

Arriving Sell Market Order (arrival rate:µ) or Cancellation (arrival rate:θ)⇒Vb

i = −1:



The Model


i = −1:



The Model

Probability of queue changes:

Increasing queue (Limit Orders):

P[V ai = 1] =

λ

λ+ µ+ θand P[V b

i = 1] =λ

λ+ µ+ θ

Decreasing queue (Market Orders and Cancellations):

P[V ai = −1] =

µ+ θ

λ+ µ+ θand P[V b

i = −1] =µ+ θ

λ+ µ+ θ

(T ai )i≥1 and (T b

i )i≥1 are exponentially distributed with parameterλ+ θ + µ.

⇒(Vai )i≥1 and (V b

i )i≥1 are independent and also independent of (T ai )i≥1

and (T bi )i≥1.



The Model


i = −1:



The Model

The bid queue is depleted ⇒qbt = 0:



The Model

Price change and simulation of new queue sizes:



The Model

Simulation of the model (Tyler Hofmeister)



Analytical results

[Cont & Larrard(2013)] give analytical results for:

the distribution of duration until the next price move

the probability of a price increase when the order flow is balanced(λ = µ+ θ)

the probability of a price increase when the order flow is asymmetric(λ < µ+ θ)

the ”efficient” price, the observed price is a noisy version of thismartingale

the diffusion limit of the price process for balanced and unbalanced orderflow



Analytical results

Diffusion limit of the price process

The price is modelled as a piecewise constant stochastic processst = Z (Nt), where

Z (n) = X1 + · · ·+ Xn

and Nt = sup{k : τ1 + · · ·+ τk ≤ t} is the number of price changes during[0, t]. Xn is the price change at time τn.For much larger time scales, prices have diffusive dynamics.



Analytical results

Theorem 1 (page 21)(balanced order flow)

If λ = µ+ θ, (stn log n√

n, t ≥ 0

)n→∞⇒

(δ

√πλ

D(f )Wt , t ≥ 0

)

where δ is the tick size and W is a standard Brownian motion. D(f ), ameasure of market depth, is given by

D(f ) =∞∑i=1

∞∑j=1

ijf (i , j).



Analytical results

Empirical test using high-frequency data

Theorem 1 suggests a linear dependency between volatilty of intradayreturns and the diffusion coefficient. [Cont & Larrard(2013)] does aregression between the standard deviation of the 10-minute price

increments and√

λD(f ) . They reached an R2 of 68%.

We used this approach as a test for our model extensions.



Analytical results

Theorem 2 (page 23)(market orders and cancellations dominate)

If λ < µ+ θ ,(snt√n, t ≥ 0

)n→∞⇒

(δ√

m(λ, θ + µ, f )Wt , t ≥ 0

).


Overview





5 References


[Swishchuk & Vadori(2015a)]: A Semi-Markovian Modeling of Limit Order Markets

Challenged Assumptions

[Swishchuk & Vadori(2015a)] extended the model of[Cont & Larrard(2013)] to:

arbitrary distributions for book events inter-arrival times (exponentialbefore)

both the nature of a new book event and its corresponding inter-arrivaltime depend on the nature of the previous book event⇒ Semi-Markovian Model

The model remains analytically tractable. They calculate:

the duration until the net price change

the probability of a price increase

the diffusion limit of the price process



Data

Used data for the numerical results:

Level 1 LOB data provided by [LOBster(2016)]: Apple, Amazon,Google, Microsoft and Intel on 2012/06/21

Different data from Deutsche Borse



Justification

The exponential distribution does not fit the data very well.

Figure: Distribution of inter-arrival times Amazon ask and bid



Justification

Dependency of book events on previous book event, e.g. for MicrosoftP(i , j) = P(V a

t+1 = j |V at = i), j , i ∈ {1,−1} and P(i) = P(V a

t = i)



Diffusion limit

Assumption 1: We assume the following inequalities to be true:

∞∑n=1

αb(n)αa(p)f (n, p) <∞,∞∑n=1

αb(n)αa(p)f (n, p) <∞,

with αa(n) := 1pa√π

(n + 2pa−1pa−1 v a

0 (1))√pa(1− pa)

√paha1 + (1− pa)ha2, h

a1 is

defined on page 7 in [Swishchuk & Vadori(2015a)], pa = Pa(1, 1) = Pa(−1,−1),αb(n) is defined accordingly.

Assumption 2: In this section we assume m(δ) <∞ and m(−δ) <∞, wherem(i) = E [τk |Xk−1 = i ], i ∈ {δ,−δ}.

