Intraday Market Making with Overnight Inventory Costs

Intraday Market Making

with Overnight Inventory Costs

Agostino Capponi

Department of Industrial Engineering and Operations Research

Columbia University

[email protected]

Joint work with T. Adrian, E. Vogt and H. Zhang

Financial/Actuarial Mathematics Seminars

Columbia University

October 12, 2016

Agostino Capponi Intraday Market Making Ann-Arbor, 2016 1

Introduction

High Frequency Trading

I HFT: Automated, high speed, low latency trading

I Majority of volume in US equity, US Treasury, and USD FX

I HFTs associated with a compression in bid-ask spreads, an increase in

volume, and smaller trade sizes, on average

I Unlike dealers HFTs unwind inventory at the end of the trading day


Introduction

What We Do

1. Develop a model of HFT market making

I Buyers and sellers arrive exogenously

I HFT intermediates

I No constraints on leverage intraday

I Exogenous overnight inventory costs

2. Confront the model with data

3. Study price stability within the model


Introduction

What We Find 1: Theoretically

1. Overnight inventory costs impact intraday price and liquidity

dynamics

2. Intraday price impact is endogenous

3. Bid and ask prices are non-increasing functions of the inventory level

4. At end of day, price sensitivity to inventory levels intensifies

I Price impact gets stronger

I Bid-ask spreads get wider


Introduction

What We Find 2: Empirically in the U.S. Treasury Market

1. Bid-ask spreads tends to rise towards the end of the day

2. Price impact tends to rise towards the end of the day

3. Price movements are negatively correlated with changes in inventory

(measured by the negative cumulative net trading volume)

4. Cumulative net volume: end users’ amount purchased (from the

HFT) minus end users’ amount sold (to the HFT)


Introduction

Related Literature

I Market making with inventory costsI Stoll (1980), Amihud and Mendelson (1980), Aıt-Sahalia and Saglam (2016)

I Asymmetric information modelsI Kyle (1985), Glosten and Milgrom (1985), Admati and Pfleiderer (1988),

Danilova and Julliard (2015), Foucault, Hombert, and Rosu (2016)

I Empirical studies of high frequency tradingI Herndeshott, Jones, and Menkveld (2011), Menkveld (2013), Brogaard,

Hendershott, and Riordan (2014), Herndeshott and Menkveld (2014),

Chaboud, Chiquoine, Hjalmarsson, and Vega (2014), Biais, Foucault, and

Moinas (2015)


HFT Inventory Costs

Outline

HFT Inventory Costs

The Model

The Control Problem

Optimal Price Policies and their Dependence on Inventory

Intertemporal Analysis of Optimal Price Policies

Endogenous Price Impact and Widening Bid-Ask Spreads

Empirical Analysis and Testable Implications

Price Stability


HFT Inventory Costs

HFT’s Desire to End the Day with Little Inventory

The SEC (2010) defines HFTs as traders that:

1. use computer programs for generating, routing, and executing orders

2. use co-location services to minimize network latencies

3. use very short time-frames to establish and liquidate positions

4. submit numerous orders that are cancelled shortly after submission

5. end the trading day in as close to a flat position as possible


HFT Inventory Costs

The Joint Staff Report (2015)

I HFTs end day flat unlike bank dealers

I Median HFT ends the trading day close to flat

I HFTs provide liquidity on both sides of the market

I Over 80 % of trading in 10-& 30-year Treasury bonds is intraday


HFT Inventory Costs

Empirical Studies of HFTs

I Jovanovic and Menkveld (2011) identify a dealer in Dutch equity

markets that trades frequently, representing a third of trades: his net

position over the trading day is zero almost half of the sample days

I Biais and Woolley (2011) reproduces the net position of that trader

showing that periods of autocorrelated positive and negative

inventory eventually end at exactly zero

I Benos and Sagade (2016) analyze data from U.K. equity markets and

find that HFTs generally end the day with a flat position


The Model

Outline

HFT Inventory Costs

The Model

The Control Problem





Price Stability


The Model

Demand and Supply

I Buy/sell orders follow a Bernoulli process with arrival probability πI buy order with probability πBO

π

I sell order with probability πSO

π

I Purchased and sold quantities depend on ask and bid prices:

