A Generalized Birth-Death Stochastic Model 4HF Order Dynamics

7/30/2019 A Generalized Birth-Death Stochastic Model 4HF Order Dynamics

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What is the limit order book Model framework First-passage times Sample computations

A generalized birth-death stochastic modelfor high-frequency order book dynamics

Alec Kercheval and He Huang

Department of MathematicsFlorida State University

July 31, 2011 Stevens Institute of Technology
http://goforward/http://find/http://goback/


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Papers

H. Huang and A. Kercheval, A generalized birth-death

stochastic model for high-frequency order book dynamics.2011, www.math.fsu.edu/~kercheva/papers/

R. Cont, S. Stoikov, R. Talreja, A stochastic model for

order book dynamics. Operations Research, 2010, 58,

549563.
http://www.math.fsu.edu/~kercheva/papers/http://www.math.fsu.edu/~kercheva/papers/http://find/http://goback/


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Outline

What is the limit order book

Model framework

Assumptions

Continuous time Markov chains

First-passage times

Generalized birth-death processes

The truncated process

Sample computations

Mid-price up move is first

Execution probability
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Trading in an order-driven market

Three kinds of actions:

limit order to buy (bids) or sell (asks) a specified number of

shares at a specified price

market order to buy or sell a specified number of shares at

best price

cancel a previously placed limit order that has not yet been

executed
http://find/


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Limit Order Book

A(j+1)

A(j)

B(i)

B(i-1)

pA

pB

A(j)=#limit

sellorders

(asks)atpricej

B(i)=#limit

buyorders

(bids)atpricei

pA=bestaskpB=bestbid
http://find/


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Sample Level II data

Table: A sample of quotes taken from raw data VOD.L. Thetime-stamp is measured in seconds from midnight, the type B refersto bids, and the level is the number of ticks from best bid (countingfrom 1 = best bid). Rows appear when an event occurs.

time-stamp type level price quantity

39301.481 B 4 134.9 203651

39301.722 B 1 135.05 10000

39302.891 B 4 134.9 193651

39302.891 B 2 135 192869

39305.192 B 1 135.05 9680

39308.359 B 4 134.9 186151

S
http://find/


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Sample Questions

Given an observed state of the order book, what is the

probability that the next move of the mid-price is upward? What is the probability that a proposed limit order placed at

best ask will be executed before the mid-price moves

downward?

Wh t i th li it d b k M d l f k Fi t ti S l t ti
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Model set-up

finite price grid = {1,2,3, . . . ,n}, allowed prices inmultiples of a tick

Z+ = {0,1,2, 3, . . . } = possible numbers of limit orders in

multiples of Sm shares, Sm = avg size of market order Ask process and Bid process, continuous time Z+-valued

Markov

A(t) = (A1(t), . . . , An(t)) B(t) = (B1(t), . . . ,Bn(t))

Ak(t) Bk(t) = 0 all k, t.

pA(t) = best ask, pB(t) = best bid price at time t

What is the limit order book Model framework First passage times Sample computations
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Model assumptions

market orders and cancellations are constant size Sm.

market buy and sell orders arrive at independent,

exponentially distributed times with rate .

cancellations at a distance j ticks from the same-side best

quote arrive at independent, exponentially distributed timeswith rate proportional to the number of outstanding shares:

for kSm shares, the cancellation rate is kj, j, k 0.

limit orders of size k = 1,2, . . . ,M (in multiples of Sm)

arrive at independent, exponentially distributed times withrate

(k)j , where j 1 is tick distance from opposite side

best quote.

parameters Sm, , j, and (k)j estimated from market data

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Continuous time Markov chains on (Z+)n

Ai(t) Ai(t) + k at rate (k)ipB(t)

for i> pB(t), k 0

Ai(t) Ai(t) 1 at rate Ai(t)ipA(t) for i pA(t)Ai(t) Ai(t) 1 at rate for i = pA(t) > 0

Bi(t) Bi(t) + k at rate (k)pA(t)i

for i< pA(t), k 0

Bi(t) Bi(t) 1 at rate Bi(t)pB(t)i for i pB(t)Bi(t) Bi(t) 1 at rate for i = pB(t) < n+ 1

