endogenous formation of limit order books: the effects of ...(joint work with r. gayduk) department...

Endogenous Formation of Limit Order Books: theEffects of Trading Frequency

Sergey Nadtochiy(joint work with R. Gayduk)

Department of MathematicsUniversity of Michigan

Dec 16, 2015

Financial and Actuarial Mathematics Seminar

University of Michigan

Sergey Nadtochiy (U of M) Endogenous Formation of Limit Order Books University of Michigan 1 / 35

Introduction

Market Microstructure

Classical models of Financial Mathematics: exogenous prices, tradingmechanism hidden.

Financial Economics: endogenous prices, trading mechanism hidden.

Market Microstructure: study trading mechanism.

Typically, two types of mechanisms are considered:

central market-maker (“quote-driven”) exchanges;

and auction-style (“order-driven”) ones.

Here, we focus on the auction-style exchanges.

The main object of our study is the Limit Order Book (LOB).


Introduction

Types of investigation

Optimize agents’ behavior, given a model for LOB.

Key empirical features of the market (such as market resilience andprice impact) are modeled exogenously.

Then, the problem of optimal execution is solved.

Literature: Almgren, Chriss, Bouchaud, Obizhaeva, Wang, Schied,Zhang, Gatheral, Alfonsi, Stoikov, Avellaneda, Cont, Talreja,Jaimungal, Cartea, Cvitanic, Shreve, Gueant, Lehalle, Pham, Bayraktar,Ludkovski, Moallemi, Carmona, Lacker, Cheridito, Guo, Pham, Ma.

Model LOB as an outcome of an equilibrium.

A fundamental price, or demand, is modeled exogenously, but LOBarises endogenously from the agents’ behavior in equilibrium.

Literature: Lachapelle-Lasry-Lehalle-Lions, Carmona-Webster,Goettler-Parlour-Rajan, Foucault, Parlour.

Market-maker and Insider: Kyle, Glosten-Milgrom, Easley-O’Hara,Muhle-Karbe-Webster.


Introduction

Our goals

1 Develop a rigorous, precise and tractable modeling framework forauction-style exchanges.

2 Study the liquidity effects that are endogenous to the exchange (due tothe agent’s interaction). In particular, how do the rules of the exchangeaffect the liquidity?


Introduction

Effects of trading frequency

Our main results are as follows.

If there exists a sequence of non-degenerate equilibria, with arbitrarilyhigh trading frequencies, then, the market efficiency increases withfrequency.

There exists such a sequence of equilibria only if the agents aremarket-neutral.

Under additional model assumptions, the market-neutrality is sufficient forthe existence of such a sequence of equilibria.


The setup

External demand

Time is discrete: n = 0, 1, . . . ,N.

Incremental demand is given by the random field D = (Dn(p))p∈R,n=1,...,N

on a filtered probability space(

Ω,F = (Fn)Nn=0 ,P)

Dn(p) denotes the total demand (both external and internal) for theasset at price p and at all more favorable price levels, in the nth time period.

D+n (p) is the maximum quantity that will be purchased at or below

price p (via market orders),

D−n (p) is the maximum quantity that will be sold at or above price p.

The fundamental price (or, “tipping point” of the demand) p0n is the

unique solution to: Dn(p) = 0.

Every agent models random field D using the same filtration F but differentprobability measures Pα P, α ∈ A, which we call beliefs.


The setup

State space and controls

State space S = R× A represents the inventory of an agent and herbeliefs.

As the beliefs do not change, the state process of an agent, Sn, representsher inventory.

The control of an agent is given by adapted processes (pn, qn, rn)N−1n=0 , with

values in R2 × 0, 1.

pn is the location of a limit order placed at time n,qn is the size of the order (with negative values corresponding to buyorders).rn indicates whether the agent submits a market order (if rn = 1) or alimit order (if rn = 0).

There are infinitely many agents, and their empirical distribution over thestate space is given by a measure-valued process (µn)Nn=0.


The setup

LOB and State process

The Limit Order Book (LOB) is a pair of processes (νn = (ν−n , ν+n ))

Nn=0,

with values in the space of measures on R.

The bid and ask prices at time n are given by

pbn = sup supp(ν−n ), pan = inf supp(ν+n ).

