1 Statistical Arbitrage and Algorithmic Trading - e-Spacio - Uned

302 Pages·2010·2.82 MB·English

Statistical Arbitrage and Algorithmic Trading : Overview and Applications

Miquel Noguer Alonso - Licenciado y Master en Administración y Dirección de Empresas –
Universitat Ramon Llull - ESADE

Facultad de Ciencias Económicas y Empresariales

Departamento de Economía Aplicada Cuantitativa II

UNED

2010

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Facultad de Ciencias Económicas y Empresariales

Departamento de Economía Aplicada Cuantitativa II

Statistical Arbitrage and Algorithmic Trading : Overview and Applications

Miquel Noguer Alonso - Licenciado y Master en Administración y Dirección de Empresas –
Universitat Ramon Llull – ESADE - UBS AG

Director : Andreu Pacheco Pages IFAE/CERN
Codirector : Manuel Jose Sanchez Sanchez UNED

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This PhD Thesis is a tribute to my family.

To my mother and brother, whose confidence, support in me has
been a constant in my life, to my wife Mima, whose love, patience,
and understanding have been the foundations of this work.

To the memory of my father, wherever he is, I love him so much.

To my son Jordi who came early this year to inspire
my work, bringing to our family such happiness
that cannot be imagined.

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Acknowledgements

This PhD thesis is the result of more than a decade of work with my talented colleagues in UBS,
Andbanc as many other people in the financial industry.

Thanks to my Thesis Director, Andreu Pacheco Pages, for his knowledge, patience, help with
ideas and devoting his precious time to my work!

Thanks to Alberto Alvarez and Jose Manuel Sanchez for their guidance, dedication and support
through all my PhD time at their university.

Thanks to Christian Mazza, Jean Pierre Gabriel and Ales Janka for their collaboration in the
research in Fribourg University. I am greatly indebted to Yi-Chen-Zhang and Damien Challet for
bringing me there.

Thanks to Lorenzo Moneta from CERN for his collaboration in the machine learning chapter of
this PhD thesis.

Thanks to Jose Miguel Dominguez for his collaboration in the factor models section of this work
and the interesting discussions we have on quantitative investing.

Thanks to Stephen Wolfram and Jason Cawley from Wolfram Research for their support and the
research we did together in the NKS summer school back in 2008.

My discussions and research with Martin Schaden have extremely useful to explore new ideas
like quantum finance.

My work wouldn’t have been the same without my conversations with Jean-Phillipe Bouchaud,
Conrad Perez Vicente and other econophysicists. Whose contribution to economics science is
changing the foundations of Finance. I am highly indebted to the Fisica i Finances research group
from the Physics department at Universitat de Barcelona for their inspiring papers and books.