Goal: compute relevant conditional probabilities withoutthe need for monte carlo simulation

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Continuous time Markov chains on (Z+)n

Ai(t) Ai(t) + k at rate (k)ipB(t)

for i> pB(t), k 0

Ai(t) Ai(t) 1 at rate Ai(t)ipA(t) for i pA(t)Ai(t) Ai(t) 1 at rate for i = pA(t) > 0

Bi(t) Bi(t) + k at rate (k)pA(t)i

for i< pA(t), k 0

Bi(t) Bi(t) 1 at rate Bi(t)pB(t)i for i pB(t)Bi(t) Bi(t) 1 at rate for i = pB(t) < n+ 1

Goal: compute relevant conditional probabilities withoutthe need for monte carlo simulation

http://find/


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First-passage times of generalized birth-death

processes

Let M = 2.Consider X(t), a Z+-valued Markov process.

X(t) has birth rates (1) and (2) of sizes 1,2, respectively

X has death rates i of size 1 at state i 1.

b = first-passage time to state zero given starting state isb Z+.

we want to compute the probability density function fb,0(t)of b.



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pdf of first passage time from state 10 to 0

0

0.005

0.01

0.015

0.02

0.025

0.03

0 20 40 60 80 100 120 140 160 180 200

probability

density

t

probability density function of the first passage time from state 10 to 0

pdf

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p g p p

First-passage time to zero

Idea: From state i 2, to reach state i 2 the process mustfirst reach state i 1.

Sob = b,b1 + b1,b2 + + 1,0

where i,i1 denotes the first-passage time from state i to statei 1, for i = 1,2, . . . b.



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p g p p

First passage from i to i 1

Let gi(t) be the pdf of i,i1, fb,0(t) the pdf of b.

Then fb,0(t) = gb(t) gb1(t) g1(t), so

fb,0

(s) =b

i=1

gi(s)

where f denotes the Laplace transform

f(s) =

0 e

sx

f(x)dx

New goal: compute gi(s), and then obtain fb,0 via Inverse

Laplace Transform of fb,0

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Recursive formulas

Let vi = (1) + (2) + i

the dwell time at state i has density vie

vit

the next transition is to state i + 1 with probability (1)/vi

to state i + 2 with probability (2)/vi

to state i 1 with probability i/vi

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Recursive formulas

gi(t) =ivi

vievit +

(1)

vivievit gi+1(t) gi(t)

+

(2)

vi vie

vit

gi+2(t) gi+1(t) gi(t)

hence

gi(s) =i

vi + s

+(1)

vi + s

gi+1(s)gi(s) +(2)

vi + s

gi+2(t)gi+1(s)gi(s)

i.e.

gi(s) =i

vi + s (1)gi+1(s) (2)gi+2(s)gi+1(s)



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Recursive formulas

gi(t) =ivi

vievit +

(1)

vivievit gi+1(t) gi(t)

+

(2)

vi vie

vit

gi+2(t) gi+1(t) gi(t)

hence

gi(s) =i

vi + s

+(1)

vi + s

gi+1(s)gi(s) +(2)

vi + s

gi+2(t)gi+1(s)gi(s)

i.e.

gi(s) =i

vi + s (1)gi+1(s) (2)gi+2(s)gi+1(s)

http://find/


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The truncated process X()

For Z+, define X() with state space {0,1, 2, . . . , } to be Xtruncated at : X has

birth rates (1) of size 1 and (

2) of size 2 at states

i 2

birth rate (1) + (2) of size 1 at state 1

death rates i of size 1 at states i 1.