State process S , starting at time m and at level s:

Sm,s,(p,q,r)m = s, ∆S

m,s,(p,q,r)n+1 = −qn1rn=1

− 1rn=0

(q+n 1D+

n+1(pn)>ν+n ((−∞,pn)) − q−n 1D−n+1(pn)>ν−n ((pn,∞))

)


The setup

State process: implicit assumptions

Sm,s,(p,q,r)m = s, ∆S

m,s,(p,q,r)n+1 = −qn1rn=1

− 1rn=0

(q+n 1D+

n+1(pn)>ν+n ((−∞,pn)) − q−n 1D−n+1(pn)>ν−n ((pn,∞))

)

In the above expression, we implicitly assume that each agent

is small so that her order is fully executed once the demand reaches it;believes that her order will be executed first among all orders withthe same priority.

The latter assumption implies a possible inconsistency with the marketclearance condition: i.e. the total executed demand may not coincide withthe total change in the cumulative inventory.

The above issue is resolved if νn(·) is continuous and r ≡ 0.


The setup

Optimization problem

An agent aims to maximize over (p, q, r):

Jν,(p,q,r)m (s, α) = Eαm

[(Sm,s,(p,q,r)N

)+

pbN −(Sm,s,(p,q,r)N

)−paN

−N−1∑n=m

(pn1rn=0 + pan1rn=1,qn<0 + pbn1rn=1,qn>0

)∆S

m,s,(p,q,r)n+1

]

The terminal payoff is due to marking to market.

A control is admissible if the above expectation is well defined.

LOB ν is admissible if Eαm|paN | ∨ |pbN | <∞.

For an admissible ν, an admissible control (p, q, r) is optimal if, for anym, s, α,

Jν,(p,q,r)m (s, α) ≥ Jν,(p,q,r)

m (s, α)

P-a.s., for any admissible (p, q, r).


The setup

Equilibrium

Consider a given empirical distribution process µ = (µn)Nn=0 and a marketmodel, as described above.

We say that a given LOB process ν and a control (p, q, r) form anequilibrium, if

1 ν is admissible, and (p, q, r) is an optimal control for ν,

2 and, for any n = 0, . . . ,N − 1, we have, P-a.s.:

ν+n ((−∞, x ]) =

∫S

1pn(s,α)≤x,rn(s,α)=0 q+n (s, α)µn(ds, dα), ∀ x ∈ R,

ν−n ([x ,∞)) =

∫S

1pn(s,α)≥x,rn(s,α)=0 q−n (s, α)µn(ds, dα), ∀ x ∈ R.


The setup

Equilibrium: comments

The equilibrium we chose is sub-game perfect.

It uses the typical assumption of a game with continuum players: eachplayer is too small to affect the LOB.

It is very similar to a Mean Field Game with purely common noise.

However, it only defines a partial equilibrium, as µ is given exogenously.

To make it a true equilibrium, we need to require, in addition, that

µn = µ0 (

(s, α) 7→(S0,s,(p,q,r)n , α

))−1

We call this an equilibrium with endogenous µ.

However, such additional restriction only makes sense if the model isconsistent with the market clearance condition.


Preliminary results

Value function and DPP

V νn (s, α) = esssupp,q,rJ

ν,(p,q,r)n (s, α) ,

V νn (s, α) = s+λan(α)− s−λbn(α), with λaN(α) = pbN and λbN(α) = paN ;

there always exists an optimal control (p, q, r), s. t. qn(s, α) ∈ 0, s;

for s > 0,

if qn(s, α) = s and rn(s, α) = 0, then the expected execution priceλa follows:

λan(α) = Eαn λan+1(α) + supp∈R

Eαn[(p − λan+1(α)

)1D+

n+1(p)>ν+n ((−∞,p))

],

and p = pn(s, α) attains the above supremum,

if qn(s, α) = 0 and rn(s, α) = 0, then λan(α) = Eαn λan+1(α),

if rn(s, α) = 1, then λan(α) = pbn .


Preliminary results

Terminal condition and LTC equilibrium

To start the backward iteration, suggested by DPP, we need to resolve thelast-time-step problem: pN−1(1, α) ∈ arg supp∈R EαN−1

[(p − pbN

)1D+

N (p)>ν+N−1((−∞,p))

],

pN−1(−1, α) ∈ arg supp∈R EαN−1

[(paN − p) 1D−N (p)>ν−N−1((−∞,p))

],

The equilibrium condition

ν+N−1((−∞, x ]) =

∫(0,∞)×A

1pN−1(1,α)≤x sµN−1(ds, dα), ∀ x ∈ R,

links νN−1 and pN−1, resulting in a fixed-point problem for νN−1.

