Trading Frictions in High Frequency MarketsTrading Frictions in High Frequency Markets R. Carmona...

Trading Frictions in High FrequencyMarkets

R. Carmona

Bendheim Center for FinanceORFE, Princeton University

joint work with Kevin Webster (Deutsche Bank

Ann Arbor

October 15, 2014

Standard Assumptions in Mathematical Finance

Black-Scholes theory

I Price given by a single number (law of one price)I Infinite liquidity

I one can buy or sell any quantity at this priceI with NO IMPACT on the asset price

I First fixes to account for liquidity frictionsI Introduction of Transaction CostsI Add Liquidity constraints ∼ transaction costs

Not satisfactory for

I Large trades (over short periods)

I High Frequency Trading (HFT)

Need Market Microstructure

I e.g. understand how are buy and sell orders executed?

The Limit Order Book (LOB) in Pictures

13.8 13.9 14.0 14.1 14.2 14.3

0e+0

02e

+04

4e+0

46e

+04

8e+0

41e

+05

DELL NASDAQ Order Book, May 18, 2013

Price

Volum

e

Time = 11 hr 42 mn

Figure : DELL Limit Order Book on May 18, 2013

LOB Dynamics in Pictures

90 95 100 105 110 115

02

46

810

12

Typical Limit Order Book

Price

Volume

90 95 100 105 110 1150

24

68

1012

LOB after the arrival of a Buy Limit Order

Price

Volume

Figure : Limit Order Book before (left) and after (right) the arrival of alimit buy order at price level p < Bt : O(t) ↪→ O(t)p+v


90 95 100 105 110 115

02

46

810

12

Typical Limit Order Book

Price

Volume

90 95 100 105 110 1150

24

68

1012

LOB after the arrival of a Buy Limit Order

Price

Volume

Figure : Limit Order Book before (left) and after (right) the arrival of alimit buy order at price level Bt < p < At : O(t) ↪→ O(t)p+v


90 95 100 105 110 115

02

46

810

12

LOB after the arrival of a Sell Limit Order

Price

Volume

90 95 100 105 110 115

02

46

810

12

LOB after the arrival of a Sell Limit Order

Price

Volume

Figure : Limit Order Book after the arrival of a limit sell order at pricelevel p = At (left) and of a limit sell order at price level Bt < p < At(right) whose effect is to lower the best ask At


90 95 100 105 110 115

02

46

810

12

LOB after the arrival of a Buy Market Order

Price

Volume

90 95 100 105 110 115

02

46

810

12

LOB after the arrival of a Buy Market Order

Price

Volume

Figure : Limit Order Book after the arrival of a market buy order withvolume smaller than what was in the book at the best ask (left)O(t) ↪→ O(t)At−v , and after the arrival of a market buy order withvolume equal to what was in the book at the best ask, lowering At .

Cancellations

90 95 100 105 110 115

02

46

810

12

LOB after the Cancellation of a Buy Limit Order

Price

Volume

90 95 100 105 110 115

02

46

810

12

LOB after the Cancellation of a Sell Limit Order

Price

Volume

Figure : Cancellation of an outstanding limit buy order at price levelp < Bt (left) and cancellation of an outstanding limit sell order at pricelevel p > At (right).

Large Fills

13.85 13.90 13.95 14.00 14.05 14.10

010000

20000

30000

40000

50000

60000


Price

Volume

Time = 15 hr 5 mn Mid-Price = 13.98

13.85 13.90 13.95 14.00 14.05 14.100

10000

20000

30000

40000

50000

60000


Price

Volume


Figure : Illustration of the market impact of a large fill.

Market Impact of Large Fills

I Current mid-pricepmid = (pBid + pAsk )/2 = 13.98

I Fill size N = 76015 (e.g. buy)

I n1 shares available at best bidp1, n2 shares at price p2 > p1,· · ·

I nk shares at price pk > pk−1N = n1 + n2 + n3 + · · · + nk

I Transaction cost

n1p1+n2p2+· · ·+nkpk = 1064578

I Effective price

peff =1

N(n1p1 + n2p2 + · · · + nkpk )

= 14.00484

I New mid-price pmid = 13.995

13.85 13.90 13.95 14.00 14.05 14.10

010000

20000

30000

40000

50000

60000


Price

Volume


13.85 13.90 13.95 14.00 14.05 14.10

010000

20000

30000

40000

50000

60000


Price

Volume


Not in this talk: Priorities and Hidden Orders

90 95 100 105 110 115

02

46

810

12

LOB with Different Bid Priorities

Price

Volume

90 95 100 105 110 115

-50

510

LOB with Hidden Liquidity

Price

Volume

Figure : Illustration (left) of one priority mechanism (if a sell marketorder comes in, the top part of the best bid bar will be executed, then themiddle part, . . .) and of the possible presence of hidden liquidity (right).

