Statistical Arbitrage Using Limit Order Book Imbalance

94 Pages·2015·2.58 MB·English

Statistical Arbitrage Using
Limit Order Book Imbalance
Anton D. Rubisov
University of Toronto Institute for Aerospace Studies
Faculty of Applied Science and Engineering
University of Toronto
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
c 2015AntonD.Rubisov
(cid:13)
Statistical Arbitrage Using
Limit Order Book Imbalance
Anton D. Rubisov
UniversityofTorontoInstituteforAerospaceStudies
FacultyofAppliedScienceandEngineering
UniversityofToronto
2015
Abstract
This dissertation demonstrates that there is high revenue potential in us-
ing limit order book imbalance as a state variable in an algorithmic trading
strategy. Beginning with the hypothesis that imbalance of bid/ask order
volumes is an indicator for future price changes, exploratory data analysis
suggests that modelling the joint distribution of imbalance and observed
price changes as a continuous-time Markov chain presents a monetizable
opportunity. The arbitrage problem is then formalized mathematically as
a stochastic optimal control problem using limit orders and market orders
withtheaimofmaximizingterminalwealth. Theproblemissolvedinboth
continuous and discrete time using the dynamic programming principle,
whichproducesbothconditionsformarketorderexecution,aswellaslimit
order posting depths, as functions of time, inventory, and imbalance. The
optimalcontrolsarecalibratedandbacktestedonhistoricalNASDAQITCH
data,whichproducesconsistentandsubstantialrevenue.
ii
Acknowledgements
I couldn’t begin to enumerate the times and ways this dissertation could have failed
to be. I am deeply grateful to several individuals for making it a reality, and I’d like to
acknowledgethemhere.
Dr. Gabriele D’Eleuterio, my supervisor at UTIAS, is the epitomical supernatural
mentoroftheherojourney,withoutwhomIwouldhaveabandonedmystudies. Gabe
was the inspirational upholder of academic values, who relentlessly stressed the sub-
tleties of scientific presentation and communication, and guided me through a pretty
toughtime. Thankyouforyoursupport,guidance,andforeverycoffee.
Dr. SebastianJaimungal,professorofstatisticsattheuniversity,tookmeonasasurro-
gate student half-way through my studies, and guided every step of this dissertation.
Despitehavingabsolutelynoobligationtodoso,hehadfaithinmeandpatiencewith
me in navigating an entirely unfamiliar subject. Thank you for repeatedly making
yourselfavailabletohelpwithallthemath.
Dr. Jack Carlyle, then a PhD student in solar physics at UCL, inspired me to quit my
job and go back to school for a graduate degree. It wasn’t an easy decision, and the
juryisoutonwhetheritwasagoodone. Butthanksforalltheteaandcrumpets.
Ted Herman, a friend that redefines friendship, has read more of my research than
anyoneeverwill. Atacrucialtime,westrucktheFran’sAccordthatinvolvedprewrit-
ten$100chequesbeingshreddedonaweeklybasisuponreceiptofaprogressupdate.
Legitimately,thisisprobablythenumberonereasonthatmydissertationgotdone.
Thank you mom for always loving, supporting, and inspiring me to my best. Thank
youdadforprettywellbeingasupervisortoo.
iii
Contents
1 Introduction 1
1.1 TheLimit-OrderBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 ITCHDataSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 OrderImbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 ExploratoryDataAnalysis 8
2.1 ModellingImbalance: ContinuousTimeMarkovChain . . . . . . . . . . 8
2.2 MaximumLikelihoodEstimateofaMarkov-ModulatedPoissonProcess 10
2.2.1 InfinitesimalGeneratorMatrix . . . . . . . . . . . . . . . . . . . . 10
2.2.2 ArrivalRates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Two-DimensionalCTMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 PredictingFuturePriceChanges . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 NaiveTradingStrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 CalibrationandBacktesting . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 ConclusionsfromtheNaiveTradingStrategies . . . . . . . . . . . . . . . 20
iv
3 MaximizingWealthviaContinuous-TimeStochasticControl 24
3.1 SystemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 DynamicProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 MaximizingTerminalWealth . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 InterpretingtheDPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 MaximizingWealthviaDiscrete-TimeStochasticControl 37
4.1 SystemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 DynamicProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 MaximizingTerminalWealth . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 SimplifyingandInterpretingtheDPE . . . . . . . . . . . . . . . . . . . . 49
5 Results 52
5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 DynamicsoftheOptimalPostingDepths . . . . . . . . . . . . . . . . . . 53
