(In)Stability

Properties

of

Limit

Order

Dynamics

Eyal

Even-Dar

Sham

M.

Kakade

Michael

Kearns

Yishay

Mansour

ABSTRACT

We

study

the

stability

properties

of

the

dynamics

of

the

standard

continuous

limit-order

mechanism

that

is

used

in

modern

equity

markets.

We

ask

whether

such

mechanisms

are

susceptible

to

butterfly

effects

the

infliction

of

large

changes

on

common

measures

of

market

activity

by

only

small

perturbations

of

the

order

sequence.

We

show

that

the

answer

depends

strongly

on

whether

the

market

consists

of

absolute

traders

(who

determine

their

prices

independent

of

the

current

order

book

state)

or

relative

traders

(who

determine

their

prices

relative

to

the

current

bid

and

ask).

We

prove

that

while

the

absolute

trader

model

enjoys

provably

strong

stability

properties,

the

relative

trader

model

is

vulnerable

to

great

instability.

Our

theoretical

results

are

supported

by

large-scale

experiments

using

limit

order

data

from

INET,

a

large

electronic

exchange

for

NASDAQ

stocks.

Categories

and

Subject

Descriptors

[Social

and

Behavioral

Sciences]:

Economics

General

Terms

Economics,

Theory

INTRODUCTION

In

recent

years

there

has

been

an

explosive

increase

in

the

automation

of

modern

equity

markets.

This

increase

has

taken

place

both

in

the

exchanges,

which

are

increasingly

computerized

and

offer

sophisticated

interfaces

for

order

placement

and

management,

and

in

the

trading

activity

itself,

which

is

ever

more

frequently

undertaken

by

software.

The

so-called

Electronic

Communication

Networks

(or

ECNs)

that

dominate

trading

in

NASDAQ

stocks

are

a

common

example

of

the

automation

of

the

exchanges.

On

the

trading

side,

computer

programs

now

are

entrusted

not

only

with

the

careful

execution

of

large

block

trades

for

clients

(sometimes

referred

to

on

Wall

Street

as

program

trading),

but

with

the

autonomous

selection

of

stocks,

direction

(long

or

short)

and

volumes

to

trade

for

profit

(commonly

referred

to

as

statistical

arbitrage).

The

vast

majority

of

equity

trading

is

done

via

the

standard

limit

order

market

mechanism.

In

this

mechanism,

continuous

trading

takes

place

via

the

arrival

of

limit

orders

specifying

whether

the

party

wishes

to

buy

or

sell,

the

volume

desired,

and

the

price

offered.

Arriving

limit

orders

that

are

entirely

or

partially

executable

with

the

best

offers

on

the

other

side

are

executed

immediately,

with

any

volume

not

immediately

executable

being

placed

in

an

queue

(or

book)

ordered

by

price

on

the

appropriate

side

(buy

or

sell).

(A

detailed

description

of

the

limit

order

mechanism

is

given

in

Section

While

traders

have

always

been

able

to

view

the

prices

at

the

top

of

the

buy

and

sell

books

(known

as

the

bid

and

ask),

a

relatively

recent

development

in

certain

exchanges

is

the

real-time

revelation

of

the

entire

order

book

the

complete

distribution

of

orders,

prices

and

volumes

on

both

sides

of

the

exchange.

With

this

revelation

has

come

the

opportunity

and

increasingly,

the

needfor

modeling

and

exploiting

limit

order

data

and

dynamics.

It

is

fair

to

say

that

market

microstructure,

as

this

area

is

generally

known,

is

a

topic

commanding

great

interest

both

in

the

real

markets

and

in

the

academic

finance

literature.

The

opportunities

and

needs

span

the

range

from

the

optimized

execution

of

large

trades

to

the

creation

of

stand-alone

proprietary

strategies

that

attempt

to

profit

from

high-frequency

microstructure

signals.

In

this

paper

we

investigate

a

previously

unexplored

but

fundamental

aspect

of

limit

order

microstructure:

the

stability

properties

of

the

dynamics.

Specifically,

we

are

interested

in

the

following

natural

question:

To

what

extent

are

simple

models

of

limit

order

markets

either

susceptible

or

immune

to

butterfly

effects

that

is,

the

infliction

of

large

changes

in

important

activity

statistics

(such

as

the

number

of

shares

traded

or

the

average

price

per

share)

by

only

minor

perturbations

of

the

order

sequence?

To

examine

this

question,

we

consider

two

stylized

but

natural

models

of

the

limit

order

arrival

process.

In

the

absolute

price

model,

buyers

and

sellers

arrive

with

limit

order

prices

that

are

determined

independently

of

the

current

state

of

the

market

(as

represented

by

the

order

books),

though

they

may

depend

on

all

manner

of

exogenous

information

or

shocks,

such

as

time,

news

events,

announcements

from

the

company

whose

shares

are

being

traded,

private

signals

or

state

of

the

individual

traders,

etc.

This

process

models

traditional

fundamentals-based

trading,

in

which

market

participants

each

have

some

inherent

but

possibly

varying

valuation

for

the

good

that

in

turn

determines

their

limit

price.

In

contrast,

in

the

relative

price

model,

traders

express

their

limit

order

prices

relative

to

the

best

price

offered

in

their

respective

book

(buy

or

sell).

Thus,

a

buyer

would

encode

their

limit

order

price

as

an

offset

(which

may

be

positive,

negative,

or

zero)

from

the

current

bid

pb,

which

is

then

translated

to

the

limit

price

pb

Again,

in

addition

to

now

depending

on

the

state

of

the

order

books,

prices

may

also

depend

on

all

manner

of

exogenous

information.

The

relative

price

model

can

be

viewed

as

modeling

traders

who,

in

addition

to

perhaps

incorporating

fundamental

external

information

on

the

stock,

may

also

position

their

orders

strategically

relative

to

the

other

orders

on

their

side

of

the

book.

A

common

example

of

such

strategic

behavior

is

known

as

penny-jumping

on

Wall

Street,

in

which

a

trader

who

has

in

interest

in

buying

shares

quickly,

but

still

at

a

discount

to

placing

a

market

order,

will

deliberately

position

their

order

just

above

the

current

bid.

More

generally,

the

entire

area

of

modern

execution

optimization

has

come

to

rely

heavily

on

the

careful

positioning

of

limit

orders

relative

to

the

current

order

book

state.

Note

that

such

positioning

may

depend

on

more

complex

features

of

the

order

books

than

just

the

current

bid

and

ask,

but

the

relative

model

is

a

natural

and

simplified

starting

point.

We

remark

that

an

alternate

view

of

the

two

models

is

that

all

traders

behave

in

a

relative

manner,

but

with

absolute

traders

able

to

act

only

on

a

considerably

slower

time

scale

than

the

faster

relative

traders.

How

do

these

two

models

differ?

Clearly,

given

any

fixed

sequence

of

arriving

limit

order

prices,

we

can

choose

to

express

these

prices

either

as

their

original

(absolute)

values,

or

we

can

run

the

order

book

dynamical

process

and

transform

each

order

into

a

relative

difference

with

the

top

of

its

book,

and

obtain

identical

results.

The

differences

arise

when

we

consider

the

stability

question

introduced

above.

Intuitively,

in

the

absolute

model

a

small

perturbation

in

the

arriving

limit

price

sequence

should

have

limited

(but

still

some)

effects

on

the

subsequent

evolution

of

the

order

books,

since

prices

are

determined

independently.

For

the

relative

model

this

intuition

is

less

clear.