However, there is no fixed-point problem for νN : we can choose νN and, inturn, (paN , p

bN) arbitrarily, as the agents do not optimize their actions at time

N.

An equilibrium with LOB ν is linear at terminal crossing (LTC) if

νN = νN−1 (x 7→ x + ∆p0N)−1, P-a.s.


Preliminary results

Degeneracy

λan(α) = Eαn λan+1(α) + supp∈R

Eαn[(p − λan+1(α)

)1D+

n+1(p)>ν+n ((−∞,p))

],

Ifsupp∈R

Eαn[p − λan+1(α)

∣∣ D+n+1(p) > ν+

n ((−∞, p))]< 0,

then, the agents at (s, α) choose to wait, in which case qn(s, α) = 0 and

λan(α) = Eαn λan+1(α)

This may indeed occur in an equilibrium, and it can be attributed to theadverse selection effect.

An equilibrium with LOB ν is non-degenerate if ν+n (R) > 0 and

ν−n (R) > 0, for all n, P-a.s..


Preliminary results

Degeneracy

Figure: Ask prices (in red) and the associated λa (in blue), as functions of time.Different curves correspond to different trading frequencies.


High-frequency limit

Fundamental price in continuous time

Consider a stochastic basis(

Ω, (Ft)t∈[0,T ] ,P, (Pα P)α∈A)

.

Under P, the limiting fundamental price process is given by

p0t = p0

0 +

∫ t

0

σsdWs , p00 ∈ R,

where 1/C ≤ σ ≤ C

Under all Pα,

p0t = p0

0 +

∫ t

0

µαs ds +

∫ t

0

σsdWαs , p0

0 ∈ R,

where |µα| ≤ C .

Given ∆t > 0, the discrete time model is defined as follows: F∆tn = Fn∆t ,

A∆t = A, P∆t = P, Pα,∆t = Pα and

p0,∆tn = p0

n∆t



Assumptions

There exists deterministic ε(∆t)→ 0, as ∆t → 0, s.t., P-a.s.,

Pαt(Eα(

(σs∨τ − στ )2 | Fτ)≤ ε(∆t)

)= 1, P− a.s.,

holds for all t ≤ s ≤ t + ∆t, all stopping times t ≤ τ ≤ s, and all α ∈ A.

For any n, there exists a strictly decreasing random function κn−1(·), suchthat κn−1(0) = 0 and∣∣Dn

(p + p0

n

)∣∣ ≥ |κn−1(p)| , ∀p ∈ R, P− a.s.

For any n, there exists a deterministic measure µ0n, s.t. µn µ0

n, P-a.s..



Asymptotic efficiency

Proposition 1

Consider a family of uniform partitions of [0,T ], with diameters∆t > 0, and with associated market models satisfying the aboveassumptions.Assume that every such model admits a non-degenerate LTCequilibrium.Then, as ∆t → 0,∣∣paN − p0

N

∣∣+∣∣pbN − p0

N

∣∣→ 0, P− a.s.

Corollary 1

Under the assumptions of Proposition 1, denote the expectedexecution prices by λa and λb.Then, as ∆t → 0,

supn=0,...,N, α∈A

(∣∣λan(α)− p0n

∣∣+∣∣λbn(α)− p0

n

∣∣)→ 0, P− a.s.



Market neutrality as a necessary condition

Additional assumption. For any α ∈ A and any t ∈ [0,T ), there exists adeterministic ε(·) ≥ 0, s.t. ε(∆t)→ 0, as ∆t → 0, and, for anyt ≤ t ′ ≤ t ′′ ≤ t + ∆t,

Pαt′

(∣∣∣∣∣Eαt′′∫ T

t

µαs ds − Eαt′∫ T

t

µαs ds

∣∣∣∣∣ ≥ ε(∆t)

)≤ ε(∆t), Pα − a.s.

Theorem 1

Consider a family of ∆t > 0, containing arbitrarily small ∆t, and theassociated market models satisfying the above assumptions.

Assume that every model admits a non-degenerate LTC equilibrium.

Then, for all α ∈ A, p0 is a martingale under Pα.