Balanced case: Pa(1, 1) = Pa(−1,−1) and Pb(1, 1) = Pb(−1,−1)Unbalanced case: Pa(1, 1) < Pa(−1,−1) or Pb(1, 1) < Pb(−1,−1)



Diffusion limit

Theorem:

Given that assumption 1 is satisfied for the balanced case and assumption 2 for the unbalanced case, we can proof the followingweak convergences in the Skorokhod topology (see [Skorokhod(1965)]):

(stnlog(n) − Ntnlog(n)s

∗

√n

, t ≥ 0

)n→∞⇒

σ∗√τ∗

Wt , for the balanced case and(stn − Ntns

∗√n

, t ≥ 0

)n→∞⇒

σ∗

√mτ

Wt , for the unbalanced case,

where Wt is a standard Brownian motion, s∗ = π∗δ + (1− π∗)(−δ)and (π∗, 1− π∗) is the stationary distribution of the

Markov chain a(X ). τ∗, mτ and (σ∗)2 are given by:

τ∗ = lim

t→+∞

t

Nt log(Nt )(see [Swishchuk & Vadori(2015a)], p.19)

mτ = π∗m(δ) + (1− π∗)m(−δ)

σ2 = 4δ2

(1− p′cont + π∗(p′cont − pcont )

(pcont + p′cont − 2)2− π∗(1− π∗)

).

pcont is the probability of two subsequent increases of the stock price, p′cont the probability of two subsequent decreases of thestock price.


Overview





5 References


Our extensions [Swishchuk, Cera, Schmidt & Hofmeister (2016)]

The general Model

We extended the model for the stock price st =∑N(t)

k=1 Xk to allow for:

magnitude of jump sizes differing from one tick taking two possiblevalue,

magnitude of jump sizes differing from one tick taking arbitrary numberof possible values.

We have:

shown results for diffusion coefficients,

implemented these results and calculated them for different data,

tested the model by doing the regression of [Cont & Larrard(2013)].



Model for the Limit Order Book with two states

[Swishchuk & Vadori(2015a)] assumed that magnitude of price jumps isequal to one tick, but:

Apple Amazon Google

Avg. up movements 1.7 1.3 3.1Avg. down movements -1.7 -1.3 -3.0Min price change -18.5 -11.5 -30.5Max price change 15.0 16.5 30.5

Table: Mid-Price Changes in ticks




We allowed the magnitude of the stock price jumps to differ from one tick,as proposed in [Swishchuk & Vadori(2016b)]:

st =

N(t)∑k=1

a(Xk),

where N(t) is the counting process for the price changes, Xk is a two stateMarkov chain with state space S = {1, 2} and a(x) is an uniformlybounded function.




Assumption 1:We assume the following inequalities to be true:

∞∑n=1


∞∑n=1


with αa(n) := 1pa√π

(n + 2pa−1pa−1 v a

0 (1))√pa(1− pa)

√paha1 + (1− pa)ha2, h

a1 is

defined on page 7 in [Swishchuk & Vadori(2015a)], pa = Pa(1, 1) = Pa(−1,−1),αb(n) is defined accordingly.

Assumption 2:In this section we assume m(1) <∞ and m(2) <∞, wherem(i) = E [τk |Xk−1 = i ], i ∈ {1, 2}.




Theorem:

Given that assumption 1 is satisfied for the balanced case and assumption 2 for the unbalanced case, we can proof the followingweak convergences in the Skorokhod topology (see [Skorokhod(1965)]):(

stnlog(n) − Ntnlog(n)a∗

√n

, t ≥ 0

)n→∞⇒

σ∗√τ∗

Wt , for the balanced case and(stn − Ntna

∗√n

, t ≥ 0

)n→∞⇒

σ∗

√mτ


where Wt is a standard Brownian motion, ai := a(i), a∗ = π∗1 a1 + π∗2 a2and (π∗1 , π∗2 ) is the stationary distribution of the

Markov chain a(X ). τ∗, mτ and (σ∗)2 are given by:

τ∗ = lim

t→+∞

t


mτ = π∗1 m(1) + π

∗2 m(2)

(σ∗)2 = π∗1 a2

1 + π∗2 a2

2 + (π∗1 a1 + π∗2 a2)[−2a1π

∗1 − 2a2π

∗2 + (π∗1 a1 + π

∗2 a2)(π∗1 + π

∗2 )]

+(π∗1 (1− pcont ) + π∗2 (1− p′cont ))(a1 − a2)2

(pcont + p′cont − 2)2

+ 2(a2 − a1) ·[π∗2 a2(1− p′cont )− π∗1 a1(1− pcont )

pcont + p′cont − 2+

(π∗1 a1 + π∗2 a2)(π∗1 − pcontπ∗1 − π

∗2 + p′contπ

∗2 )

pcont + p′cont − 2

]

pcont is the probability of two subsequent increases of the stock price, p′cont the probability of two subsequent decreases of thestock price.