It

seems

possible

that

a

small

perturbation

could

(for

example)

slightly

modify

the

current

bid,

which

in

turn

could

slightly

modify

the

price

of

the

next

arriving

order,

which

could

then

slightly

modify

the

price

of

the

subsequent

order,

and

so

on,

leading

to

an

amplifying

sequence

of

events.

Our

main

results

demonstrate

that

these

two

models

do

indeed

have

dramatically

different

stability

properties.

We

first

show

that

for

any

fixed

sequence

of

prices

in

the

absolute

model,

the

modification

of

a

single

order

has

a

bounded

and

extremely

limited

impact

on

the

subsequent

evolution

of

the

books.

In

particular,

we

define

a

natural

notion

of

distance

between

order

books

and

show

that

small

modifications

can

result

in

only

constant

distance

to

the

original

books

for

all

subsequent

time

steps.

We

then

show

that

this

implies

that

for

almost

any

standard

statistic

of

market

activity

the

executed

volume,

the

average

price

execution

price,

and

many

others

the

statistic

can

be

influenced

only

infinitesimally

by

small

perturbations.

In

contrast,

we

show

that

the

relative

model

enjoys

no

such

stability

properties.

After

giving

specific

(worst-case)

relative

price

sequences

in

which

small

perturbations

generate

large

changes

in

basic

statistics

(for

example,

altering

the

number

of

shares

traded

by

a

factor

of

two),

we

proceed

to

demonstrate

that

the

difference

in

stability

properties

of

the

two

models

is

more

than

merely

theoretical.

Using

extensive

INET

(a

major

ECN

for

NASDAQ

stocks)

limit

order

data

and

order

book

reconstruction

code,

we

investigate

the

empirical

stability

properties

when

the

data

is

interpreted

as

containing

either

absolute

prices,

relative

prices,

or

mixtures

of

the

two.

The

theoretical

predictions

of

stability

and

instability

are

strongly

borne

out

by

the

subsequent

experiments.

In

addition

to

stability

being

of

fundamental

interest

in

any

important

dynamical

system,

we

believe

that

the

results

described

here

provide

food

for

thought

on

the

topics

of

market

impact

and

the

backtesting

of

quantitative

trading

strategies

(the

attempt

to

determine

hypothetical

past

performance

using

historical

data).

They

suggest

that

one"s

confidence

that

trading

quietly

and

in

small

volumes

will

have

minimal

market

impact

is

linked

to

an

implicit

belief

in

an

absolute

price

model.

Our

results

and

the

fact

that

in

the

real

markets

there

is

a

large

and

increasing

amount

of

relative

behavior

such

as

penny-jumping

would

seem

to

cast

doubts

on

such

beliefs.

Similarly,

in

a

purely

or

largely

relative-price

world,

backtesting

even

low-frequency,

low-volume

strategies

could

result

in

historical

estimates

of

performance

that

are

not

only

unrelated

to

future

performance

(the

usual

concern),

but

are

not

even

accurate

measures

of

a

hypothetical

past.

The

outline

of

the

paper

follows.

In

Section

we

briefly

review

the

large

literature

on

market

microstructure.

In

Section

we

describe

the

limit

order

mechanism

and

our

formal

models.

Section

presents

our

most

important

theoretical

results,

the

Theorem

for

the

absolute

price

model.

This

theorem

is

applied

in

Section

to

derive

a

number

of

strong

stability

properties

in

the

absolute

model.

Section

presents

specific

examples

establishing

the

worstcase

instability

of

the

relative

model.

Section

contains

the

simulation

studies

that

largely

confirm

our

theoretical

findings

on

INET

market

data.

RELATED

WORK

As

was

mentioned

in

the

Introduction,

market

microstructure

is

an

important

and

timely

topic

both

in

academic

finance

and

on

Wall

Street,

and

consequently

has

a

large

and

varied

recent

literature.

Here

we

have

space

only

to

summarize

the

main

themes

of

this

literature

and

to

provide

pointers

to

further

readings.

To

our

knowledge

the

stability

properties

of

detailed

limit

order

microstructure

dynamics

have

not

been

previously

considered.

(However,

see

Farmer

and

Joshi

for

an

example

and

survey

of

other

price

dynamic

stability

studies.)

On

the

more

theoretical

side,

there

is

a

rich

line

of

work

examining

what

might

be

considered

the

game-theoretic

properties

of

limit

order

markets.

These

works

model

traders

and

market-makers

(who

provide

liquidity

by

offering

both

buy

and

sell

quotes,

and

profit

on

the

difference)

by

utility

functions

incorporating

tolerance

for

risks

of

price

movement,

large

positions

and

other

factors,

and

examine

the

resulting

equilibrium

prices

and

behaviors.

Common

findings

predict

negative

price

impacts

for

large

trades,

and

price

effects

for

large

inventory

holdings

by

market-makers.

An

excellent

and

comprehensive

survey

of

results

in

this

area

can

be

found

in

There

is

a

similarly

large

body

of

empirical

work

on

microstructure.

Major

themes

include

the

measurement

of

price

impacts,

statistical

properties

of

limit

order

books,

and

attempts

to

establish

the

informational

value

of

order

books

A

good

overview

of

the

empirical

work

can

be

found

in

Of

particular

note

for

our

interests

is

which

empirically

studies

the

distribution

of

arriving

limit

order

prices

in

several

prominent

markets.

This

work

takes

a

view

of

arriving

prices

analogous

to

our

relative

model,

and

establishes

a

power-law

form

for

the

resulting

distributions.

There

is

also

a

small

but

growing

number

of

works

examining

market

microstructure

topics

from

a

computer

science

perspective,

including

some

focused

on

the

use

of

microstructure

in

algorithms

for

optimized

trade

execution.

Kakade

et

al.

introduced

limit

order

dynamics

in

competitive

analysis

for

one-way

and

volume-weighted

average

price

(VWAP)

trading.

Some

recent

papers

have

applied

reinforcement

learning

methods

to

trade

execution

using

order

book

properties

as

state

variables

MICROSTRUCTURE

PRELIMINARIES

The

following

expository

background

material

is

adapted

from

The

market

mechanism

we

examine

in

this

paper

is

driven

by

the

simple

and

standard

concept

of

a

limit

order.

Suppose

we

wish

to

purchase

shares

of

Microsoft

(MSFT)

stock.

In

a

limit

order,

we

specify

not

only

the

desired

volume

shares),

but

also

the

desired

price.

Suppose

that

MSFT

is

currently

trading

at

roughly

a

share

(see

Figure

which

shows

an

actual

snapshot

of

an

MSFT

order

book

on

INET),

but

we

are

only

willing

to

buy

the

shares

at

a

share

or

lower.

We

can

choose

to

submit

a

limit

order

with

this

specification,

and

our

order

will

be

placed

in

a

queue

called

the

buy

order

book,

which

is

ordered

by

price,

with

the

highest

offered

unexecuted

buy

price

at

the

top

(often

referred

to

as

the

bid).

If

there

are

multiple

limit

orders

at

the

same

price,

they

are

ordered

by

time

of

arrival

(with

older

orders

higher

in

the

book).

In

the

example

provided

by

Figure

our

order

would

be

placed

immediately

after

the

extant

order

for

shares

at

though

we

offer

the

same

price,

this

order

has

arrived

before

ours.

Similarly,

a

sell

order

book

for

sell

limit

orders

is

maintained,

this

time

with

the

lowest

sell

price

offered

(often

referred

to

as

the

ask)

at

its

top.

Thus,

the

order

books

are

sorted

from

the

most

competitive

limit

orders

at

the

top

(high

buy

prices

and

low

sell

prices)

down

to

less

competitive

limit

orders.