Moreover, for all α ∈ A, pbn < p0n < pan, Pα-a.s..



Lemma

Xt =

∫ t

0

µudu +

∫ t

0

σudBu, t ∈ [0, 1]

Assume ∃C > 1, s. t. |στ | ≤ C and |µτ | ≤ C for any stopping time τ .

Then the following holds.

1 ∀c > 0 ∃C1 > 0, s.t.

P

(sup

t∈[0,1]

Xt > x + z

)≤ C1e

−czP

(sup

t∈[0,1]

Xt > x

), ∀x , z ≥ 0.

2 ∀c > 0 ∃C2, ε > 0, s.t.

P

(sup

t∈[0,1]

Xt > x

)≤ C2P(X1 > x), ∀x ≥ 0,

provided |στ | ≥ c , µ2τ ≤ ε and E

((σs∨τ − στ )2 | Fτ

)≤ ε, for any

s ∈ [0, 1] and any stopping time τ .



Corollary

Xt =

∫ t

0

µudu +

∫ t

0

σudBu, t ≥ 0, X0 = 0

Assume that |µ| ≤ C , 1/C ≤ σ ≤ C , and that there exists a deterministicε(∆t)→ 0, as ∆t → 0, s.t.

E(

(σs∨τ − στ )2 | Fτ)≤ ε(∆t)

holds a.s. for all 0 ≤ s ≤ ∆t and all stopping times 0 ≤ τ ≤ s.

Then, ∃C1 > 0 s.t., for all small enough ∆t > 0,

P (X∆t > x + z | X∆t > x) ≤ C1e−z/√

∆t , ∀x , z ≥ 0.


Existence

Assumptions

A is a singleton, 1/C ≤ σ ≤ C .

There exists deterministic ε(∆t)→ 0, as ∆t → 0, s.t.

Pt

(E(

(σs∨τ − στ )2 | Fτ)≤ ε(∆t)

)= 1,

holds for all t ≤ s ≤ t + ∆t, all stopping times t ≤ τ ≤ s.

There exists a constant c > 0, s.t., for some δ > 0,

log σs/σt ≤ c , ∀t ∈ [0,T ], s ∈ [T − δ,T ]

The following holds for all n = 1, . . . ,N − 1:

supp∈R

D+n (p) < µ1

n, supp∈R

D−n (p) < µ2n,

where

µ1n =

∫(0,∞)

sµn(ds), µ2n =

∫(−∞,0)

|s|µn(ds)


Existence

Existence: exogenous µ

Theorem 2

Consider a market model with ∆t > 0 and an empirical distribution processµ.

Let the above assumptions hold.

Assume that p0 is a martingale under P.

Then, whenever c and ∆t are small enough, the market model and µ admita non-degenerate LTC equilibrium.

Moreover, this equilibrium can be constructed so that all agents who arelong the asset post limit sell orders at the ask price, while all agents whoare short the asset post limit buy orders at the bid price.


Existence

Existence: endogenous µ

A is a singleton, σt is deterministic and non-increasing in t ∈ [0,T ], and p0

is a martingale.

For any n, there exists a strictly decreasing continuous (deterministic)function κn(·), with κn(0) = 0, s.t. Dn

(p0n + p

)= κn(p).

Theorem 3

Consider a market model and an initial empirical distribution µ0.Let the above assumptions hold, and assume that, in addition,

µ1,c0 >

N−1∑n=1

supp∈R

D+n (p), µ2,c

0 >

N−1∑n=1

supp∈R

D−n (p)

Then, there exists an empirical distribution process µ, with theprescribed µ0, s.t. the associated market model and µ admit anon-degenerate LTC equilibrium, in which the agents do not postmarket orders, the LOB is continuous (i.e. has no mass points in R),and

µn = µ0 (

(s, α) 7→(S0,s,(p,q,r)n , α

))−1


example

Brownian motion with drift

A = α ⊂ R.

p0t = p0

0 + αt + σWt , σ, p00 ∈ R.

supp∈R D+n (p) < µ1

n(A), supp∈R D−n (p) < µ2n(A).

Shifted variables: pa,b := pa,b−p0, λa,b := λa,b−p0, ξ ∼ N (α∆t, σ√

∆t).