5.2.1 ComparingOptimalControlPerformance . . . . . . . . . . . . . 62
5.3 In-SampleBacktesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Same-DayCalibration . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.2 WeekOffsetCalibration . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.3 AnnualCalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 Out-of-SampleBacktesting . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Conclusion 79
v
List of Figures
1.1 Structureandmechanicsofthelimit-orderbook . . . . . . . . . . . . . . 4
2.1 Hypotheticaltimelineofmarketordersandimbalanceregimeswitches . 10
2.2 Time intervals for time-weighted averaging of imbalance and for price
change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Comparisonofthenaivetradingstrategies . . . . . . . . . . . . . . . . . 21
5.1 OptimalbuyLOdepthsforsellpressureimbalance . . . . . . . . . . . . 55
5.2 OptimalbuyLOdepthsforneutralimbalance . . . . . . . . . . . . . . . 56
5.3 OptimalbuyLOdepthsforbuypressureimbalance . . . . . . . . . . . . 57
5.4 OptimalsellLOdepthsforsellpressureimbalance . . . . . . . . . . . . . 59
5.5 OptimalsellLOdepthsforneutralimbalance . . . . . . . . . . . . . . . . 60
5.6 OptimalsellLOdepthsforbuypressureimbalance . . . . . . . . . . . . 61
5.7 Comparisonofthefourstochasticoptimalcontrolmethods . . . . . . . . 63
5.8 Cointegrationrelationofthefourstochasticcontrolmethods . . . . . . . 64
5.9 Detailedsamplepathsofthestochasticoptimalcontrolstrategies . . . . 66
5.10 Comparisonofoptimalpostingdepthsonthesamplepath . . . . . . . . 67
5.11 ComparisonofP&Landinventoryonthesamplepath . . . . . . . . . . 68
vi
5.12 In-samplebacktestingperformanceusingsame-daycalibration . . . . . 70
5.13 In-samplebacktestingperformanceusingweek-offsetcalibration . . . . 73
5.14 In-samplebacktestingperformanceusingannualcalibration . . . . . . . 76
5.15 Out-of-samplebacktestingperformanceusingannualcalibration . . . . 78
vii
List of Tables
2.1 Stocksusedintheexploratorydataanalysis . . . . . . . . . . . . . . . . . 8
2.2 One-dimensionalencodingoftwo-dimensionalCTMC . . . . . . . . . . 13
2.3 p-valuesfortestingthetimehomogeneityhypothesis . . . . . . . . . . . 15
2.4 Probabilitiesoffuturepricechanges . . . . . . . . . . . . . . . . . . . . . 18
2.5 Hypotheticaltimelineofadverseselectionwithmarketorders . . . . . . 23
5.1 Stocksusedinthestochasticoptimalcontrolbacktesting . . . . . . . . . 52
5.2 Fixedparametersforbacktesting . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Parameterformulaeforbacktesting . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Correlationmatrixofstochasticoptimalcontrolreturns . . . . . . . . . . 62
5.5 Numberoftradescomparisonofthefourstochasticcontrolmethods . . 64
5.6 In-samplebacktestingperformanceusingsame-daycalibration . . . . . 71
5.7 In-samplebacktestingperformanceusingweek-offsetcalibration . . . . 74
5.8 In-samplebacktestingperformanceusingannualcalibration . . . . . . . 76
5.9 Out-of-samplebacktestingperformanceusingannualcalibration . . . . 78
viii
Nomenclature
# Numberofstatesintowhichthesmoothedimbalanceisdiscretized
bins
Filtrationofaprobabilityspace
F
Infinitesimalgeneratoroperator
L
∆S(t) Signoftheobservedpricechangeattime t
∆
t Sizeofrollingwindowforimbalancesmoothing
I
∆
t Sizeofwindowforpricechangecalculation
S
δ Thedepthatwhichwepostabuy(δ+)andsell(δ )limitorder
± −
η Magnitudeofthemidpricechange(arandomvariable)
0,z
1
Indicatorfunction
κ Limitorderfillprobabilityconstant
λ Arrivalratesofbuy(λ+)andsell(λ )orders
± −
G Continuous-timeMarkovchaingeneratormatrix
P Continuous-timeMarkovchaintransitionprobabilitymatrix
µ Rateofarrivalofotheragents’buy(µ+)andsell(µ )marketorders
± −
∂ Partialderivativewithrespecttox
x
ρ(t) Imbalanceratio,smoothedbyaveragingoverarollingwindow
τ Astoppingtimeatwhichweexecuteamarketorder
ξ Bid-AskHalf-Spread
ix
H(t,x,s,z,q) Continuous-timevaluefunction
Hτ,δ±(t,x,s,z,q) Continuous-timeperformancecriterionofthecontrolsτ and δ
±
I(t) Imbalanceratio,raw
K Otheragents’buy(K+)andsell(K )marketorders
± −
L A counting process of how many of our buy (L+) and sell (L ) limit orders are
± −
filled
M Executionofourbuy(M+)andsell(M )marketorders
± −
Qτ,δ Ourinventoryattime t havingusedcontrolsτ and δ
t
S(t) Midprice
V (x,s,z,q) Discrete-timevaluefunction
k
Vδ±(x,s,z,q) Discrete-timeperformancecriterionofthecontrolsτ and δ
k ±
τ,δ
W Ourwealth(cash,valueofassets,andliquidationpenalty)attimethavingused
t
controlsτ and δ
Xτ,δ Ourcashattime t havingusedcontrolsτ and δ
t
Z(t) Continuous-timeMarkovchainvariable
AAPL StocktickerforAppleInc.
FARO StocktickerforFAROTechnologiesInc.
INTC StocktickerforIntelCorporation
MMM Stocktickerfor3MCompany
NTAP StocktickerforNetApp,Inc.
ORCL StocktickerforOracleCorporation
CTMC Continuous-timeMarkovchain
DPE Dynamicprogrammingequation
ITCH MarketdatafeedprovidedbyNASDAQ
x