QBO(x) = c (p − x)+ , QSO(x) = c (x − q)+

I p: the maximum price at which a buy order is placed

I q: minimum price at which a sell order is placed

I Equilibrium price: the price at which the market clears in a frictionless

market with asynchronous trades:

minx

E[(

QBO(x)NBOt − QSO(x)NSO

t

)2], p =

πBO p + πSO q

π


The Model

Demand and Supply

QSO(x)QBO(x)

q p

��

�

��

��

��

�� (�� )

��


The Model

The HFT

I The HFT optimally chooses bid and ask prices to:

max(a·,b·)∈(R2

+)TE[WT − λI 2

T + pIT]

I subject to:

Wt = W0 +t∑

s=1

as QBO(as) ∆NBO

s −t∑

s=1

bs QSO(bs) ∆NSO

s

It =t∑

s=1

QSO(bs) ∆NSOs︸︷︷︸

Shares boughtfrom sell investors

−t∑

s=1

QBO(as) ∆NBOs︸︷︷︸

Shares soldto buy investors


The Control Problem

Outline

HFT Inventory Costs

The Model

The Control Problem





Price Stability


The Control Problem

Dynamic Programming Formulation

I We consider the class of Markov control strategies.

I The value function is given by

V (t,w , i) := sup(a·,b·)

E[WT − λI 2

T + pIT |Wt = w , It = i]

I Using the dynamic programming principle, we obtain

V (t − 1,w , i) = V (t,w , i) + sup(a·,b·)∈(R2

+)TH(t, a, b)

where the Hamiltonian H is given by

H(t, a, b) := πBO [V (t,w + a QBO(a), i − QBO(a))− V (t,w , i)]

+πSO [V (t,w − b QSO(b), i + QSO(b))− V (t,w , i)]


The Control Problem

The Simplified Optimization Problem

I The linearity of value function in the wealth variable w suggests

V (t,w , i) = w + F (t, i)

I All we need to solve for is

F (t − 1, i) = F (t, i) + Ht(i)

Ht(i) := sup(a,b)

πBO[

1

ca(p − a)+ + F (t, i − (p − a)+)− F (t, i)

]+ πSO

[−1

cb(b − q)+ + F (t, i + (b − q)+)− F (t, i)

]F (T , i) = −λi2 + pi ,

where a = ca, b = cb, p = cp, q = cqAgostino Capponi Intraday Market Making Ann-Arbor, 2016 17

The Control Problem Optimal Price Policies and their Dependence on Inventory

First Order Conditions

I The first order conditions (FOC) are

∂iF (t, i − p + at(i)) +1

c(p − 2at(i)) = 0

∂iF (t, i + bt(i)− q) +1

c(q − 2bt(i)) = 0

I The solutions at(i), bt(i) are the candidate ask and bid prices at time

t, decided at t − 1 based on the inventory level i at t − 1

I If F (t, i) is strictly concave in i , the mappings i : 7→ at(i) and

i :7→ bt(i) are all strictly decreasing, continuous, and mapping onto R



Impact of End-of Day Inventory Motives on HFT Intraday

First-Order Conditions

∂iF (t, i − p + a(i))

0

2c a(i)− p

a(i)a∗(i)myopic

a∗(i)forward looking



Candidate Bid Ask Prices

I Fix t and assume F (t, i) to be strictly concave in i . Then the function

Gt(i) := ∂iF (t, i)− 2i/c

is strictly decreasing and admits an i-inverse G−1t .

I We can then obtain explicit representations for bid and ask prices

at(i) = G−1t

(p − 2i

c

)− i + p, bt(i) = G−1

t

(q − 2i

c

)− i + q

I Explicit, but computationally efficient? Yes, if we can efficiently

compute the inverse function G−1t (more later)



Trading Boundaries I

I Optimal ask and bid prices are a∗t (i) = a∗t (i)c and b∗t (i) = b∗t (i)

c ,

a∗t (i) = max(at(i), 0), b∗t (i) = max(bt(i), 0)

I Define the critical inventory boundaries L1t and L2

t as solutions to

b∗t (L1t ) = q, a∗t (L2

t ) = p

I Plugging into the FOCs

∂iF (t, i)∣∣i=L1

t− q = 0, ∂iF (t, i)

∣∣i=L2

t− p = 0



Trading Boundaries II

I Hence the optimal bid price b∗t (i) is always lower than or equal to q

when the inventory level i ≥ L1t

I Likewise, the optimal ask price a∗t (i) is always higher than or equal to

p when the inventory level i ≤ L2t



The Optimal Price Policy Functions

Lt2,p

Lt1,q

Ask a˜t* (i)

Bid b˜t*(i)

-�� -�� -��

��

��

��

��

��

��

�� (�� )

��(��)


The Control Problem Intertemporal Analysis of Optimal Price Policies

The Critical Inventory Thresholds

Proposition 4.1

The sequence (L1t )Tt=1 is positive, and strictly decreasing, while the

sequences (L2t )Tt=1 is negative, and strictly increasing. In particular,

L1T = p−q

2λ and L2T = − p−p

2λ .