What happened on July 19, 2012?

0 100000 200000

19

31

96

IBM nb of trades

IBM

_M

id

Figure : Mid-price of IBM throughout the day on July 19, 2014.

SCREAMING for

FORENSIC ANALYSIS

Some Remarks (relevant to the talk)

I “There is no question that the goal of many HFT strategies isto profit from LFTs’ mistakes. [...] Part of HFT’ success isdue to the reluctance of LFT to adopt (or even torecognize) their paradigm.” Maureen O’ Hara

I HFT is not going away

I Speed is important, but not the fundamental differencebetween HFT and LFT.

I Microstructure matters! It drives trades and prices, notfundamentals.

I Key ’tool’ differentiating HFT from LFT: event-based clockvs calendar clock.

I Academia, and some LFTs, should learn from incorporatingthe HFT paradigm.

Ambitious (Pretentious?) Research Program

1. Understand, at the microscopic level, structuralrelationships and strategies that HFTs exploit.

2. Identify which microscopic features matter at themacroscopic level, and provide models on that scale.

3. Use these models to update LFT models and providemonitoring tools: transaction cost analysis, measure oftoxicity of order flow...

Today’s Talk: Search for Structural Relationships

From LOB Models to Low Frequency Models

I Understand transition from discrete to continuous models

I Incorporate market microstructure into low-frequency models.

I Differentiate between liquidity takers and providers

I Identify self-financing conditions

Searching for Answers in the Data

I Nasdaq ITCH data includes all limit and market orders.

I Perfect reconstruction of visible limit order book.

I Example: KO (Coca Cola) on 18/04/13.

Midprice

Conventions

I Trade clock, n = 1, ...N corresponds to the timest1 < ... < tN at which trades occur

I Notation: ∆nx = xn+1 − xn.I pn: mid-price just before the trade at time tn (i.e. ptn−)

Bid-Ask Spread

Convention, Notation, Assumption

I All executions (100%) happen at the best bid or best askexecution = trade ? answer requires reconstruction of large orders !

I sn: bid-ask spread just before the trade.

I sn ≈ |∆np|.

Aggregate inventory

Conventions and comments

I Inventory of the aggregate liquidity provider

I Ln: Inventory just before the trade.

I ∆nL < 0 means that a market order bought at the ask.

Price Impact

Empirical Fact

I ∆nL∆np ≤ 0 holds 99.1% of the time.I Prices move in favor of market orders (adverse selection)

Statistical Test of the Hypothesis

Assume mid-price p and inventory L are Itô processes{dpt = µtdt + σtdWt

dLt = btdt + ltdW′t

with d [W ,W ′]t = ρtdt. Let pN and LN be discrete samplings

pNn = pn/N and LNn = Ln/N

Define{CNt =

∑bNtc−1n=1 ∆np

N ∆nLN

V Nt = N∑bNtc−2

n=1

((∆np

N ∆n+1LN)2

+ ∆npN ∆nL

N ∆n+1pN ∆n+1L

N)

Then

L

(CNt − [p, L]t√

N−1|V Nt |

)→ N(0, 1)

Confidence intervals for [W ,W ′]t and test of H0 : ∃t ∈ [0, 1], ρt > 0

Tests Results

Stock proba reject nb wrong trades total nb trades percent false

MSFT 0.7868301 19 27540 0.06899056KO 0.9876695 72 20362 0.3535998BA 0.9999383 222 4824 4.60199GPS 0.9999044 97 7378 1.314719GE 0.9991448 4 12969 0.03084278CS 0.8971721 132 3621 3.645402CPB 0.9421457 129 3578 3.605366BCS 0.9625842 43 1613 2.66584JNJ 0.9550316 152 16114 0.9432791UPS 0.9983282 237 5608 4.226106CLX 0.9563385 118 1381 8.544533T 0.9996831 27 13287 0.2032061DELL 0.9893074 1 3742 0.02672368XOM 0.9998707 340 20714 1.641402CAT 0.9814122 397 13456 2.950357COF 0.8973841 131 6103 2.146485AAPL 0.9999987 2347 46710 5.02462PG 0.9998587 189 18616 1.015256GOOG 0.9929220 609 8595 7.085515HSY 0.9615380 177 1807 9.795241WFC 0.9129410 13 17672 0.0735627DTV 0.6174753 117 9334 1.253482BBY 0.9999374 85 7181 1.183679MT 0.8870935 18 2273 0.791905GM 0.9774693 19 5963 0.3186316CL 0.9833529 187 3006 6.220892MA 0.9996761 113 1435 7.874564KSU 0.9945635 118 1756 6.719818GIS 0.9735843 68 3624 1.87638

Constant Correlation (99% conf. int.)