The

bid

and

ask

prices

together

are

sometimes

referred

to

as

the

inside

market,

and

the

difference

between

them

as

the

spread.

By

definition,

the

order

books

always

consist

exclusively

of

unexecuted

orders

they

are

queues

of

orders

hopefully

waiting

for

the

price

to

move

in

their

direction.

Figure

Sample

INET

order

books

for

MSFT.

How

then

do

orders

get

(partially)

executed?

If

a

buy

(sell,

respectively)

limit

order

comes

in

above

the

ask

(below

the

bid,

respectively)

price,

then

the

order

is

matched

with

orders

on

the

opposing

books

until

either

the

incoming

order"s

volume

is

filled,

or

no

further

matching

is

possible,

in

which

case

the

remaining

incoming

volume

is

placed

in

the

books.

For

instance,

suppose

in

the

example

of

Figure

a

buy

order

for

shares

arrived

with

a

limit

price

of

This

order

would

be

partially

filled

by

the

two

sell

orders

at

in

the

sell

books,

the

sell

order

at

and

the

sell

order

at

for

a

total

of

shares

executed.

The

remaining

shares

of

the

incoming

buy

order

would

become

the

new

bid

of

the

buy

book

at

It

is

important

to

note

that

the

prices

of

executions

are

the

prices

specified

in

the

limit

orders

already

in

the

books,

not

the

prices

of

the

incoming

order

that

is

immediately

executed.

Thus

in

this

example,

the

executed

shares

would

be

at

different

prices.

Note

that

this

also

means

that

in

a

pure

limit

order

exchange

such

as

INET,

market

orders

can

be

simulated

by

limit

orders

with

extreme

price

values.

In

exchanges

such

as

INET,

any

order

can

be

withdrawn

or

canceled

by

the

party

that

placed

it

any

time

prior

to

execution.

Every

limit

order

arrives

atomically

and

instantaneously

there

is

a

strict

temporal

sequence

in

which

orders

arrive,

and

two

orders

can

never

arrive

simultaneously.

This

gives

rise

to

the

definition

of

the

last

price

of

the

exchange,

which

is

simply

the

last

price

at

which

the

exchange

executed

an

order.

It

is

this

quantity

that

is

usually

meant

when

people

casually

refer

to

the

(ticker)

price

of

a

stock.

Formal

Definitions

We

now

provide

a

formal

model

for

the

limit

order

cess

described

above.

In

this

model,

limit

orders

arrive

in

a

temporal

sequence,

with

each

order

specifying

its

limit

price

and

an

indication

of

its

type

(buy

or

sell).

Like

the

actual

exchanges,

we

also

allow

cancellation

of

a

standing

(unexecuted)

order

in

the

books

any

time

prior

to

its

execution.

Without

loss

of

generality

we

limit

attention

to

a

model

in

which

every

order

is

for

a

single

share;

large

order

volumes

can

be

represented

by

sequences.

Definition

Let

Σ

...σn

be

a

sequence

of

limit

orders,

where

each

σi

has

the

form

ni,

ti,

vi

Here

ni

is

an

order

identifier,

ti

is

the

order

type

(buy,

sell,

or

cancel),

and

vi

is

the

limit

order

value.

In

the

case

that

ti

is

a

cancel,

ni

matches

a

previously

placed

order

and

vi

is

ignored.

We

have

deliberately

called

vi

in

the

definition

above

the

limit

order

value

rather

than

price,

since

our

two

models

will

differ

in

their

interpretation

of

vi

(as

being

absolute

or

relative).

In

the

absolute

model,

we

do

indeed

interpret

vi

as

simply

being

the

price

of

the

limit

order.

In

the

relative

model,

if

the

current

order

book

configuration

is

(A,

B)

(where

A

is

the

sell

and

B

the

buy

book),

the

price

of

the

order

is

ask(A)

vi

if

ti

is

sell,

and

bid(B)

vi

if

ti

is

buy,

where

by

ask(X)

and

bid(X)

we

denote

the

price

of

the

order

at

the

top

of

the

book

X.

(Note

vi

can

be

negative.)

Our

main

interest

in

this

paper

is

the

effects

that

the

modification

of

a

small

number

of

limit

orders

can

have

on

the

resulting

dynamics.

For

simplicity

we

consider

only

modifications

to

the

limit

order

values,

but

our

results

generalize

to

any

modification.

Definition

A

k-modification

of

Σ

is

a

sequence

Σ

such

that

for

exactly

k

indices

ik

vij

vij

tij

tij

and

nij

nij

For

every

ij

j

k}

σ

σ

We

now

define

the

various

quantities

whose

stability

properties

we

examine

in

the

absolute

and

relative

models.

All

of

these

are

standard

quantities

of

common

interest

in

financial

markets.

volume(Σ):

Number

of

shares

executed

(traded)

in

the

sequence

Σ.

average(Σ):

Average

execution

price.

close(Σ):

Price

of

the

last

(closing)

execution.

lastbid(Σ):

Bid

at

the

end

of

the

sequence.

lastask(Σ):

Ask

at

end

of

the

sequence.

THE

THEOREM

In

this

section

we

provide

our

most

important

technical

result.

It

shows

that

in

the

absolute

model,

the

effects

that

the

modification

of

a

single

order

has

on

the

resulting

evolution

of

the

order

books

is

extremely

limited.

We

then

apply

this

result

to

derive

strong

stability

results

for

all

of

the

aforementioned

quantities

in

the

absolute

model.

Throughout

this

section,

we

consider

an

arbitrary

order

sequence

Σ

in

the

absolute

model,

and

any

Σ

of

Σ.

At

any

point

(index)

i

in

the

two

sequences

we

shall

use

to

denote

the

sell

and

buy

books

(respectively)

in

Σ,

and

to

denote

the

sell

and

buy

books

in

Σ

for

notational

convenience

we

omit

explicitly

superscripting

by

the

current

index

i.

We

will

shortly

establish

that

at

all

times

i,

and

are

very

close.

Although

the

order

books

are

sorted

by

price,

we

will

use

(for

example)

to

indicate

that

contains

an

order

at

some

price

that

is

not

present

in

but

that

otherwise

and

are

identical;

thus

deleting

the

order

at

in

would

render

the

books

the

same.

Similarly,

means

contains

an

order

at

price

not

present

in

contains

an

order

at

price

not

present

in

and

that

otherwise

and

are

identical.

Using

this

notation,

we

now

define

a

set

of

stable

system

states,

where

each

state

is

composed

from

the

order

books

of

the

original

and

the

modified

sequences.

Shortly

we

show

that

if

we

change

only

one

order"s

value

(price),

we

remain

in

this

set

for

any

sequence

of

limit

orders.

Definition

Let

ab

be

the

set

of

all

states

and

such

that

and

Let

¯ab

be

the

set

of

states

such

that

where

and

Let

a¯b

be

the

set

of

states

such

that

where

and

Let

¯a¯b

be

the

set

of

states

in

which

and

or

in

which

and

Finally

we

define

S

ab

¯ab

¯ba

¯a¯b

as

the

set

of

stable

states.

Theorem

Theorem)

Consider

any

sequence

of

orders

Σ

and

any

Σ

of

Σ.

Then

the

order

books

and

determined

by

Σ

and

Σ

lie

in

the

set

S

of

stable

states

at

all

times.

ab

¯a¯b

a¯b¯ab

Figure

Diagram

representing

the

set

S

of

stable

states

and

the

possible

movements

transitions

in

it

after

the

change.

The

idea

of

the

proof

of

this

theorem

is

contained

in

Figure

which

shows

a

state

transition

diagram

labeled

by

the

categories

of

stable

states.