I For i ∈ [L2T , L

1T ], we have the end of day or time-T optimal price

policy functions

a∗T (i) =1

1 + λc

(p

(1

2+ λc

)+

p

2− λi

)+

,

b∗T (i) =1

1 + λc

(q

(1

2+ λc

)+

p

2− λi

)+



The Critical Inventory Thresholds

Lt1

Lt2

Only sell

Only buy

Buy & sell

� ��

-��

-��

�

��

��

��

��

(��)



Time Invariant Concave Structure of F

I Assume that F (T , i) is C 1 and strictly i-concave. Then, for anyt = 1, 2, . . . ,T , F (t − 1, i) is strictly i-concave, continuouslydifferentiable, and with a i-derivative mapping onto R, which admitsthe following recursive representation

∂iF (t − 1, i) =

(1 − πBO )∂iF (t, i) + π

BO∂iF (t, i − p), i ≥ L0

t

(1 − πBO )∂iF (t, i) +

πBO

c(2at (i) − p), L0

t > i ≥ L1t

(1 − πBO − π

SO )∂iF (t, i) +πBO

c(2at (i) − p) +

πSO

c(2bt (i) − q), L1

t > i > L2t

(1 − πSO )∂iF (t, i) +

πSO

c(2bt (i) − q), L2

t > i,

.

where L0t is the inventory level such that at (L0

t ) = 0, i.e. ∂iF (t, L0t − p) + p = 0



An equivalent representation of ∂iF

I Assume that F (T , i) is C 1 and strictly i-concave. Then, for any

t = 1, 2, . . . ,T , F (t − 1, i) admits the equivalent representation

∂iF (t − 1, i) = E[∂iF (t, I

(a∗,b∗)t )|I (a∗,b∗)

t−1 = i]

=

(1− πBO)∂iF (t, i) + πBO∂iF (t, i − p + a∗t (i)), i ≥ L1t

(1− πBO − πSO)∂iF (t, i)

+ πBO∂iF (t, i − p + a∗t (i))

+ πSO∂iF (t, i + b∗t (i)− q), L1t > i > L2

t

(1− πSO)∂iF (t, i) + πSO∂iF (t, i + b∗t (i)− q), L2t ≥ i



Computational Efficiency

I Assume F (T , i) to be quadratic in i . The equivalent representation of

∂iF yields that F (t, i) is piecewise quadratic in i , for any t.

I Recall that

at(i) = G−1t

(p − 2i

c

)− i + p, bt(i) = G−1

t

(q − 2i

c

)− i + q

Gt(i) = ∂iF (t, i)− 2i/c

I The inverse of a piecewise linear function is piecewise linear, hence

at(i) and bt(i) can be efficiently computed

I Using the piecewise linear representations of at(i) and bt(i), we can

regress backward and obtain the piecewise linear representation of

∂iF (t − 1, i) from the piecewise linear representation of ∂iF (t, i).


The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads

Bid-Ask Spreads and Price Sensitivities

Proposition 4.2Let (λ1

t )Tt=1 and (λ2t )Tt=1 be sequences of positive numbers: λ1

T = λ2T = λ,

λ1t−1 = λ1

t

(1−min{πBO , πSO}

λ1t c

1 + λ1t c

), t = 2, 3, . . . ,T ,

λ2t−1 = λ2

t

(1− (πBO + πSO)

λ2t c

1 + λ2t c

), t = 2, 3, . . . ,T ,

For any t = t0, t0 + 1, . . . ,T ,

−λ1t

1 + λ1t c≤

a∗t (i1)− a∗t (i2)

i1 − i2,b∗t (i1)− b∗t (i2)

i1 − i2≤ −

λ2t

1 + λ2t c, for any L1

t ≥ i1 > i2 ≥ L2t ,

B(λ2t ) ≤ a∗t (i)− b∗t (i) ≤ B(λ1

t ), L1t ≥ i ≥ L2

t , B(λ) =12

+ λc

1 + λc(p − q),

−2λ1t ≤

∂iF (t, i1)− ∂iF (t, i2)

i1 − i2≤ −2λ2

t , i1 > i2



Increasing Sensitivities

I The concavity of F (t, i) depends on the optimal trading behavior of

the market maker for t < T

I The trading activities of the HFT reduce the concavity: the concavity

is reduced the most when the HFT trades with both counter-parties

I Sensitivity of prices/value function to the inventory level, and bid-ask

spread, are lower when the time-to-close T − t increases.