Stock correlation conf int LB conf int UB

MSFT -0.07092524 -0.08635133 -0.055465144KO -0.17980857 -0.19721907 -0.162284686BA -0.24564343 -0.28017226 -0.210479802GPS -0.20956508 -0.23805431 -0.180715538GE -0.10889870 -0.13119283 -0.086494473CS -0.26149646 -0.30092609 -0.221174300CPB -0.30102519 -0.33967660 -0.261358758BCS -0.12981970 -0.19232648 -0.066263777JNJ -0.28789591 -0.30639672 -0.269177673UPS -0.31015068 -0.34090637 -0.278731788CLX -0.08433734 -0.15272316 -0.015147704T -0.17896560 -0.20050877 -0.157249443DELL -0.04412542 -0.08606558 -0.002029145XOM -0.24353023 -0.26029237 -0.226621346CAT -0.27501096 -0.29541130 -0.254359932COF -0.35245848 -0.38099946 -0.323246409AAPL -0.17270704 -0.18424586 -0.161120624PG -0.24705819 -0.26470169 -0.229249351GOOG -0.23689058 -0.26294176 -0.210494224HSY -0.38699253 -0.43731292 -0.334256416WFC -0.13639039 -0.15535551 -0.117324764DTV -0.16045534 -0.18631773 -0.134370753BBY -0.22586854 -0.25451487 -0.196826185MT -0.20132473 -0.25258903 -0.148933300GM -0.23327661 -0.26457097 -0.201491456CL -0.27786344 -0.32064850 -0.233946821MA -0.32520881 -0.38467018 -0.263059565KSU -0.41253170 -0.46225846 -0.360218465GIS -0.21370970 -0.25416570 -0.172507145

(Integrated) Quadratic Covariations

Cash Account

Convention and Comment

I Kn: cash holdings just before the trade.

I Self-financing by construction: changes in cash are theamounts exchanged during trades. No more, no less

Wealth Definition

Xn = Lnpn + Kn

Accounting rule: wealth is the value of the inventory marked tothe mid-price plus the cash holdings.

Self-financing equations

Possible wealth dynamics

∆nX = Ln∆np (1)

∆nX = Ln∆np +sn2|∆nL| (2)

∆nX = Ln∆np +sn2|∆nL|+ ∆np∆nL (3)

Corresponding relationships for self-financing cash

∆nK = −pn+1∆nL (1)

∆nK = −pn+1∆nL +sn2|∆nL| (2)

∆nK = −pn∆nL +sn2|∆nL| (3)

Which one is Right? (Best Bid / Ask executions)

1. trader triggers a buy with a market order;

2. trader triggers a sell with a market order;

3. trader has his buy limit order executed;

4. trader has his sell limit order executed;

5. trader is not part of the current trade.

In case 1, the trader buys at the ask (Ln+1 − Ln > 0)

Kn+1 − Kn = − (pn + sn/2) (Ln+1 − Ln)

In case 2, the trader sells at the bid (Ln+1 − Ln < 0)

Kn+1 − Kn = − (pn − sn/2) (Ln+1 − Ln)

In case 3, the trader buys at the bid (Ln+1 − Ln > 0)

Kn+1 − Kn = − (pn − sn/2) (Ln+1 − Ln)

In case 4, the trader sells at the ask (Ln+1 − Ln < 0)

Kn+1 − Kn = − (pn + sn/2) (Ln+1 − Ln)

Finally, in case 5, Kn+1 = Kn and Ln+1 = Ln.

Net Result

These five cases can be summarized in

∆nK = −pn∆nL±sn2|∆nL|

where ”±” meansI ”+” when trading with limit orders

I ”−” when trading with market orders.Using the definition Xn = Lnpn + Kn. of the wealth we get

∆nX = Ln∆np ±sn2|∆nL|+ ∆nL∆np

Comparing the three self-financing equations

Comments

I True wealth coincides with (3).I Difference between (1) and (2) (transaction costs) is large.I Difference between (2) and (3) (price impact) cannot be neglected

Did We Miss Something?