This

diagram

describes

all

transitions

that

can

take

place

after

the

arrival

of

the

order

on

which

Σ

and

Σ

differ.

The

following

establishes

that

immediately

after

the

arrival

of

this

differing

order,

the

state

lies

in

S.

Lemma

If

at

any

time

the

current

books

and

are

in

the

set

ab

(and

thus

identical),

then

modifying

the

price

of

the

next

order

keeps

the

state

in

S.

Proof.

Suppose

the

arriving

order

is

a

sell

order

and

we

change

it

from

to

assume

without

loss

of

generality

that

If

neither

order

is

executed

immediately,

then

we

move

to

state

¯ab;

if

both

of

them

are

executed

then

we

stay

in

state

ab;

and

if

only

is

executed

then

we

move

to

state

¯a¯b.

The

analysis

of

an

arriving

buy

order

is

similar.

Following

the

arrival

of

their

only

differing

order,

Σ

and

Σ

are

identical.

We

now

give

a

sequence

of

lemmas

showing

Executed

with

two

orders

Not

executed

in

both

Arrivng

buy

order

Arriving

buy

order

Arriving

buy

order

Arriving

sell

order

¯ab

ab

¯a¯b

Executed

only

with

(not

and

Executed

with

and

Figure

The

state

diagram

when

starting

at

state

¯ab.

This

diagram

provides

the

intuition

of

Lemma

that

following

the

initial

difference

covered

by

Lemma

the

state

remains

in

S

forever

on

the

remaining

(identical)

sequence.

We

first

show

that

from

state

¯ab

we

remain

in

S

regardless

the

next

order.

The

intuition

of

this

lemma

is

demonstrated

in

Figure

Lemma

If

the

current

state

is

in

the

set

¯ab,

then

for

any

order

the

state

will

remain

in

S.

Proof.

We

first

provide

the

analysis

for

the

case

of

an

arriving

sell

order.

Note

that

in

¯ab

the

buy

books

are

identical

Thus

either

the

arriving

sell

order

is

executed

with

the

same

buy

order

in

both

buy

books,

or

it

is

not

executed

in

both

buy

books.

For

the

first

case,

the

buy

books

remain

identical

(the

bid

is

executed

in

both)

and

the

sell

books

remain

unchanged.

For

the

second

case,

the

buy

books

remain

unchanged

and

identical,

and

the

sell

books

have

the

new

sell

order

added

to

both

of

them

(and

thus

still

differ

by

one

order).

Next

we

provide

an

analysis

of

the

more

subtle

case

where

the

arriving

item

is

a

buy

order.

For

this

case

we

need

to

take

care

of

several

different

scenarios.

The

first

is

when

the

top

of

both

sell

books

(the

ask)

is

identical.

Then

regardless

of

whether

the

new

buy

order

is

executed

or

not,

the

state

remains

in

¯ab

(the

analysis

is

similar

to

an

arriving

sell

order).

We

are

left

to

deal

with

case

where

and

are

different.

Here

we

discuss

two

subcases:

(a)

and

and

(b)

and

a

Here

and

are

as

in

the

definition

of

¯ab

in

Definition

and

a

is

some

other

price.

For

subcase

(a),

by

our

assumption

then

either

both

asks

get

executed,

the

sell

books

become

identical,

and

we

move

to

state

ab;

neither

ask

is

executed

and

we

remain

in

state

¯ab;

or

only

is

executed,

in

which

case

we

move

to

state

¯a¯b

with

and

where

is

the

arriving

buy

order

price.

For

subcase

(b),

either

buy

order

is

executed

in

neither

sell

book

we

remain

in

state

¯ab;

or

the

buy

order

is

executed

in

both

sell

books

and

stay

in

state

¯ab

with

{a

or

only

is

executed

and

we

move

to

state

¯a¯b.

Lemma

If

the

current

state

is

in

the

set

a¯b,

then

for

any

order

the

state

will

remain

in

S.

Lemma

If

the

current

configuration

is

in

the

set

¯a¯b,

then

for

any

order

the

state

will

remain

in

S

The

proofs

of

these

two

lemmas

are

omitted,

but

are

similar

in

spirit

to

that

of

Lemma

The

next

and

final

lemma

deals

with

cancellations.

Lemma

If

the

current

order

book

state

lies

in

S,

then

following

the

arrival

of

a

cancellation

it

remains

in

S.

Proof.

When

a

cancellation

order

arrives,

one

of

the

following

possibilities

holds:

the

order

is

still

in

both

sets

of

books,

it

is

not

in

either

of

them

and

it

is

only

in

one

of

them.

For

the

first

two

cases

it

is

easy

to

see

that

the

cancellation

effect

is

identical

on

both

sets

of

books,

and

thus

the

state

remains

unchanged.

For

the

case

when

the

order

appears

only

in

one

set

of

books,

without

loss

of

generality

we

assume

that

the

cancellation

cancels

a

buy

order

at

Rather

than

removing

from

the

book

we

can

change

it

to

have

price

meaning

this

buy

order

will

never

be

executed

and

is

effectively

canceled.

Now

regardless

the

state

that

we

were

in,

is

still

only

in

one

buy

book

(but

with

a

different

price),

and

thus

we

remain

in

the

same

state

in

S.

The

proof

of

Theorem

follows

from

the

above

lemmas.

ABSOLUTE

MODEL

STABILITY

In

this

section

we

apply

the

Theorem

to

show

strong

stability

properties

for

the

absolute

model.

We

begin

with

an

examination

of

the

executed

volume.

Lemma

Let

Σ

be

any

sequence

and

Σ

be

any

of

Σ.

Then

the

set

of

the

executed

orders

(ID

numbers)

generated

by

the

two

sequences

differs

by

at

most

Proof.

By

Theorem

we

know

that

at

each

stage

the

books

differ

by

at

most

two

orders.

Now

since

the

union

of

the

IDs

of

the

executed

orders

and

the

order

books

is

always

identical

for

both

sequences,

this

implies

that

the

executed

orders

can

differ

by

at

most

two.

Corollary

Let

Σ

be

any

sequence

and

Σ

be

any

kmodification

of

Σ.

Then

the

set

of

the

executed

orders

(ID

numbers)

generated

by

the

two

sequences

differs

by

at

most

An

order

sequence

Σ

is

a

k-extension

of

Σ

if

Σ

can

be

obtained

by

deleting

any

k

orders

in

Σ

Lemma

Let

Σ

be

any

sequence

and

let

Σ

be

any

kextension

of

Σ.

Then

the

set

of

the

executed

orders

generated

by

Σ

and

Σ

differ

by

at

most

This

lemma

is

the

key

to

obtain

our

main

absolute

model

volume

result

below.

We

use

edit(Σ,

Σ

to

denote

the

standard

edit

distance

between

the

sequences

Σ

and

Σ

the

minimal

number

of

substitutions,

insertions

and

deletions

or

orders

needed

to

change

Σ

to

Σ

Theorem

Let

Σ

and

Σ

be

any

absolute

model

order

sequences.

Then

if

edit(Σ,

Σ

k,

the

set

of

the

executed

orders

generated

by

Σ

and

Σ

differ

by

at

most

In

particular,

|volume(Σ)

volume(Σ

Proof.

We

first

define

the

sequence

˜Σ

which

is

the

intersection

of

Σ

and

Σ

Since

Σ

and

Σ

are

at

most

k

apart,we

have

that

by

k

insertions

we

change

˜Σ

to

either

Σ

or

Σ

and

by

Lemma

its

set

of

executed

orders

is

at

most

from

each.

Thus

the

set

of

executed

orders

in

Σ

and

Σ

is

at

most

apart.