I The larger πBO and πSO are, the faster these two sequences decrease,

and the faster bid-ask spread and sensitivities decay



Cost-Benefit Tradeoff

I More frequent orders make markets more liquid, i.e. with lower

bid-ask spreads

I A more liquid market makes the inventory constraint fade away faster

I As time approaches the end of the day, the growing concern about

the inventory constraint discourages the HFT from trading actively

I Hence bid-ask spreads get larger

I Tradeoff between making trading profits and holding a non-zero

inventory at the end of day



Optimal Bid-Ask Spread at Zero Inventory Level

Lower πBO ,πSO

Benchmark

Lower λ

� ��

��

��

��

��

��

��

��

��

��(��)



Flash Events

I Flash rally: a much larger number of buyers, relative to sellers, arrives

during a period of time. Then the price impact generated from trades

of the HFT with buyers will quickly push ask and bid prices upward

I Flash crash: a much larger number of sellers, relative to buyers,

arrives during a period of time. Then the the price impact generated

from the asset sales will quickly drive ask and bid prices down



Simulated Inventory and Midquote Paths

��

��

��

��

��

��

��(��)

� ��

-��

-��

�

��

��

��

��



Endogenous Price Impact

I Time varying and endogenous nature of price-impact and bid-ask

spreads are distinguishing features due to end-of-day constraint

I If we are sufficiently far from day-end, ask and bid prices are roughly

linear in the inventory:

a∗t = β0 + β1It−1, β1 < 0

I The difference between consecutive ask prices is

a∗t+1 − a∗t = β1(It − It−1) = β1

(QSO(b∗t )∆NSO

t − QBO(a∗t )∆NBOt

)I Buy order increases ask to discourage a subsequent trade with buyer

I Sell order decreases ask to invite a subsequent trade with buyer



Outline

HFT Inventory Costs

The Model

The Control Problem





Price Stability



Testable Implications of the Model

(TI-1) There is a significant positive relationship between both bid and ask

prices and the negative of the HFT’s inventory

(TI-2) The dependence of bid and ask prices on HFT’s inventory becomes

stronger as time approaches the day’s end. Thus price impact is

largest at day’s end.

(TI-3) Flash events: Endogenous price impact and one-sided trading during

short window, followed by reversal of trading

(TI-4) The bid-ask spread tends to increase as time approaches the day’s end



US Treasury Data

I High-frequency intraday data from BrokerTec

I accounts for 60% of electronic trading activity in the cash market

I Trade and limit order book data time-stamped to the millisecond

I Construct HFT inventory proxy as the negative cumulative net

volume: buy from HFT minus sell to HFTI Brogaard, Hendershott, and Riordan (2014) point out that if HFTs’

inventory positions are close to zero overnight, then their inventories can be

measured by accumulating their buying and selling activity in each security

from opening to each point in time



10-Year Treasury Prices and Cumulative Net Volume

-5000 0 5000100 Lots of $1 million

-3

-2

-1

0

1

2

3

Dai

ly P

rice

Cha

nge

(Per

cent

of P

ar)

bols = 0.0001t-stat = 12.93R2 = 0.17

Cumulative net dollar volume change of $1 billion corresponds to an

increase in 10-year Treasury prices of about 0.01 percent of par



Intraday Impact of Inventory on Pricing

I The model suggests that the HFT’s desire to end the day flat causes

the relationship between quoted prices and inventory to steepen near

the close of trading

I Run the regression of the negative of the cumulative net dollar

volume (proxing inventory) on price changes each hour of the active

trading day



Inventories and 10-Year Treasury Prices by Hour

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00-0.03

-0.02

-0.01

0

0.01

0.02

0.03

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00-0.03

-0.02

-0.01

0

0.01

0.02

0.03



Price Changes and Inventories:

