Approximate methods for dynamic portfolio allocation under transaction costs

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Nabeel Butt, The University of Western Ontario
Supervisor: Dr Matt Davison, The University of Western Ontario
Co-Supervisor: Dr Greg Reid, The University of Western Ontario
A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree
in Applied Mathematics
© Nabeel Butt 2012
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APPROXIMATE METHODS FOR DYNAMIC PORTFOLIO ALLOCATION
UNDER TRANSACTION COSTS
(Spine title: Dynamic Portfolio Allocation under transaction costs )
(Thesis format: Monograph)
by
Nabeel Butt
Graduate Program in Applied Mathematics
A thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
The School of Graduate and Postdoctoral Studies
The University of Western Ontario
London, Ontario, Canada
(cid:13)c Nabeel Butt 2012
THE UNIVERSITY OF WESTERN ONTARIO
School of Graduate and Postdoctoral Studies
CERTIFICATE OF EXAMINATION
Supervisor:
Examiners:
.....................
Dr. MattDavison .....................
Dr. AdamMetzler
JointSupervisor:
.....................
..................... Dr. IanMcleod
Dr. GregReid
SupervisoryCommittee: .....................
Dr. MattThompson
.....................
Dr. AdamMetzler .....................
Dr. MarkReesor
Thethesisby
Nabeel Butt
entitled:
APPROXIMATEMETHODSFORDYNAMICPORTFOLIOALLOCATIONUNDER
TRANSACTIONCOSTS
isacceptedinpartialfulfillmentofthe
requirementsforthedegreeof
DoctorofPhilosophy
............... ..............................
Date ChairoftheThesisExaminationBoard
ii
Acknowledgments
I would like to start by expressing my deepest gratitude towards God for all His help and
support all along. Next I thank my beloved parents for their efforts in helping me get the
best education possible. I would also like to thank all my thesis examiners for their insightful
commentsleadingtoamuchimprovedversionofthethesis.
MyPhDyearsatUWOweresomeofthebestyearsofmylife. MyprimaryPhdsupervisor
Dr Matt Davison was an ever present support and a great source of guidance. Matt was very
helpful in all our meetings and gave me complete freedom to pursue novel ideas. It was Matt
whoinitiallydirectedmetowardsaMITACS2008industrialproblemsolvingworkshop. Many
of the ideas in the thesis were inspired by different aspects of the hedge fund problem the
workshop involved. My co-supervisor Dr Greg Reid was a great source of advice and sparked
my interest in Homotopy methods in applied mathematics. Greg also helped me develop an
interestinexperimentalmathematics.
Last but not the least I would like to thank my imaginary friend Mathematica for all its
support! :-)
iii
Abstract
The thesisprovides simpleand intuitivelattice based algorithmsfor solvingdynamic port-
folio allocation problems under transaction costs. The early part of the thesis concentrates
upon developing a toolbox based on discrete probability approximations. The discrete ap-
proximations are shown to provide a reasonable approximation for most popular transaction
cost models in the academic literature. The tool, once forged, is implemented in the powerful
Mathematica based parallel computing environment. In the second part of the thesis we pro-
vide applications of our framework to real world problems. We show re-balancing portfolios
is more valuable in an investment environment where the growth and volatility of risky assets
is non-constant over the time horizon. We also provide a framework for modeling random
transaction costs and compute the loss of expected utility of an investor faced with random
transaction costs. Approximate methods are provided to solve portfolio constraints such as
portfolio insurance and draw-down. Finally, we also highlight a lattice based framework for
pairstrading.