Spread

Bounds

Theorem

establishes

strong

stability

for

executed

volume

in

the

absolute

model.

We

now

turn

to

the

quantities

that

involve

execution

prices

as

opposed

to

volume

alone

namely,

average(Σ),

close(Σ),

lastbid(Σ)

and

lastask(Σ).

For

these

results,

unlike

executed

volume,

a

condition

must

hold

on

Σ

in

order

for

stability

to

occur.

This

condition

is

expressed

in

terms

of

a

natural

measure

of

the

spread

of

the

market,

or

the

gap

between

the

buyers

and

sellers.

We

motivate

this

condition

by

first

showing

that

without

it,

by

changing

one

order,

we

can

change

average(Σ)

by

any

positive

value

x.

Lemma

There

exists

Σ

such

that

for

any

x

there

is

a

Σ

of

Σ

such

that

average(Σ

average(Σ)

x.

Proof.

Let

Σ

be

a

sequence

of

alternating

sell

and

buy

orders

in

which

each

seller

offers

p

and

each

buyer

p

x,

and

the

first

order

is

a

sell.

Then

all

executions

take

place

at

the

ask,

which

is

always

p,

and

thus

average(Σ)

p.

Now

suppose

we

modify

only

the

first

sell

order

to

be

at

price

This

initial

sell

order

will

never

be

executed,

and

now

all

executions

take

place

at

the

bid,

which

is

always

p

x.

Similar

instability

results

can

be

shown

to

hold

for

the

other

price-based

quantities.

This

motivates

the

introduction

of

a

quantity

we

call

the

second

spread

of

the

order

books,

which

is

defined

as

the

difference

between

the

prices

of

the

second

order

in

the

sell

book

and

the

second

order

in

the

buy

book

(as

opposed

to

the

bid-ask

difference,

which

is

commonly

called

the

spread).

We

note

that

in

a

liquid

stock,

such

as

those

we

examine

experimentally

in

Section

the

second

spread

will

typically

be

quite

small

and

in

fact

almost

always

equal

to

the

spread.

In

this

subsection

we

consider

changes

in

the

sequence

only

after

an

initialization

period,

and

sequences

such

that

the

second

spread

is

always

defined

after

the

time

we

make

a

change.

We

define

to

be

the

maximum

second

spread

in

the

sequence

Σ

following

the

change.

Theorem

Let

Σ

be

a

sequence

and

let

Σ

be

any

of

Σ.

Then

|lastbid(Σ)

lastbid(Σ

|lastask(Σ)

lastask(Σ

where

is

the

maximum

over

the

second

spread

in

Σ

following

the

Proof.

We

provide

the

proof

for

the

last

bid;

the

proof

for

the

last

ask

is

similar.

The

proof

relies

on

Theorem

and

considers

states

in

the

stable

set

S.

For

states

ab

and

¯ab,

we

have

that

the

bid

is

identical.

Let

bid(X),

sb(X),

ask(X),

be

the

bid,

the

second

highest

buy

order,

and

the

ask

of

a

sequence

X.

Now

recall

that

in

state

a¯b

we

have

that

the

sell

books

are

identical,

and

that

the

two

buy

books

are

identical

except

one

different

order.

Thus

ask(Σ)

ask(Σ

bid(Σ

Now

it

remains

to

bound

bid(Σ).

Here

we

use

the

fact

that

the

bid

of

the

modified

sequence

is

at

least

the

second

highest

buy

order

in

the

original

sequence,

due

to

the

fact

that

the

books

are

different

only

in

one

order.

Since

bid(Σ

sb(Σ)

ask(Σ)

bid(Σ)

we

have

that

|bid(Σ)

bid(Σ

as

desired.

In

state

¯a¯b

we

have

that

for

one

sequence

the

books

contain

an

additional

buy

order

and

an

additional

sell

order.

First

suppose

that

the

books

containing

the

additional

orders

are

the

original

sequence

Σ.

Now

if

the

bid

is

not

the

additional

order

we

are

done,

otherwise

we

have

the

following:

bid(Σ)

ask(Σ)

sb(Σ)

bid(Σ

where

sb(Σ)

bid(Σ

since

the

original

buy

book

has

only

one

additional

order.

Now

assume

that

the

books

with

the

additional

orders

are

for

the

modified

sequence

Σ

We

have

bid(Σ)

ask(Σ)

ask(Σ

bid(Σ

where

we

used

the

fact

that

ask(Σ)

ask(Σ

since

the

modified

sequence

has

an

additional

order.

Similarly

we

have

that

bid(Σ)

bid(Σ

since

the

modified

buy

book

contains

an

additional

order.

We

note

that

the

proof

of

Theorem

actually

establishes

that

the

bid

and

ask

of

the

original

and

modified

sequences

are

within

at

all

times.

Next

we

provide

a

technical

lemma

which

relates

the

(first)

spread

of

the

modified

sequence

to

the

second

spread

of

the

original

sequence.

Lemma

Let

Σ

be

a

sequence

and

let

Σ

be

any

of

Σ.

Then

the

spread

of

Σ

is

bounded

by

Proof.

By

the

Theorem,

we

know

that

the

books

of

the

modified

sequence

and

the

original

sequence

can

differ

by

at

most

one

order

in

each

book

(buy

and

sell).

Therefore,

the

second-highest

buy

order

in

the

original

sequence

is

always

at

most

the

bid

in

the

modified

sequence,

and

the

second-lowest

sell

order

in

the

original

sequence

is

always

at

least

the

ask

of

the

modified

sequence.

We

are

now

ready

to

state

a

stability

result

for

the

average

execution

price

in

the

absolute

model.

It

establishes

that

in

highly

liquid

markets,

where

the

executed

volume

is

large

and

the

spread

small,

the

average

price

is

highly

stable.

Theorem

Let

Σ

be

a

sequence

and

let

Σ

be

any

of

Σ.

Then

|average(Σ)

average(Σ

volume(Σ)

where

pmax

is

the

highest

execution

price

in

Σ.

Proof.

The

proof

will

show

that

every

execution

in

Σ

besides

the

execution

of

the

modified

order

and

the

last

execution

has

a

matching

execution

in

Σ

with

a

price

different

by

at

most

and

will

use

the

fact

that

pmax

is

a

bound

on

the

price

in

Σ

Referring

to

the

proof

of

the

Theorem,

suppose

we

are

in

state

¯a¯b,

where

we

have

in

one

sequence

(which

can

be

either

Σ

or

Σ

an

additional

buy

order

b

and

an

additional

sell

order

a.

Without

loss

of

generality

we

assume

that

the

sequence

with

the

additional

orders

is

Σ.

If

the

next

execution

does

not

involve

a

or

b

then

clearly

we

have

the

same

execution

in

both

Σ

and

Σ

Suppose

that

it

involves

a;

there

are

two

possibilities.

Either

a

is

the

modified

order,

in

which

case

we

change

the

average

price

difference

by

(pmax

and

this

can

happen

only

once;

or

a

was

executed

before

in

Σ

and

the

executions

both

involve

an

order

whose

limit

price

is

a.

By

Lemma

the

spread

of

both

sequences

is

bounded

by

which

implies

that

the

price

of

the

execution

in

Σ

was

at

most

a

while

execution

is

in

Σ

is

at

price

a,

and

thus

the

prices

are

different

by

at

most

In

states

¯ab,

a¯b

as

long

as

we

have

concurrent

executions

in

the

two

sequences,

we

know

that

the

prices

can

differ

by

at

most

If

we

have

an

execution

only

in

one

sequence,

we

either

match

it

in

state

¯a¯b,

or

charge

it

by

(pmax

if

we

end

at

state

¯a¯b.