A single-step problem becomes:

pa ∈ arg maxp≤pa E[(p − pb − ξ)1ξ>p

],

λa = pb + E[(pa − pb − ξ)1ξ>pa

],

pb ∈ arg maxp≥pb E[(pa + ξ − p)1ξ<p

],

λb = pa − E[(pa − pb + ξ)1ξ<pb

],

pb ≤ λa ≤ pa, pb ≤ λb ≤ pa,

For any α ∈ R, solution exists for all small enough ∆t > 0.


example

Degeneracy

pan ∈ arg maxp≤pa

nE[(p − λan+1 − ξ

)1ξ>p

],

λan = λan+1 + α∆t + E[(pan − λan+1 − ξ

)1ξ>pa

n)],

If α = 0, then

λan < 0, hence, pan is well defined by the above;higher frequency leads to smaller |λa,b| and |pa,b|: i.e. higher efficiency.

If α > 0, then,

we may obtain λan ≥ 0, and, hence, pan is not well defined. In thiscase, agents choose to wait: i.e. stop providing liquidity.higher frequency leads to smaller tolerance for the value of |α|.


example

Zero-drift case

Figure: Bid and ask prices (left) and the associated expected execution prices(right), as functions of time horizon. Different curves correspond to differenttrading frequencies. Zero drift case.


example

Zero-drift case

Figure: Time zero bid-ask spread as a function of trading frequency (measured inthe number of steps).


example

Positive drift

Figure: Ask prices (in red) and the associated λa (in blue), as functions of timehorizon. Different curves correspond to different trading frequencies. Positivedrift case.


example

Positive drift

Figure: The maximum value of drift that admits a non-degenerate equilibrium, asa function of trading frequency (measured in the number of steps).


Proof of the main result

Sketch of the proof of Thm 1

We need to show that, if the agents have a signal about future pricemovements, then, they will choose not to post limit orders.

Consider the agents who are long the asset (i.e. they are trying to sell) andpost limit orders around the ask price.

There are two reasons why they may choose not to post limit orders:

1 they are bearish – then, they submit a market order;2 they are bullish – then, they submit nothing and wait.

If the long agents are bearish, eventually

λan = λan+1 + α∆t + Eαn[(pan − λan+1 − ξ

)1ξ>pa

n)]≤ 0,

whilepbn ≈ λbn ≈ 0

Hence,λan < pbn ,

and the agents submit a market order.


Proof of the main result

Sketch of the proof: adverse selection

If the long agents are bullish,

λan = λan+1 + α∆t + Eαn[(pan − λan+1 − ξ

)1ξ>pa

n)]≥ 0,

and, in turn, for any p ∈ R,

Eαn−1

[(p − λan − ξ)1ξ>p

]< 0,

Hence, wherever the agents post their limit orders, they will regret doing itonce the orders are executed, as, in that case, they would have been ableto get more for their shares.

This is precisely the adverse selection effect.


Conclusions and extensions

Summary

I have presented a new framework for modeling market microstructure inauction-style exchanges, in which the LOB arises endogenously, as a resultof equilibrium in a game between multiple strategic players.

The mechanics of the exchange are reproduced closely enough, so that theycan be used to analyze the liquidity role of market participants, and itsdependence on the rules of the exchange, such as the trading frequency.

The proposed framework is based on the theory of mean field games, butdoes not belong to any known classes of mean field games.



Summary

The main economic conclusions are as follows.

Even in the absence of any real shortage of the asset (i.e. when theexternal demand is large), the agents may choose not to provideliquidity in equilibrium, causing the market to degenerate, or, in otherwords, creating a liquidity crisis.

The occurrence or non-occurrence of such degeneracy (or, crisis)depends on the rules of the exchange, and, in particular, on thetrading frequency.

Increasing the trading frequency has a dual effect on the market. Onthe one hand, if the market does not degenerate in the high frequencyregime, then, it becomes more efficient. On the other hand, thehigher frequency makes markets more sensitive to the deviation of theagents from market-neutrality.



Future work/work in progress

Show that the continuum-player game can be viewed as a limit offinite-player games.

A more general existence result is needed.

Take into account the time priority of limit orders.

Include risk aversion of the agents, or limit their admissible inventory levels,in order to observe a predatory behavior in the equilibrium.

Continuous time models.

Describe a specific class of models that can be calibrated to the market data.


endogenous formation of limit order books: the effects of ...(joint work with r. gayduk) department...

Documents