Trajectories below 2 are the same for X and X()

.

http://find/


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Truncated first passage times

Let g()i (s) be the Laplace transform of the first-passage time

()i,i1 of X

() from state i to i 1.

g()

(s) =

s+

g()1(s) =

1

s+ (1) + (2) + 1 ((1) + (2))g() (s)

g()i (s) = i

s+ (1) + (2) + i (1)g()i+1(s)

(2)g()i+2(s)g

()i+1(s)

for i = 2, 3, . . . , 1



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Truncated first passage times

Let g()i (s) be the Laplace transform of the first-passage time

()i,i1 of X

() from state i to i 1.

g()

(s) =

s+

g()1(s) =

1

s+ (1) + (2) + 1 ((1) + (2))g() (s)

g()i (s) = i

s+ (1) + (2) + i (1)g()i+1(s)

(2)g()i+2(s)g

()i+1(s)

for i = 2, 3, . . . , 1

http://find/


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Convergence of truncated process

Proposition

For each state i, the first passage time(

)i,i1 converges inprobability toi,i1 as .

Therefore g()i (s) converges togi(s), for all s with[s] > 0,

as .

http://find/


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Convergence of truncated process

Numerical experiments show the convergence is very rapid as

: = 30 is already large enough to make the firstpassage times from i = 10 of X() and X indistinguishable to 7

digits with our parameters = 3.16

(0) = 0.71

(1)0 = 7.46

(2)0 = 0.80



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Example: P(mid-price moves up before down)

Assume at t = 0:

There are aorders at best ask pA(0); a = the first time allorders at price pA(0) disappear

There are b orders at best bid pB(0); b = the first time allorders at price pB(0) disappear

Bid-Ask spread is S ticks

iA is the first time an ask arrives i ticks away from best bid

iB is the first time a bid arrives i ticks away from best ask,i = 1, . . . ,S 1.



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Direction of the next price move

Let B = 1B

S1B , arrival time of first bid between

pA(0) and pB(0)

Let A = 1A

S1A , arrival time of first ask between

pA(0) and pB(0)

Both A and B are exponential with rate

S =S

i=1

((1)i +

(2)i )

aB = a B and bA = b A

we want P[aB bA < 0]

http://find/


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pdf of aB

aB = a B

Let fa, faB be the pdf of a, aB, resp. Then

faB(s) = fa(S + s) +

SS + s

(1 fa(S + s))

Similarly

fbA (s) = fb(S + s) +S

S + s(1 fb(S + s)).

http://find/


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pdf of aB

aB = a B

Let fa, faB be the pdf of a, aB, resp. Then

faB(s) = fa(S + s) +

SS + s(

1 fa(S + s))

Similarly

fbA (s) = fb(S + s) +S

S + s(1 fb(S + s)).

http://find/


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Probability that next move is up

After computing inverse Laplace transforms,

P[aB bA < 0] =

0

0

faB(u)fbA (u z)dudz

http://find/


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Sample outcomes

Table: Probability that mid-price increases at its next move,

= 3.16, (0) = 0.71, (1)0 = 7.46,

(2)0 = 0.80,Sm = 8127. Column

labels indicate the number of initial shares (in multiples of Sm) at best

bid, row labels indicate the number of initial shares at best ask.

b = 1 2 3 4

a= 1 0.50 0.64 0.69 0.72

2 0.35 0.50 0.55 0.59

3 0.30 0.43 0.50 0.534 0.28 0.40 0.47 0.50

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Prob. of executing an ask before price moves down

Table: Probability of executing an ask order at current best ask beforethe mid-price moves downward, S = 1, when the order size is 1Sm.

b = 1 2 3 4

a= 1 0.59 0.83 0.92 0.96

2 0.56 0.79 0.89 0.94

3 0.54 0.77 0.87 0.93

4 0.52 0.75 0.86 0.92

Table: Same, when the order size is 2Sm.

b = 1 2 3 4

a= 1 0.55 0.78 0.88 0.94

2 0.53 0.76 0.86 0.92

3 0.52 0.74 0.85 0.91

4 0.51 0.73 0.84 0.90

http://find/


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Papers

H. Huang and A. Kercheval, A generalized birth-death

stochastic model for high-frequency order book dynamics.

2011, www.math.fsu.edu/~kercheva/papers/

R. Cont, S. Stoikov, R. Talreja, A stochastic model for

order book dynamics. Operations Research, 2010, 58,

549563.
http://www.math.fsu.edu/~kercheva/papers/http://www.math.fsu.edu/~kercheva/papers/http://find/

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A Generalized Birth-Death Stochastic Model 4HF Order Dynamics

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