If

we

end

in

state

ab,

¯ab

or

a¯b,

then

every

execution

in

states

¯ab

or

a¯b

were

matched

to

an

execution

in

state

¯a¯b.

If

we

end

up

in

state

¯a¯b,

we

have

the

one

execution

that

is

not

matched

and

thus

we

charge

it

(pmax

We

next

give

a

stability

result

for

the

closing

price.

We

first

provide

a

technical

lemma

regarding

the

prices

of

consecutive

executions.

Lemma

Let

Σ

be

any

sequence.

Then

the

prices

of

two

consecutive

executions

in

Σ

differ

by

at

most

Proof.

Suppose

the

first

execution

is

taken

at

time

t;

its

price

is

bounded

below

by

the

current

bid

and

above

by

the

current

ask.

Now

after

this

execution

the

bid

is

at

least

the

second

highest

buy

order

at

time

t,

if

the

former

bid

was

executed

and

no

higher

buy

orders

arrived,

and

higher

otherwise.

Similarly,

the

ask

is

at

most

the

second

lowest

sell

order

at

time

t.

Therefore,

the

next

execution

price

is

at

least

the

second

bid

at

time

t

and

at

most

the

second

ask

at

time

t,

which

is

at

most

away

from

the

bid/ask

at

time

t.

Lemma

Let

Σ

be

any

sequence

and

let

Σ

be

a

of

Σ.

If

the

volume(Σ)

then

|close(Σ)

close(Σ

Proof.

We

first

deal

with

case

where

the

last

execution

occurs

in

both

sequences

simultaneously.

By

Theorem

both

the

ask

and

the

bid

of

Σ

and

Σ

are

at

most

apart

at

every

time

t.

Since

the

price

of

the

last

execution

is

their

asks

(bids)

at

time

t

we

are

done.

Next

we

deal

with

the

case

where

the

last

execution

among

the

two

sequences

occurs

only

in

Σ.

In

this

case

we

know

that

either

the

previous

execution

happened

simultaneously

in

both

sequences

at

time

t,

and

thus

all

three

executions

are

within

the

second

spread

of

Σ

at

time

t

(the

first

execution

in

Σ

by

definition,

the

execution

at

Σ

from

identical

arguments

as

in

the

former

case,

and

the

third

by

Lemma

Otherwise

the

previous

execution

happened

only

in

Σ

at

time

t,

in

which

case

the

two

executions

are

within

the

the

spread

of

Σ

at

time

t

(the

execution

of

Σ

from

the

same

arguments

as

before,

and

the

execution

in

Σ

must

be

inside

its

spread

in

time

t).

If

the

last

execution

happens

only

in

Σ

we

know

that

the

next

execution

of

Σ

will

be

at

most

away

from

its

previous

execution

by

Lemma

Together

with

the

fact

that

if

an

execution

happens

only

in

one

sequence

it

implies

that

the

order

is

in

the

spread

of

the

second

sequence

as

long

as

the

sequences

are

the

proof

is

completed.

Spread

Bounds

for

k-Modifications

As

in

the

case

of

executed

volume,

we

would

like

to

extend

the

absolute

model

stability

results

for

price-based

quantities

to

the

case

where

multiple

orders

are

modified.

Here

our

results

are

weaker

and

depend

on

the

k-spread,

the

distance

between

the

kth

highest

buy

order

and

the

kth

lowest

sell

order,

instead

of

the

second

spread.

(Looking

ahead

to

Section

we

note

that

in

actual

market

data

for

liquid

stocks,

this

quantity

is

often

very

small

as

well.)

We

use

sk(Σ)

to

denote

the

k-spread.

As

before,

we

assume

that

the

k-spread

is

always

defined

after

an

initialization

period.

We

first

state

the

following

generalization

of

Lemma

Lemma

Let

Σ

be

a

sequence

and

let

Σ

be

any

of

Σ.

For

if

s

is

always

defined

after

the

change,

then

s

(Σ

s

The

proof

is

similar

to

the

proof

of

Lemma

and

omitted.

A

simple

application

of

this

lemma

is

the

following:

Let

Σ

be

any

sequence

which

is

an

-modification

of

Σ.

Then

we

have

s

Now

using

the

above

lemma

and

by

simple

induction

we

can

obtain

the

following

theorem.

Theorem

Let

Σ

be

a

sequence

and

let

Σ

be

any

k-modification

of

Σ.

Then

|lastbid(Σ)

lastbid(Σ

Pk

s

|lastask(Σ)−lastask(Σ

Pk

s

|close(Σ)

close(Σ

Pk

s

|average(Σ)

average(Σ

Pk

+s

volume(Σ)

s

where

s

(Σ)

is

the

maximum

over

the

-spread

in

Σ

following

the

first

modification.

We

note

that

while

these

bounds

depend

on

deeper

measures

of

spread

for

more

modifications,

we

are

working

in

a

order

model.

Thus

in

an

actual

market,

where

single

orders

contain

hundreds

or

thousands

of

shares,

the

k-spread

even

for

large

k

might

be

quite

small

and

close

to

the

standard

in

liquid

stocks.

RELATIVE

MODEL

INSTABILITY

In

the

relative

model

the

underlying

assumption

is

that

traders

try

to

exploit

their

knowledge

of

the

books

to

strategically

place

their

orders.

Thus

if

a

trader

wants

her

buy

order

to

be

executed

quickly,

she

may

position

it

above

the

current

bid

and

be

the

first

in

the

queue;

if

the

trader

is

patient

and

believes

that

the

price

trend

is

going

to

be

downward

she

will

place

orders

deeper

in

the

buy

book,

and

so

on.

While

in

the

previous

sections

we

showed

stability

results

for

the

absolute

model,

here

we

provide

simple

examples

which

show

instability

in

the

relative

model

for

the

executed

volume,

last

bid,

last

ask,

average

execution

price

and

the

last

execution

price.

In

Section

we

provide

many

simulations

on

actual

market

data

that

demonstrate

that

this

instability

is

inherent

to

the

relative

model,

and

not

due

to

artificial

constructions.

In

the

relative

model

we

assume

that

for

every

sequence

the

ask

and

bid

are

always

defined,

so

the

books

have

a

non-empty

initial

configuration.

We

begin

by

showing

that

in

the

relative

model,

even

a

single

modification

can

double

the

number

of

shares

executed.

Theorem

There

is

a

sequence

Σ

and

a

Σ

of

Σ

such

that

volume(Σ

Proof.

For

concreteness

we

assume

that

at

the

beginning

the

ask

is

and

the

bid

is

The

sequence

Σ

is

composed

from

n

buy

orders

with

followed

by

n

sell

orders

with

and

finally

an

alternating

sequence

of

buy

orders

with

and

sell

orders

with

−1

of

length

Since

the

books

before

the

alternating

sequence

contain

n

sell

orders

at

and

n

buy

orders

at

we

have

that

each

pair

of

buy

sell

order

in

the

alternating

part

is

matched

and

executed,

but

of

the

initial

orders

is

executed,

and

thus

volume(Σ)

n.

Now

we

change

the

first

buy

order

to

have

After

the

first

orders

there

are

still

no

executions;

however,

the

books

are

different.

Now

there

are

n

sell

orders

at

n

buy

orders

at

and

one

buy

order

at

Now

each

order

in

the

alternating

sequence

is

executed

with

one

of

the

former

orders

and

we

have

volume(Σ

The

next

theorem

shows

that

the

spread-based

stability

results

of

Section

do

not

also

hold

in

the

relative

model.