Keywords: PortfolioAllocation,Transactioncosts
iv
Contents
CertificateofExamination ii
Acknowlegements iii
Abstract iv
ListofFigures x
ListofTables xxix
ListofAppendices xxxii
1 Aquantitativeanalysisofcontinuoustimeportfoliostrategies 1
2 Literaturereviewandnotations 4
2.1 Anoverviewofdynamicportfoliotheory . . . . . . . . . . . . . . . . . . . . . 4
2.2 Briefreviewoftransactioncostliterature . . . . . . . . . . . . . . . . . . . . . 8
2.3 Towardsdiscretetimemodeling . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Notationusedinthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Latticeframeworkofthethesisusingnotationsabove . . . . . . . . . . . . . . 14
2.5.1 Bermudanputoptionpricingintheframework . . . . . . . . . . . . . 14
2.5.2 Growthratemaximizationportfolioproblemintheframework . . . . . 15
3 Introduction to discrete probability approximation and sketch of modeling ap-
proach 19
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Analogybetweendiscretetimeandcontinuoustimeportfoliotheory . . . . . . 20
3.3 Bellmanprinciplefordiscretetimefinitehorizonproblems . . . . . . . . . . . 21
3.4 Utilityofterminalwealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Anillustrativeexample: deformationsolutionforadynamicinvestor . . . . . 22
3.6 Transferofwealth,transactioncoststructureandno-transactionregion . . . . 24
3.6.1 Transactioncostmodels . . . . . . . . . . . . . . . . . . . . . . . . . 25
Transfer of wealth between risky assets and trading cost proportional
totheamounttransferred . . . . . . . . . . . . . . . . . . . 25
Risk-free asset banker for buying/selling risky assets and trading cost
proportionaltotheamounttraded . . . . . . . . . . . . . . . 28
v
Buying/sellingriskyassetsandtradingcostproportionaltotheamount
ofwealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Asynopsisofapproximatelatticemethods . . . . . . . . . . . . . . . . . . . 31
3.8 Ondiscreteprobabilityapproximations . . . . . . . . . . . . . . . . . . . . . 33
3.8.1 Theexampleofasimplemodel . . . . . . . . . . . . . . . . . . . . . . 33
3.8.2 Binomialdiscreteprobabilityapproximation . . . . . . . . . . . . . . 36
3.8.3 Overviewofbasicdiscreteprobabilityapproximationconstructionpro-
cedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Treein1-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Treein2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Treein3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Generalframeworkforadiscreteprobabilityapproximationinℵ-D . . 40
3.9 Onthephilosophyofprobabilitydeformationcontinuation . . . . . . . . . . . 42
3.10 Towardsrobustandefficientlatticealgorithms . . . . . . . . . . . . . . . . . . 46
3.11 Analysisofcontinuoustimedynamictradingstrategies . . . . . . . . . . . . . 51
3.11.1 Riskanalysisofstrategies . . . . . . . . . . . . . . . . . . . . . . . . 51
3.11.2 Onthevalueofre-balancing . . . . . . . . . . . . . . . . . . . . . . . 53
4 OverviewofMathematicaImplementations 59
4.1 Treeconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Moment/Cross-momentmatching . . . . . . . . . . . . . . . . . . . . 60
4.1.2 Treesviamoregeneralprobabilitydeformation . . . . . . . . . . . . . 60
4.2 Dynamicprogrammingcomputations . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 ParallelcomputinginMathematica . . . . . . . . . . . . . . . . . . . 60
4.2.2 Recursionviadynamicprogramming . . . . . . . . . . . . . . . . . . 61
4.2.3 Analysisoftheoptimalcontrollaw . . . . . . . . . . . . . . . . . . . . 62
5 Probabilitydeformationcontinuationschemes 64
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Probabilitydeformationschemes . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Deformationschemesforaportfoliomodelin1-D . . . . . . . . . . . . . . . 66
5.3.1 Modeldescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.2 Numericalanalysisofprobabilitydeformationschemes . . . . . . . . . 67
5.4 Deformationschemesforaportfoliomodelin2-D . . . . . . . . . . . . . . . 70
5.4.1 Modeldescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4.2 Numericalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Someremarksonmomentdivisiondeformation . . . . . . . . . . . . . . . . . 73
5.6 Applicabilitytoawideclassofstochasticprocessesforriskygrowth . . . . . . 76
5.7 Towardsadistribution-freeapproach . . . . . . . . . . . . . . . . . . . . . . . 76
5.8 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 Momentbaseddiscreteprobabilityapproximationoftransactioncostmodels 78
6.1 Treeapproximationsforfixedtransactioncostmodel . . . . . . . . . . . . . . 78
6.1.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.1.2 Approximationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 86
vi
6.1.3 Modeloutputandvalidation . . . . . . . . . . . . . . . . . . . . . . . 88
N = 1riskyassets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Resultsfor N ≥ 2riskyassets . . . . . . . . . . . . . . . . . . . . . . 90
6.1.4 Model risk: optimal policies when risky portfolio growth follows an
arbitrarydistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1.5 Analyzingfinitehorizonboundaries . . . . . . . . . . . . . . . . . . . 94
6.1.6 Computationalcomplexityanderroranalysis . . . . . . . . . . . . . . 95
6.2 Approximate dynamic mean-variance portfolio optimization under transaction
costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.2 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.3 Numericalmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.4 No-transactionregionswithtimeandefficiencyfrontiers . . . . . . . . 110