Nature of NASDAQ Messaging

I All messages point to a trades at Best-Bid or Best-Ask

I Could this be the result of large order splitting?

I Easy to develop simple algorithms reconstructingparent orders from child orders

SAME RESULT!

More General LOB Models

I Typically a LOB is the superposition of two histograms

Order Book Shape Function

Transaction Cost Function

Legendre transform of γ:

c(l) = supu

(ul − γ(u)) .

c is convex and satisfies c(0) = 0.Cash

∆K = −p∆L + c (−∆L) ,

Wealth

∆X = L∆p + c(−∆L) + ∆p∆L.

The continuous limit

What are the issues?

I 3 self-financing wealth conditions to choose from;

I Adverse Selection constraint;

I Choice of assumptions on p and L: jumps? finite variation?

I Bid-ask spread: fixed? Vanishing?

Informally

I Want (3) to include transaction costs and price impact

I pNn = pbn/Nc and LNn = Lbn/Nc from pt and Lt continuous Itô

processes sampled at 1/N, 2/N, ..., 1 (trade clock)

I So ∆npN = O(1/

√N) and ∆nL

N = O(1/√N) as N →∞

I We will also wantsNn = O(1/

√N)

Continuous setup

Continuous dataAssume {

dpt = µtdt + σtdWt

dLt = btdt + `tdW′t

for some µt , bt , σt > 0, `t > 0, adapted, and

[W ,W ′]t =

∫ t0ρsds for some ρt ∈ [−1, 1]

and assume st is continuous and adapted

Discretization choice

pNn = pbn/Nc; LNn = Lbn/Nc

and

sNn =1√Nsbn/Nc

Proposed discrete equations

Wealth dynamics

∆nXN = LNn ∆np

N +sbn/Nc

2√N|∆nLN |+ ∆npN∆nLN

Price impact constraint

∆npN∆nL

N ≤ 0

If we want the discretization to mimic the micro structure of a LOB

Back to the diffusion limitXt = limN→∞ X

NbNtc in u.c.p. satisfies:

Wealth dynamics

dXt = Ltdpt +st`t√

2πdt + d [L, p]t

for the Best-Bid / Best-Ask order book model, and in the general case

dXt = Ltdpt + Φlt (ct)dt + d [L, p]t

with

Φσ(F ) =

∫F (y)φσ2(y)dy

Price impact constraintA necessary condition for an inventory obtained by limit orders is:

d [L, p]t ≤ 0

Applications: I. Option Hedging in Complete ModelExogenous model for midprice pt

dpt = µ(pt)dt + σ(pt)dWt , (4)

Assume γt continuous and Markovian (in price level p)

γt(α) = γ(pt , α).

Admissible inventory ANY F-adapted Itô process

Lt = L0 +

∫ t0budu +

∫ t0ludWu (5)

lt is signed.

I lt < 0 when trading with limit orders

I lt ≥ 0 when trading with market orders.

Defineg(p, l) = sign(l)Φl (c(p, ·)) (6)

Applications: I. Hedging

I K0 trader’s initial cash endowment

I Hedging portfolio value (self - financing condition)

Xt = L0p0+K0+

∫ t0

Ludpu+

∫ t0

(σ(pu)lu − g(pu, lu)) du+r∫ t0

(Xu−puLu)du

I f ∈ C 0 payoff function of a European option with maturity T

DefinitionAn initial cash endowment K0 and an inventory process Lt replicate theEuropean payoff f (pT ) at maturity T if

XT = f (pT ) and X0 = K0 + p0L0. (7)

NB: When quoting a price for the option, the trader needs to quote an

initial delta asked of the buyer of the option !

ResultIf f ∈ C 0, T > 0 and v ∈ C 1,3 satisfies:

∂v

∂t(t, p)+g

(p, σ(p)

∂2v

∂p2(t, p)

)−σ

2(p)

2

∂2v

∂p2(t, p)+rp

∂v

∂p(t, p) = rv(t, p)

with terminal condition v(T , p) = f (p), then

Lt =∂v

∂p(t, pt), and K0 = v(0, p0)−

∂v

∂p(0, p0)p0

replicate the payoff f (pT ) at maturity T , its volatility is given by

lt = σ(pt)∂2v

∂p2(t, pt),

and the replication price of the option is X0 = v(0, p0).