Before

providing

the

proof,

we

give

its

intuition.

At

the

beginning

the

sell

book

contains

only

two

prices

which

are

far

apart

and

both

contain

only

two

orders,

now

several

buy

orders

arrive,

at

the

original

sequence

they

are

not

being

executed,

while

in

the

modified

sequence

they

will

be

executed

and

leave

the

sell

book

with

only

the

orders

at

the

high

price.

Now

many

sell

orders

followed

by

many

buy

orders

will

arrive,

such

that

in

the

original

sequence

they

will

be

executed

only

at

the

low

price

and

in

the

modified

sequence

they

will

executed

at

the

high

price.

Theorem

For

any

positive

numbers

s

and

x,

there

is

sequence

Σ

such

that

s

and

a

Σ

of

Σ

such

that

|close(Σ)

close(Σ

x

|average(Σ)

average(Σ

x

|lastbid(Σ)

lastbid(Σ

x

|lastask(Σ)

lastask(Σ

x

Proof.

Without

loss

of

generality

let

us

consider

sequences

in

which

all

prices

are

integer-valued,

in

which

case

the

smallest

possible

value

for

the

second

spread

is

we

provide

the

proof

for

the

case

but

the

case

is

similar.

We

consider

a

sequence

Σ

such

that

after

an

initialization

period

there

have

been

no

executions,

the

buy

book

has

orders

at

price

and

the

sell

book

has

two

orders

at

price

and

orders

with

value

where

y

is

a

positive

integer

that

will

be

determined

by

the

analysis.

The

original

sequence

Σ

is

a

buy

order

with

followed

by

two

buy

orders

with

then

sell

orders

with

and

then

buy

orders

with

We

first

note

that

there

are

executions,

all

at

price

the

last

bid

is

and

the

last

ask

is

Next

we

analyze

a

modified

sequence.

We

change

the

first

buy

order

from

to

Therefore,

the

next

two

buy

orders

with

are

executed,

and

afterwards

we

have

that

the

bid

is

and

the

ask

is

y.

Now

the

sell

orders

are

accumulated

at

and

after

the

next

y

buy

orders

the

bid

is

at

Therefore,

at

the

end

we

have

that

lastbid(Σ

y

lastask(Σ

y,

close(Σ

y,

and

average(Σ

y

y)

Setting

y

x

we

obtain

the

lemma

for

every

property.

We

note

that

while

this

proof

was

based

on

the

fact

that

there

are

two

consecutive

orders

in

the

books

which

are

far

(y)

apart,

we

can

provide

a

slightly

more

complicated

example

in

which

all

orders

are

close

(at

most

apart),

yet

still

one

change

results

in

large

differences.

SIMULATION

STUDIES

The

results

presented

so

far

paint

a

striking

contrast

between

the

absolute

and

relative

price

models:

while

the

absolute

model

enjoys

provably

strong

stability

over

any

fixed

event

sequence,

there

exist

at

least

specific

sequences

demonstrating

great

instability

in

the

relative

model.

The

worstcase

nature

of

these

results

raises

the

question

of

the

extent

to

which

such

differences

could

actually

occur

in

real

markets.

In

this

section

we

provide

indirect

evidence

on

this

question

by

presenting

simulation

results

exploiting

a

rich

source

of

real-market

historical

limit

order

sequence

data.

By

interpreting

arriving

limit

order

prices

as

either

absolute

values,

or

by

transforming

them

into

differences

with

the

current

bid

and

ask

(relative

model),

we

can

perform

small

modifications

on

the

sequences

and

examine

how

different

various

outcomes

(volume

traded,

average

price,

etc.)

would

be

from

what

actually

occurred

in

the

market.

These

simulations

provide

an

empirical

counterpart

to

the

theory

we

have

developed.

We

emphasize

that

all

such

simulations

interpret

the

actual

historical

data

as

falling

into

either

the

absolute

or

relative

model,

and

are

meaningful

only

within

the

confines

of

such

an

interpretation.

Nevertheless,

we

feel

they

provide

valuable

empirical

insight

into

the

potential

(in)stability

properties

of

modern

equity

limit

order

markets,

and

demonstrate

that

one"s

belief

or

hope

in

stability

largely

relies

on

an

absolute

model

interpretation.

We

also

investigate

the

empirical

behavior

of

mixtures

of

absolute

and

relative

prices.

Data

The

historical

data

used

in

our

simulations

is

commercially

available

limit

order

data

from

INET,

the

previously

mentioned

electronic

exchange

for

NASDAQ

stocks.

Broadly

speaking,

this

data

consists

of

practically

every

single

event

on

INET

regarding

the

trading

of

an

individual

stockevery

arriving

limit

order

(price,

volume,

and

sequence

ID

number),

every

execution,

and

every

cancellation

of

a

standing

order

all

timestamped

in

milliseconds.

It

is

data

sufficient

to

recreate

the

precise

INET

order

book

in

a

given

stock

on

a

given

day

and

time.

We

will

report

stability

properties

for

three

stocks:

Amazon,

Nvidia,

and

Qualcomm

(identified

in

the

sequel

by

their

tickers,

AMZN,

NVDA

and

QCOM).

These

three

provide

some

range

of

liquidities

(with

QCOM

having

the

greatest

and

NVDA

the

least

liquidity

on

INET)

and

other

trading

properties.

We

note

that

the

qualitative

results

of

our

simulations

were

similar

for

several

other

stocks

we

examined.

Methodology

For

our

simulations

we

employed

order-book

reconstruction

code

operating

on

the

underlying

raw

data.

The

basic

format

of

each

experiment

was

the

following:

Run

the

order

book

reconstruction

code

on

the

original

INET

data

and

compute

the

quantity

of

interest

(volume

traded,

average

price,

etc.)

Make

a

small

modification

to

a

single

order,

and

recompute

the

resulting

value

of

the

quantity

of

interest.

In

the

absolute

model

case,

Step

is

as

simple

as

modifying

the

order

in

the

original

data

and

re-running

the

order

book

reconstruction.

For

the

relative

model,

we

must

first

pre-process

the

raw

data

and

convert

its

prices

to

relative

values,

then

make

the

modification

and

re-run

the

order

book

reconstruction

on

the

relative

values.

The

type

of

modification

we

examined

was

extremely

small

compared

to

the

volume

of

orders

placed

in

these

stocks:

namely,

the

deletion

of

a

single

randomly

chosen

order

from

the

sequence.

Although

a

deletion

is

not

its

edit

distance

is

and

we

can

apply

Theorem

For

each

trading

day

examined,this

single

deleted

order

was

selected

among

those

arriving

between

AM

and

PM,

and

the

quantities

of

interest

were

measured

and

compared

at

PM.

These

times

were

chosen

to

include

the

busiest

part

of

the

trading

day

but

avoid

the

half

hour

around

the

opening

and

closing

of

the

official

NASDAQ

market

AM

and

PM

respectively),

which

are

known

to

have

different

dynamics

than

the

central

portion

of

the

day.

We

run

the

absolute

and

relative

model

simulations

on

both

the

raw

INET

data

and

on

a

cleaned

version

of

this

data.

In

the

cleaned

we

remove

all

limit

orders

that

were

canceled

in

the

actual

market

prior

to

their

execution

(along

with

the

cancellations

themselves).

The

reason

is

that

such

cancellations

may

often

be

the

first

step

in

the

repositioning

of

orders

that

is,

cancellations

of

the

order

that

are

followed

by

the

submission

of

a

replacement

order

at

a

different

price.

Not

removing

canceled

orders

allows

the

possibility

of

modified

simulations

in

which

the

same

order

is

executed

twice,

which

may

magnify

instability

effects.

