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
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Intraday Market Making with Overnight Inventory Costs
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Intraday Market Making
with Overnight Inventory Costs
Agostino Capponi
Department of Industrial Engineering and Operations Research
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 2
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 3
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 4
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)
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 5
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)
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 6
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 7
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 8
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 9
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 10
The Model
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 11
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 12
The Model
Demand and Supply
QSO(x)QBO(x)
q p
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Agostino Capponi Intraday Market Making Ann-Arbor, 2016 13
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 14
The Control Problem
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 15
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)]
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 16
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 18
The Control Problem Optimal Price Policies and their Dependence on Inventory
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 19
The Control Problem Optimal Price Policies and their Dependence on Inventory
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)
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 20
The Control Problem Optimal Price Policies and their Dependence on Inventory
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 21
The Control Problem Optimal Price Policies and their Dependence on Inventory
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 22
The Control Problem Optimal Price Policies and their Dependence on Inventory
The Optimal Price Policy Functions
Lt2,p
Lt1,q
Ask a˜t* (i)
Bid b˜t*(i)
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Agostino Capponi Intraday Market Making Ann-Arbor, 2016 23
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
)+
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 24
The Control Problem Intertemporal Analysis of Optimal Price Policies
The Critical Inventory Thresholds
Lt1
Lt2
Only sell
Only buy
Buy & sell
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Agostino Capponi Intraday Market Making Ann-Arbor, 2016 25
The Control Problem Intertemporal Analysis of Optimal Price Policies
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 26
The Control Problem Intertemporal Analysis of Optimal Price Policies
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 27
The Control Problem Intertemporal Analysis of Optimal Price Policies
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).
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 28
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 29
The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 30
The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 31
The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads
Optimal Bid-Ask Spread at Zero Inventory Level
Lower πBO ,πSO
Benchmark
Lower λ
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Agostino Capponi Intraday Market Making Ann-Arbor, 2016 32
The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 33
The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads
Simulated Inventory and Midquote Paths
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Agostino Capponi Intraday Market Making Ann-Arbor, 2016 34
The Control Problem Endogenous Price Impact and Widening Bid-Ask Spreads
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 35
Empirical Analysis and Testable Implications
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 36
Empirical Analysis and Testable Implications
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 37
Empirical Analysis and Testable Implications
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 38
Empirical Analysis and Testable Implications
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 39
Empirical Analysis and Testable Implications
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
Agostino Capponi Intraday Market Making Ann-Arbor, 2016 40