Tests of Equal Slopes on Intraday vs Close

Time of Day i βclose − βi t-stat p-value

9:00 - 10:00 -0.01 [-6.43] (0.000)

10:00 - 11:00 -0.01 [-6.43] (0.000)

11:00 - 12:00 -0.01 [-6.67] (0.000)

12:00 - 13:00 -0.01 [-6.34] (0.000)

13:00 - 14:00 -0.01 [-4.76] (0.000)

14:00 - 15:00 -0.01 [-5.76] (0.000)

15:00 - 16:00 -0.01 [-7.40] (0.000)

16:00 - 17:00 -0.01 [-6.81] (0.000)



Flash Events

I The model suggests that in a period in which there is intense buying

pressure relative to selling pressure, cumulative net volumes increase

sharply (and HFT inventories decline)

I ↑ buying pressure ⇒ positive cumulative net volume ⇒ ↓ HFT

inventory ⇒ ↑ quotes to avoid further inventory declines ⇒ ↑ price

impact

I Conversely, in phases of intense selling pressure, cumulative net

volume declines, and HFT inventories rise

I Flash event: price spike and reversal in short period

I Plot the relationship between quoted prices and cumulative net

volume during the October 15 flash event



Cumulative Net Volume and Prices during a Flash Event

09:30 09:33 09:36 09:38 09:41 09:44-600

-400

-200

0

200

400

600

800

1000C

umul

ativ

e N

et V

olum

e ($

milli

ons)

102

102.5

103

103.5

104

104.5

105



Intraday Bid-Ask Spreads

I The model predicts bid-ask spreads to increase towards the close



Bid-Ask Spreads Intraday and over Time

06:00 09:00 12:00 15:00 18:00Time of Day

1.6

1.7

1.8

1.9

2

2.1

Bas

is P

oint

s of

Par

2012 2013 2014 2015 2016Year

0

5

10

15

Bas

is P

oint

s of

Par

Bid-Ask Spread at 17:25Bid-Ask Spread at 9:00


Price Stability

Outline

HFT Inventory Costs

The Model

The Control Problem





Price Stability


Price Stability

Price Stability

I Are price paths more volatile when the severity of the inventory

constraint goes up or arrivals are more frequent?

I Comparative statics with respect to the two key parameters

I The overnight funding cost λ

I The arrival probabilities πBO , πSO


Price Stability

Measuring Price Stability

I We analyze three measures of price stability

1. The maximum price deviation from the equilibrium price p is

max{p − minimum traded bid price, maximum traded ask price − p}

2. The maximum drawdown of the mid-price is

max1≤t≤T (max1≤s≤t Ss − St)

where St = 12 (a∗t + b∗t ) is the mid-price

3. The maximum bid-ask spread at trading times, i.e., the spread

whenever a trade occurs either with a buy or sell investor


Price Stability

Comparative StaticsPanel A: Max Spread λ = 0.002 λ = 0.01 λ = 0.02 λ = 0.04 λ = 0.08

πBO = πSO = 5% 1.7880 (2.3033e-05) 2.0064 ( 2.0108e-04) 2.1691 (3.7665e-04) 2.3700 (5.8740e-04) 2.5909 (7.6314e-04)

πBO = πSO = 7% 1.7881 (2.2992e-05) 2.0076 (2.0184e-04) 2.1708 (3.8002e-04) 2.3707 (5.9166e-04) 2.5922 (7.7200e-04)

πBO = πSO = 10% 1.7884 (2.2898e-05) 2.0093 (2.0300e-04) 2.1733 (3.8408e-04) 2.3748 (5.9812e-04) 2.5954 (7.8088e-04)

πBO = πSO = 20% 1.7891 (2.2188e-05) 2.0149 (2.0571e-04) 2.1832 (3.9414e-04) 2.3872 (6.1458e-04) 2.6047 (7.9631e-04)

πBO = πSO = 30% 1.7899 (2.1403e-05) 2.0217 (2.0702e-04) 2.1946 (4.0415e-04) 2.4036 (6.3242e-04) 2.6256 (8.1021e-04)

Theoretical spread at T 1.7962 2.0923 2.3375 2.6273 2.9000

Panel B: Max deviation λ = 0.002 λ = 0.01 λ = 0.02 λ = 0.04 λ = 0.08

πBO = πSO = 5% 0.3565 (9.7284e-07) 0.4350 (2.0269e-04) 0.4784 (2.7300e-04) 0.5267 (3.9184e-04) 0.5767 (6.1118e-04)