I Positive gamma options can only be hedged with marketorders

I Negative gamma options can only be hedged with limit orders.

Case of Best-Bid / Best-Ask LOB

Model assumptionsPrice model:

dpt = µ(t, pt)dt + σ(t, pt)dWt

Inventory model:dLt = btdt − `tdWt

Spread model: (empirical studies)

st =√

2πλσt , with λ > 1/2

No dividend or interest rate.

NB: λ = 1 implies dXt = Ltdpt (frictionless case)

Objective

Given the model for p and s, find L such that X hedges aEuropean option with payoff f (pT ).

Replication argument

Markovian setup, so price of the option given by function v(t, p).Itô’s formula:

d(Xt − v(t, pt)) = (Lt −∆t) dpt − (Θt +1

2Γtσ

2(t, pt))dt

+st√2π`tdt + d [p, L]t

Matching Itô decompositions

Lt = ∆t

which also implies−`t = Γtσ(t, pt)

Final Solution

Delta hedging {Lt = ∆t

`t = −Γtσ(t, pt)

Only negative Gamma options can be replicated via limit orders!

Pricing PDE

∂tv(t, p) +

(λ− 1

2

)σ2(t, p)∂2pv(t, p) = 0

local volatility multiplied by a factor of√

2λ− 1

Applications. II Market making

Setting

Still, aggregate market maker.

I Price pt exogenously given

I Sole control: bid-ask spread st .

I Affects inventory Lt .

I Price impact included through correlation between inventoryand price.

Objectives

Mathematical ProblemSolve the optimal control problem of a risk-neutralrepresentative market maker.

Model Insights

I What macro-quantities the market maker is long.

I What factors affect the optimal bid-ask spread.

Microscopic model

Modified Almgren & Chriss model

∆nL = −λn+1∆np

Modified Avellaneda & Stoikov model

E[λn+1| Fn] = ρn(sn)fn(sn)

E[λ2n+1∣∣Fn] = fn(sn)2

Macroscopic model

Inventory

Recall, pt given exogenously,

dLt = −ρt(st)ft(st)dpt + ft(st)√

1− ρ2t (st)σtdW⊥t

First term is standard linear price impact. Second term is thenon-toxic order flow (in the terminology of O’ Hara et.al).

Objective function

EXT(risk neutral market maker)

Solution

(Pontryagin) stochastic maximum principle

Solution can be reduced via martingale methods to finding themaximum of the function:

Ft : s 7→s√

2πσtft(s)− αtρt(s)ft(s)

where

αt = E [pT − pt | Ft ]µtσ2t

+Ztσt

with Zt the volatility of E [pT | Ft ].

Extra assumption

Homogeinization

Assume ft and ρt to be of the form

ρt(s) = ρ(t/σt); ft(s) = f (st/σt)

Consequence

Optimal spread:s∗t = σtm(αt)

P&L:

EXT = E[∫ T

0M(αt)σ

2t dt

]for some functions m and M. M is always decreasing. m isincreasing under certain uniqueness assumptions.

Special case 1: martingale market

Price modelIf

dpt = σtdWt

thenαt = 1

Consequence

Benchmark case. linear relationship between spread andvolatility.Market maker is long volatility as long as he is profitable(M(1) > 0).

Special case 2: momentum market

Price model

dpt = µptdt + σptdWt

GBM (Samuelson) with µ > 0, then

αt =µ

σ2

(eµ(T−t) − 1

)+ eµ(T−t)

Consequence

αt > 1, αt is a deterministic, decreasing function of t.The market maker quotes larger spreads, expects less profit andcaptures less volume in the ’momentum’ Black-Scholes model.

Special case 3: mean-reverting market

Price model

dpt = −ρ(pt − p0)dt + σdWtmean reverting OU with ρ > 0, then

αt = −ρ

σ2(pt − p0)2

(e−ρ(T−t) − 1

)+ e−ρ(T−t)

Consequence

αt < 1 iff (pt − p0)2 < σ2

ρUnless the price is significantly away from its long-term trend, themarket maker quotes smaller spreads, expects more profit andcaptures more volume in the ’mean reverting’ Ornstein-Uhlenbeckmodel.

Summary of the three testbed cases

Martingale market

st/σt is a constant. Market maker is on average long theintegrated volatility.

Momentum marketst/σt is an increasing function of T − t. Profits are smaller andspreads larger than in the martingale market.