Again,

it

is

clear

that

neither

the

raw

nor

the

cleaned

data

can

perfectly

reflect

what

would

have

happened

under

the

deleted

orders

in

the

actual

market.

However,

the

results

both

from

the

raw

data

and

the

clean

data

are

qualitatively

similar.

The

results

mainly

differ,

as

expected,

in

the

executed

volume,

where

the

instability

results

for

the

relative

model

are

much

more

dramatic

in

the

raw

data.

Results

We

begin

with

summary

statistics

capturing

our

overall

stability

findings.

Each

row

of

the

tables

below

contains

a

ticker

(e.g.

AMZN)

followed

by

either

-R

(for

the

uncleaned

or

raw

data)

or

-C

(for

the

data

with

canceled

orders

removed).

For

each

of

the

approximately

trading

days

in

trials

were

run

in

which

a

randomly

selected

order

was

deleted

from

the

INET

event

sequence.

For

each

quantity

of

interest

(volume

executed,

average

price,

closing

price

and

last

bid),

we

show

for

the

both

the

absolute

and

Here

same

is

in

quotes

since

the

two

orders

will

actually

have

different

sequence

ID

numbers,

which

is

what

makes

such

repositioning

activity

impossible

to

reliably

detect

in

the

data.

relative

model

the

average

percentage

change

in

the

quantity

induced

by

the

deletion.

The

results

confirm

rather

strikingly

the

qualitative

conclusions

of

the

theory

we

have

developed.

In

virtually

every

case

(stock,

raw

or

cleaned

data,

and

quantity)

the

percentage

change

induced

by

a

single

deletion

in

the

relative

model

is

many

orders

of

magnitude

greater

than

in

the

absolute

model,

and

shows

that

indeed

butterfly

effects

may

occur

in

a

relative

model

market.

As

just

one

specific

representative

example,

notice

that

for

QCOM

on

the

cleaned

data,

the

relative

model

effect

of

just

a

single

deletion

on

the

closing

price

is

in

excess

of

a

full

percentage

point.

This

is

a

variety

of

market

impact

entirely

separate

from

the

more

traditional

and

expected

kind

generated

by

trading

a

large

volume

of

shares.

Stock

Date

volume

average

Rel

Abs

Rel

Abs

AMZN-R

AMZN-C

NVDA-R

NVDA-C

QCOM-R

QCOM-C

Stock

Date

close

lastbid

Rel

Abs

Rel

Abs

AMZN-R

AMZN-C

NVDA-R

NVDA-C

QCOM-R

QCOM-C

In

Figure

we

examine

how

the

change

to

one

the

quantities,

the

average

execution

price,

grows

with

the

introduction

of

greater

perturbations

of

the

event

sequence

in

the

two

models.

Rather

than

deleting

only

a

single

order

between

AM

and

PM,

in

these

experiments

a

growing

number

of

randomly

chosen

deletions

was

performed,

and

the

percentage

change

to

the

average

price

measured.

As

suggested

by

the

theory

we

have

developed,

for

the

absolute

model

the

change

to

the

average

price

grows

linearly

with

the

number

of

deletions

and

remains

very

small

(note

the

vastly

different

scales

of

the

y-axis

in

the

panels

for

the

absolute

and

relative

models

in

the

figure).

For

the

relative

model,

it

is

interesting

to

note

that

while

small

numbers

of

changes

have

large

effects

(often

causing

average

execution

price

changes

well

in

excess

of

percent),

the

effects

of

large

numbers

of

changes

levels

off

quite

rapidly

and

consistently.

We

conclude

with

an

examination

of

experiments

with

a

mixture

model.

Even

if

one

accepts

a

world

in

which

traders

behave

in

either

an

absolute

or

relative

manner,

one

would

be

likely

to

claim

that

the

market

contains

a

mixture

of

both.

We

thus

ran

simulations

in

which

each

arriving

order

in

the

INET

event

streams

was

treated

as

an

absolute

price

with

probability

α,

and

as

a

relative

price

with

probability

1−α.

Representative

results

for

the

average

execution

price

in

this

mixture

model

are

shown

in

Figure

for

AMZN

and

NVDA.

Perhaps

as

expected,

we

see

a

monotonic

decrease

in

the

percentage

change

(instability)

as

the

fraction

of

absolute

traders

increases,

with

most

of

the

reduction

already

being

realized

by

the

introduction

of

just

a

small

population

of

absolute

traders.

Thus

even

in

a

largely

relative-price

world,

a

x

−3

QCOM−R

June

Absolute

Number

of

changes

Averageprice

QCOM−R

June

Relative

Number

of

changes

Averageprice

Figure

Percentage

change

to

the

average

execution

price

(y-axis)

as

a

function

of

the

number

of

deletions

to

the

sequence

(x-axis).

The

left

panel

is

for

the

absolute

model,

the

right

panel

for

the

relative

model,

and

each

curve

corresponds

to

a

single

day

of

QCOM

trading

in

June

Curves

represent

averages

over

trials.

small

minority

of

absolute

traders

can

have

a

greatly

stabilizing

effect.

Similar

behavior

is

found

for

closing

price

and

last

bid.

AMZN−R

Feburary

α

Averageprice

NVDA−R

June

α

Averageprice

Figure

Percentage

change

to

the

average

execution

price

(y-axis)

vs.

probability

of

treating

arriving

INET

orders

as

absolute

prices

(x-axis).

Each

curve

corresponds

to

a

single

day

of

trading

during

a

month

of

Curves

represent

averages

over

trials.

For

the

executed

volume

in

the

mixture

model,

however,

the

findings

are

more

curious.

In

Figure

we

show

how

the

percentage

change

to

the

executed

volume

varies

with

the

absolute

trader

fraction

α,

for

NVDA

data

that

is

both

raw

and

cleaned

of

cancellations.

We

first

see

that

for

this

quantity,

unlike

the

others,

the

difference

induced

by

the

cleaned

and

uncleaned

data

is

indeed

dramatic,

as

already

suggested

by

the

summary

statistics

table

above.

But

most

intriguing

is

the

fact

that

the

stability

is

not

monotonically

increasing

with

α

for

either

the

cleaned

or

uncleaned

datathe

market

with

maximum

instability

is

not

a

pure

relative

price

market,

but

occurs

at

some

nonzero

value

for

α.

It

was

in

fact

not

obvious

to

us

that

sequences

with

this

property

could

even

be

artificially

constructed,

much

less

that

they

would

occur

as

actual

market

data.

We

have

yet

to

find

a

satisfying

explanation

for

this

phenomenon

and

leave

it

to

future

research.

ACKNOWLEDGMENTS

We

are

grateful

to

Yuriy

Nevmyvaka

of

Lehman

Brothers

in

New

York

for

the

use

of

his

INET

order

book

reconstruction

code,

and

for

valuable

comments

on

the

work

presented

NVDA−C

June

α

Volume

NVDA−R

June

α

Volume

Figure

Percentage

change

to

the

executed

volume

(y-axis)

vs.

probability

of

treating

arriving

INET

orders

as

absolute

prices

(x-axis).

The

left

panel

is

for

NVDA

using

the

raw

data

that

includes

cancellations,

while

the

right

panel

is

on

the

cleaned

data.

Each

curve

corresponds

to

a

single

day

of

trading

during

June

Curves

represent

averages

over

trials.

here.

Yishay

Mansour

was

supported

in

part

by

the

IST

Programme

of

the

European

Community,

under

the

PASCAL

Network

of

Excellence,

by

a

grant

from

the

Israel

Science

Foundation

and

an

IBM

faculty

award.

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