πBO = πSO = 7% 0.3583 (9.9658e-07) 0.4361 (2.0533e-04) 0.4795 (2.7873e-04) 0.5279 (4.0312e-04) 0.5793 (6.3825e-04)

πBO = πSO = 10% 0.3599 (1.0191e-04) 0.4376 (2.0893e-04) 0.4812 (2.8595e-04) 0.5300 (4.1296e-04) 0.5813 (6.4236e-04)

πBO = πSO = 20% 0.3619 (1.0486e-04) 0.4398 (2.1502e-04) 0.4846(2.9775e-04) 0.5346 (4.3024e-04) 0.5869 (6.6160e-04)

πBO = πSO = 30% 0.3626 (1.0532e-04) 0.4414 (2.1752e-04) 0.4876 (3.0546e-04) 0.5394 (4.4557e-04) 0.5934 (6.7936e-04)

Panel C: Max drawdown λ = 0.002 λ = 0.01 λ = 0.02 λ = 0.04 λ = 0.08

πBO = πSO = 5% 0.2142 (2.9570e-04) 0.4450 (5.9802e-04) 0.5704 (8.0682e-04) 0.7030 (1.2477e-03) 0.8408 (2.2192e-03)

πBO = πSO = 7% 0.2202 (3.0202e-04) 0.4483 (6.0427e-04) 0.5729 (8.2453e-04) 0.7050 (1.2941e-03) 0.8454 (2.3461e-03)

πBO = πSO = 10% 0.2246 (3.0812e-04) 0.4494 (6.0748e-04) 0.5727 (8.2378e-04) 0.7076 (1.2656e-03) 0.8402 (2.1197e-03)

πBO = πSO = 20% 0.2288 (3.1391e-04) 0.4451 (6.1501e-04) 0.5655 (8.4144e-04) 0.6951 (1.2806e-03) 0.8423 (2.2391e-03)

πBO = πSO = 30% 0.2293 (3.1616e-04) 0.4377 (6.1930e-04) 0.5527 (8.4952e-04) 0.6851 (1.3058e-03) 0.8289 (2.1909e-03)


Price Stability

Distribution of Maximum Bid-Ask Spread

λ = 0.002 and π = 5% λ = 0.002 and π = 30%

1.74 1.75 1.76 1.77 1.78 1.79 1.8 1.810

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Fre

qu

en

cy

Max spread (basepoint of par)1.74 1.75 1.76 1.77 1.78 1.79 1.8 1.810

0.5

1

1.5

2

2.5

3

3.5x 10

4

Fre

qu

en

cy

Max spread (basepoint of par)

λ = 0.08 and π = 5% λ = 0.08 and π = 30%

1.6 1.8 2 2.2 2.4 2.6 2.8 30

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Fre

qu

en

cy

Max spread (basepoint of par)1.8 2 2.2 2.4 2.6 2.8 30

0.5

1

1.5

2

2.5

3

3.5x 10

4

Fre

qu

en

cy

Max spread (basepoint of par)


Price Stability

Distribution of Maximum Deviation from Equilibrium

λ = 0.002 and π = 5% λ = 0.002 and π = 30%

0.35 0.4 0.45 0.5 0.550

200

400

600

800

1000

1200

Fre

qu

en

cy

Max deviation (basepoint of par)0.35 0.4 0.45 0.5 0.55 0.6 0.65

0

200

400

600

800

1000

1200

1400

Fre

qu

en

cy

Max deviation (basepoint of par)

λ = 0.08 and π = 5% λ = 0.08 and π = 30%

0 5 10 150

0.5

1

1.5

2

2.5x 10

4

Fre

qu

en

cy

Max deviation (basepoint of par)0 2 4 6 8 10 12 14

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Fre

qu

en

cy

Max deviation (basepoint of par)


Price Stability

Distribution of Maximum Price Drawdown

λ = 0.002 and π = 5% λ = 0.002 and π = 30%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

200

400

600

800

1000

1200

1400

Fre

qu

en

cy

Maximum drawdown (basepoint of par)0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

1000

1200

1400

Fre

qu

en

cy

Maximum drawdown (basepoint of par)