Used data for the numerical results:

Level 1 LOB data provided by [LOBster(2016)]: Apple, Amazon,Google, Microsoft and Intel on 2012/06/21

LOB data provided in [Cartera, Jaimungal & Penalva(2015)]: Cisco,Facebook, Intel, Liberty Global, Liberty Interactive, Microsoft, Vodafonefrom 2014/11/03 to 2014/11/07




Figure: Linear relationship of coefficients and standard deviation of 10 minutesmid-price changes for two states balanced case (left) and unbalanced case (right)

Balanced case: adjusted R2 of 0.9788Unbalanced case: adjusted R2 of 0.9821




As a comparison we did the same regression for one tick price jumps:

Figure: Linear relationship of coefficients and standard deviation of 10 minutesmid-price changes for one tick jumps balanced case (left) and unbalanced case(right)




Model for the Limit Order Book with arbitrary number of states

Figure: Jump sizes Apple, Amazon and Google Midprices




We will consider the following model:

st =

N(t)∑k=1

a(Xk),

where Xk is a Markov chain with n states, meaning that the state space isextended to S = {1, ..., n}.




Theorem:Given that assumption 1 is satisfied for the balanced case and m(i) <∞ for all i = 1, 2, . . . n is satisfied for the unbalancedcase, we can proof the following weak convergences in the Skorokhod topology (see [Skorokhod(1965)]):(

stnlog(n) − Ntnlog(n)a∗

√n

, t ≥ 0

)n→∞⇒

σ∗√τ∗

Wt , for the balanced case and(stn − Ntna

∗√n

, t ≥ 0

)n→∞⇒

σ∗

√mτ


where Wt is a standard Brownian motion, a∗ =∑

i∈S π∗i a(i) and mτ =

∑i∈S π

∗i m(i). τ∗ and (σ∗)2 are given by:

τ∗ = lim

t→+∞

t


(σ∗)2 =∑i∈S

πi v(i)

v(i) = b(i)2 +∑j∈S

(g(j)− g(i))2P(i, j)− 2b(i)∑j∈S

(g(j)− g(i))P(i, j),

where

b = (b(1), b(2), ..., b(n))′,

b(i) : = a(Xi )− a∗ := a(i)− a∗ and

g : = (P + Π∗ − I )−1b.

P is a transition probability matrix, where P(i, j) = P(Xk+1 = j|Xk = i). Π∗ denotes the stationary distribution of P and g(j)

is the jth entry of g .




How our algorithm chooses the states:




Figure: Linear relationship of coefficients and standard deviation of 10 minutesmid-price changes for many states, Balanced case (left) and unbalanced case(right)



Overview





5 References


References

A. Cartera, S. Jaimungal & J. Penalva (2015) Algorithmic and High-Frequency Trading. Cambridge University Press.

R. Cont & A. de Larrard (2013) Price dynamics in a Markovian Limit Order Book market, SIAM Journal for Financial

Mathematics 4 (1): 1–25.

R. Cont, S. Stoikov & R. Talreja (2010) A stochastic model for order book dynamics, Operations Research 58 (3),

549–563.

M.D. Gould, M.A. Porter, S. Williams, M. McDonald, D.J. Fenn, S.D. Howison (2013) Limit Order Books, Quantitative

Finance 13 (10), 1709–1742.

Humbold Universitt zu Berlin, Germany (2013) LOBSTER: Limit Order Book System - The Efficient Reconstructor.

http://LOBSTER.wiwi.hu-berlin.de. Accessed: 2016-07-20

J.R. Norris (1997) Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics.

A. Skorokhod (1965) Studies in the Theory of Random Processes. Reading: Addison Wesley. Reprinted by Dover

Publivations, NY.

A. Swishchuk, K. Cera, J.Schmidt & T. Hofmeister (2016) General Semi-Markov Model for Limit Order Books: Theory,

Implementation and Numerics. SSRN: http://papers.ssrn.com/sol3/papers.cfm?abstract id=2820221.

A. Swishchuk & N. Vadori (2015a) Semi-Markov Model for the price dynamics in limit order markets. SSRN:

papers.ssrn.com/sol3/papers.cfm?abstract id=2579865.

A. Swishchuk & N. Vadori (2015b) Strong Law of Large Numbers and Central Limit Theorems for functionals of

inhomogeneous Semi-Markov processes, Stochastic Analysis and Applications 13 (2), 213–243.

A. Swishchuk & N. Vadori (2016a) A semi-markovian modeling of limit order markets. arXiv: 1601.01710.

A. Swishchuk & N. Vadori (2016b) A semi-markovian modeling of limit order markets. Submitted to SIAM Journal of

Financial Mathematics, under review.Katharina Cera & Julia Schmidt Limit Order Books September 25th 2016 48 / 49

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Banff, September 23-25, 2016