Mean-reverting market

st/σt is an increasing function of (pt − p0)2. Profits are typicallylarger and spreads typically smaller than in the martingale case.

Toxicity index

MotivationO’Hara defines a toxicity index as a measure of the adverseselection limit orders are subject to. Useful for:

I Deciding whether to use limit or market orders.

I Market making.

I Understanding flash crashes.

Different interpretations

O’Hara’s toxicity index is based on an informed trader model andonly looks at trade volumes. We propose within our price impactframework, an index which takes into account both trade volumesand price changes.

Two forms of Toxicity

Instantaneous toxicity

ρt = −corr(∆p,∆L)tis an estimate of the instantaneous correlation between price and

inventory variations. It represents the proportion of incomingmarket orders that move the price.

Integrated toxicity

r = −2∑

∆np∆nL∑sn|∆nL|

measures the ratio between the money lost to price impact, andthe money collected through spread. De facto, Market Makershold an option on this ratio.

Stock correlation r toxicityAAPL -0.17270704 0.19904208GOOG -0.23689058 0.32856196BRCM -0.19237560 0.29776003CELG -0.26835355 0.48287317CTSH -0.33887494 0.51758560CSCO -0.08393210 0.09300757BIIB -0.27832205 0.40193651AMZN -0.23614694 0.30494250GPS -0.20956508 0.48908889SFG -0.24173454 0.57253111INTC -0.05301259 0.05574866GE -0.10889870 0.11888714JKHY -0.33407745 0.56987813PFE -0.15849674 0.15958849CBT -0.34887086 0.74490980AGN -0.35890531 0.78020785CB -0.38667565 0.58090719AA -0.08046277 0.08406282FPO -0.49598056 1.14964119

General Case

Empirical / Econometric Data Analysis

I Reasonable procedure to reconstruct large orders fromNASADQ ITCH messages

I Self-financing condition needs to be modified

I Adverse selection constraint even ”more true !”

General LOB Model

Theoretical / Mathematical Analysis

I At each time t, order book given by two positive measures(bids & ask distributions, say bt & at)

I Equivalently, by the mid-price pt and a shape function γts.t. γ′′t = bt + at , γt(0) = γ

′t(0) = 0

I Equivalently, by the mid-price pt and a transaction costfunction ct Legendre transform of γt

I Previous case: bt = δpt−st/2, at = δpt+st/2, ct(`) =st2 |`|

Low Frequency Model

Input Data

I Diffusion processes for pt and LtI A continuous process γt with values in a space of convex

functions, γt(0) = γ′t(0) = 0

I Its Legendre transform ` ↪→ ct(`)

Sef-financing Equation

I Discretization & Microstructure AnalysisI For each integer N, pNn = pn/N and L

Nn = Ln/N

I Discretize and rescale the order book γNn (x) =1N γn/N

(√Nx)

I Then cNn (`) =1N cn/N (

√N`)

I Define trade clock wealth (thru s.f.e.)∆nX

N = LNn ∆npN + 1N cn/N (−

√N∆nL

N ) + ∆npN ∆nL

N

I Macrostructure Analysis via Limit N →∞I Self-financing equation

dXt = Ltdpt + Φlt (ct)dt + d [L, p]t

with

Φσ(F ) =

∫F (y)φσ2(y)dy .

Conclusions (for now)

I Standard self-financing equations miss certain features ofhigh-frequency microstructure.

I We propose an equation which takes into account:

I transaction costsI price impactI differentiates between limit orders and market orders

I Same level of complexity

I Similar equation for market orders

I Fits data very well(tested on a pool of 120 stocks selected for an ECB study of HFT).

I Generalizes to a full LOB

I Needs more attention

I Relate ρt ≤ 0 to the queuing systems in LOBsI Which limit order strategies produce a given inventory Lt?I ............................

Conclusions (for now)

I Standard self-financing equations miss certain features ofhigh-frequency microstructure.

I We propose an equation which takes into account:

I transaction costsI price impactI differentiates between limit orders and market orders

I Same level of complexity

I Similar equation for market orders

I Fits data very well(tested on a pool of 120 stocks selected for an ECB study of HFT).

I Generalizes to a full LOB

I Needs more attention

I Relate ρt ≤ 0 to the queuing systems in LOBsI Which limit order strategies produce a given inventory Lt?I ............................

Limiting argument

Trading Frictions in High Frequency MarketsTrading Frictions in High Frequency Markets R. Carmona...

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