6.2.5 Sharperatiotimeseries . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.6 Comparisonofsolutionwithmodelusingtheexactdistribution . . . . . 111
6.2.7 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Treeapproximationofproportionaltransactioncostmodel . . . . . . . . . . . 113
6.4 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7 Valueofre-balancingportfoliosundertransactioncosts 117
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.2 Investmentmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3 Numericalanalysisofthevalueofre-balancing . . . . . . . . . . . . . . . . . 120
7.3.1 Log-utilitycase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.2 CRRAcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.3 CARAcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3.4 Mean-variancecase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.4 IntuitiveexplanationofresultsusingthestatevariableSDE . . . . . . . . . . . 126
7.5 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8 Latticeapproximationforadynamicstochastictransactioncostmodel 140
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.2 Transactioncostmodelwithstochasticvolatility . . . . . . . . . . . . . . . . . 140
8.3 Investmentmodelundertransactioncosts . . . . . . . . . . . . . . . . . . . . 141
8.4 Formulationasastochastictransactioncostmodel . . . . . . . . . . . . . . . 142
8.5 Latticeformulationofthemodel . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.6 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9 Portfoliooptimizationundertransactioncostsincorporatingrealisticconstraints 151
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.3 Solutionmethodologyforconstraints . . . . . . . . . . . . . . . . . . . . . . . 152
9.4 Numericalresultsforrealisticproblems . . . . . . . . . . . . . . . . . . . . . 154
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
vii
10 Latticemethodsforpairstrading 161
10.1 Dynamicpairstradingbasedupondiscretetimesignals . . . . . . . . . . . . . 161
10.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.1.2 Latticebasedsolutionmethodology . . . . . . . . . . . . . . . . . . . 164
10.2 Latticemethodfordynamicpairstradingundertransactioncosts . . . . . . . . 168
10.2.1 Intuitionbehinddynamicpairstrading . . . . . . . . . . . . . . . . . . 168
10.2.2 Dynamicprogrammingformulationofthetradingmodel . . . . . . . . 170
10.2.3 Evolutionofportfoliostateprocessesunderapairstradingmodel . . . 172
Position1-A− > 0,A− < 0withA− (cid:31) |A− |: . . . . . . . . . . . . 172
1,k 2,k 1,k 2,k
10.2.4 Ageneralizedtradingmodel . . . . . . . . . . . . . . . . . . . . . . . 174
Evolutionofstateparticleswithoutanypre-determinedtradingrule . . 174
10.2.5 Solutionmethodology . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.2.6 Numericalresultsforcontrollaw . . . . . . . . . . . . . . . . . . . . 175
10.2.7 Mean-varianceoptimalityofdynamicpairstrading: . . . . . . . . . . . 176
10.2.8 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11 CONCLUSION 179
11.1 Developingmethodsforimprovedcomputationalspeed . . . . . . . . . . . . . 182
11.2 Extendingourmodelingframeworktoawiderrangeofassetclasses . . . . . . 183
11.3 Analyzetheoreticaleconomicproblems . . . . . . . . . . . . . . . . . . . . . 185
11.4 Incorporatingparameteruncertaintyintoourdecisionmakingmethodology . . 185
11.5 Incorporatingmacro-economicfactorsintoourdecisionmakingmethodology . 187
11.6 Arigorousanalysisofdifferentdynamictradingstrategies . . . . . . . . . . . 187
Bibliography 188
A Mathematicacodeforchapter4 194
A.1 Treeconstructioncode: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.1.1 Treeconstructionin2-D . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.1.2 Treeconstructionin3-D . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.2 Treesviamoregeneralprobabilitydeformationcode: . . . . . . . . . . . . . . 196
A.2.1 SQIDschemein1-D . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
A.2.2 SQIDschemein2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
A.3 Recursionviadynamicprogrammingcode . . . . . . . . . . . . . . . . . . . . 197
A.3.1 Initialrecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.3.2 Subsequentrecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.3.3 Analysisofoptimalcontrolsobtained-sayconstructingtheboundaries
ofno-transactionregionsusingConvexHull[] . . . . . . . . . . . . . . 198
A.3.4 Code that uses creation of small ‘Balls’ to create the no-transaction
regioninchapter6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
A.4 Analysisoftheoptimalcontrollaw . . . . . . . . . . . . . . . . . . . . . . . . 198
A.4.1 Codesnippetshowingcontrolstorage . . . . . . . . . . . . . . . . . . 198
A.4.2 Codesnippetshowinguseofstoredcontrolsforfurtheranalysistogen-
erateefficientfrontierforbenchmarkprobleminsection3.4. . . . . . . 199
viii
B PairTradingModels 200
B.1 AlternatePairtradingmodelsinsection1ofchapter11 . . . . . . . . . . . . . 200
B.1.1 Modelusing Log(Z ) = A −φ −φ B signal . . . . . . . . . . . . . . 200
k k 1 2 k
B.2 DynamicPairTradingModelinsection2ofchapter11 . . . . . . . . . . . . . 201
B.2.1 Position2: A− > 0,A− < 0withA− ≺ |A− |: . . . . . . . . . . . . 201
1,k 2,k 1,k 2,k
B.2.2 Position3: A− < 0,A− > 0with|A− | ≺ A− : . . . . . . . . . . . . 203
1,k 2,k 1,k 2,k
B.2.3 Position4: A− < 0,A− > 0with|A− | (cid:31) A− : . . . . . . . . . . . . 205
1,k 2,k 1,k 2,k
B.3 Generaltradingmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
CurriculumVitae 211
ix