λ = 0.08 and π = 5% λ = 0.08 and π = 30%

0 10 20 30 40 500

0.5

1

1.5

2

2.5x 10

4

Fre

qu

en

cy

Maximum drawdown (basepoint of par)0 5 10 15 20 25 30 35 40

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Fre

qu

en

cy

Maximum drawdown (basepoint of par)


Price Stability

Simulated Price Trajectories for Different Overnight

Inventory Costs

λ=0.1λ0

λ=λ0

λ=4λ0

� ��

��

��

��

��

��

��

��

��

��(��)


Conclusion

Conclusion

I We study the importance of the overnight inventory cost for the

determination of intraday price and liquidity dynamics

I Optimal price setting strategy of the HFT gives rise to bid-ask spreads

and price impact metrics that tend to rise towards the end of the day

I Both bid-ask spread and price impact arise endogenously as functions

of inventory, time of day, and magnitude of the overnight inventory

cost

I Even though trading is costless intraday, the overnight inventory cost

impacts bid-ask spreads and price impact at all times during the day

I The steepening of price impact due to the end of day constraint lead

to more volatile price paths intraday


Conclusion

Admati, A., and P. Pfleiderer (1988): “A Theory of Intraday Patterns:

Volume and Price Variability,” Review of Financial Studies, 1(1), 3–40.

Aıt-Sahalia, Y., and M. Saglam (2016): “High Frequency Traders: Taking

Advantage of Speed,” NBER Working Paper, (19531).

Amihud, Y., and H. Mendelson (1980): “Dealership Market:

Market-making with Inventory,” Journal of Financial Economics, 8(1), 31–53.

Benos, E., and S. Sagade (2016): “Price Discovery and the Cross-Section of

High-Frequency Trading,” Journal of Financial Markets, forthcoming.

Biais, B., T. Foucault, and S. Moinas (2015): “Equilibrium Fast

Trading,” Journal of Financial Economics, 116(2), 292–313.

Biais, B., and P. Woolley (2011): “High Frequency Trading,” Working

Paper, London School of Economics and Toulouse School of Economics.

Brogaard, J., T. Hendershott, and R. Riordan (2014):

“High-Frequency Trading and Price Discovery,” Review of Financial Studies,

27(8), 2267–2306.


Conclusion

Chaboud, A., B. Chiquoine, E. Hjalmarsson, and C. Vega (2014):

“Rise of the machines: Algorithmic trading in the foreign exchange market,”

Journal of Finance, 69(5), 2045–2084.

Danilova, A., and C. Julliard (2015): “Information Asymmetries, Volatility,

Liquidity, and the Tobin Tax,” Working Paper, London School of Economics.

Foucault, T., J. Hombert, and I. Rosu (2016): “News Trading and

Speed,” Journal of Finance, 71(1), 335–382.

Glosten, L., and P. Milgrom (1985): “Bid, Ask and Transaction Prices in a

Specialist Market with Heterogeneously Informed Traders,” Journal of

Financial Economics, 14, 71–100.

Herndeshott, T., C. Jones, and A. Menkveld (2011): “Does

algorithmic trading improve liquidity?,” Journal of Finance, 66, 1–33.

Herndeshott, T., and A. Menkveld (2014): “Price pressures,” Journal of

Financial Economics, 114, 405–423.


Conclusion

Joint Staff Report (2015): “The U.S. Treasury Market on October 15,

2014,” U.S. Department of the Treasury, Board of Governors of the Federal

Reserve System, U.S. Securities and Exchange Commission, U.S. Commodity

Futures Trading Commission.

Jovanovic, B., and A. J. Menkveld (2011): “Middlemen in Limit-Order

Markets,” Working paper, VU University of Amsterdam.

Kyle, A. (1985): “Continuous Auctions and Insider Trading,” Econometrica,

53(6), 1315–1335.

Menkveld, A. (2013): “High Frequency Trading and the New-Market Makers,”

Journal of Financial Markets, 16(4), 712–740.

Stoll, H. (1980): “The Supply of Dealer Services in Securities Markets,”

Journal of Finance, 33(4), 1133–1151.


Conclusion

Paper available at

I T. Adrian, A. Capponi, E. Vogt, and H. Zhang. Intraday market

making with overnight inventory costs. Preprint available at http:

//papers.ssrn.com/sol3/papers.cfm?abstract_id=2844881


http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2844881

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2844881

Documents

Intraday Market Making with Overnight Inventory Costs