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1. Introduction................................................................................................................14
1.1. Scope....................................................................................................................................14
1.2. Definitions...........................................................................................................................16
1.2.1 Forecasting.................................................................................................................................16
1.2.2 The ultimate goal: achieving high Sharpe ratios...................................................................18
2. Algorithmic Trading strategies: track record, categories and disciplines18
2.1. The industry of systematic traders: Barclay Systematic Traders Index............18
2.2. Trading strategies categories : Mean reversion, momentum / Regime
switching / Factor models..........................................................................................................20
2.2.1 Mean-reverting versus momentum strategies.......................................................................20
2.2.2 Regime switching......................................................................................................................25
2.2.3 Stationarity and cointegration..................................................................................................27
2.2.4 Factor models............................................................................................................................28
2.2.5 High-frequency trading strategies...........................................................................................30
2.2.6 What is your exit strategy?.......................................................................................................32
2.2.7 Event trading..............................................................................................................................33
2.2.8 Volatility arbitrage......................................................................................................................34
2.3. Statistics and Finance: Econophysics and behavioral finance............................43
2.3.1 Statistics, econophysics and behavioural finance................................................................43
2.3.2 Agent-Based Modelling of Financial Markets........................................................................46
2.3.3 Game theory..............................................................................................................................47
2.3.4 Microstructure – Are the dynamics of financial markets endogenous or exogenous ?..48
2.3.5 Endogenous-Exogenous Market model.................................................................................50
2.3.6 Statistics and Finance...............................................................................................................50
3. P: discrete-time processes....................................................................................52
3.1. Random walk......................................................................................................................53
3.1.1 Continuous invariants...............................................................................................................54
3.1.2 Discrete invariants.....................................................................................................................55
3.1.3 Generalized representations....................................................................................................56
3.1.4 Heavy tails..................................................................................................................................57
3.2. ARMA processes...............................................................................................................59
3.3. Long memory.....................................................................................................................61
3.4. Volatility clustering...........................................................................................................63
4. Part II Q: continuous-time processes..................................................................64
4.1. Levy processes..................................................................................................................65
4.1.1 Diffusion......................................................................................................................................65
4.1.2 Jumps..........................................................................................................................................66
4.1.3 Generalized representations....................................................................................................68
4.1.4 Notable examples......................................................................................................................69
4.2. Autocorrelated processes..............................................................................................71
4.3. Long memory.....................................................................................................................72
4.4. Volatility clustering...........................................................................................................74
4.4.1 Stochastic volatility....................................................................................................................74
4.4.2 Subordination.............................................................................................................................75
4.5. Markov Switching Models...............................................................................................77
4.6. Fractals and multifractals in finance............................................................................78
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4.6.1 Basic definitions.........................................................................................................................78
4.6.2 Multifractals................................................................................................................................81
4.6.3 Multifractal model of asset returns..........................................................................................84
4.6.4 Markov switching multifractal...................................................................................................85
4.7. Quantum Finance..............................................................................................................86
4.8. State Space representation............................................................................................86
4.9. Bayesian statistics............................................................................................................89
4.9.1 The likelihood function..............................................................................................................91
4.9.2 The Poisson Distribution Likelihood Function.......................................................................91
4.9.3 Bayes theorem...........................................................................................................................92
5. Statistical Arbitrage, Cointegration, and Multivariate Ornstein-Uhlenbeck
93
5.1. The multivariate Ornstein-Uhlenbeck process..........................................................94
5.2. The geometry of the Ornstein-Uhlenbeck dynamics...............................................97
5.3. Cointegration....................................................................................................................101
5.4. Vector autoregressive model.......................................................................................103
5.4.1 Definition...................................................................................................................................103
5.5. Principal Components Analysis Statistical Arbitrage...........................................104
6. Statistical Model Estimation................................................................................107
6.1. Statistical Estimation and Testing..............................................................................107
6.2. Estimation Methods........................................................................................................108
6.2.1 The Least-Squares Estimation Method................................................................................109
6.2.2 The Maximum-Likelihood Estimation Method.....................................................................109
6.2.3 Bayesian Estimation Methods...............................................................................................109
6.2.4 Robust Estimation...................................................................................................................110
6.3. Estimation of Matrices...................................................................................................111
6.4. Estimation of Regression Models...............................................................................111
6.4.1 Linear regression is the workhorse of equity modelling....................................................111
6.5. Estimation of Vector Autoregressive Models..........................................................113
6.6. Estimation of Linear Hidden-Variable Models.........................................................114
6.7. Estimation of Nonlinear Hidden-Variable Models...................................................115
7. Nonlinear Dynamical Systems............................................................................115
7.1. Motivation..........................................................................................................................115
7.2. Discrete systems: the logistic map............................................................................117
7.3. Continuous systems......................................................................................................119
7.4. Lorenz model....................................................................................................................120
7.5. Pathways to chaos..........................................................................................................122
7.6. Measuring chaos.............................................................................................................123
8. Technical analysis..................................................................................................124
8.1. History................................................................................................................................124
8.2. General description........................................................................................................125
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8.3. Characteristics.................................................................................................................126
8.4. Prices move in trends ?.................................................................................................127
8.5. Rule-based trading..........................................................................................................127
8.6. Indicators...........................................................................................................................128
9. Statistical Arbitrage Applications – Momentum and value analysis........129
9.1. The historical performance of value and price momentum.................................130
9.1.1 The big picture: Value and price momentum in the US.....................................................130
9.1.2 A closer look at historical performance Portfolio returns...................................................132
9.1.3 A formal test: Has alpha disappeared?................................................................................136
9.2. Systematic strategies and market risk: What has changed?..............................138
9.2.1 Correlation with market returns.............................................................................................138
9.2.2 Market neutrality beyond correlation....................................................................................141
9.2.3 The quest for market neutrality..............................................................................................144
9.3. Momentum in a multiasset class context.................................................................152
9.4. Moving average model SP500......................................................................................154
9.5. Moving Average + RSI bund 1 minute.......................................................................155
9.6. Statistical analysis..........................................................................................................156
9.6.1 SP500-Descriptive statistics + ACF + PACF + GARCH modelling..................................156
9.6.2 Single stock-Microsoft: Variance ratio test..........................................................................158
9.6.3 Time series analysis SP500..................................................................................................159
10. Machine Learning review......................................................................................160
10.1. Supervised, unsupervised and statistical learning............................................161
10.2. Artificial Neural Networks.........................................................................................161
10.2.1 Background..............................................................................................................................163
10.2.2 Models.......................................................................................................................................163
10.2.3 Choosing a cost function........................................................................................................165
10.2.4 Learning paradigms................................................................................................................166
10.2.5 Learning algorithms.................................................................................................................167
10.2.6 Applications..............................................................................................................................168
10.2.7 Types of neural networks.......................................................................................................168
10.2.8 Theoretical properties.............................................................................................................174
10.2.9 Support vector machines....................................................................................................175
10.3. Classification and regression trees........................................................................182
10.4. Genetic Algorithms.....................................................................................................182
10.4.1 Initialization...............................................................................................................................184
10.4.2 Selection...................................................................................................................................184
10.4.3 Reproduction............................................................................................................................184
10.4.4 Termination..............................................................................................................................185
10.4.5 Solutions...................................................................................................................................185
10.4.6 Variants.....................................................................................................................................187
10.4.7 Problem domains.....................................................................................................................188
11. Statistical analysis of genetic algorithms in discovering technical
analysis trading strategies.............................................................................................188
11.1. Introduction...................................................................................................................189
11.2. Trading with gas..........................................................................................................190
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11.3. Testing different models............................................................................................192
11.3.1 Linear Time Series..................................................................................................................193
11.3.2 Bilinear Process.......................................................................................................................194
11.3.3 ARCH Processes.....................................................................................................................194
11.3.4 GARCH Processes.................................................................................................................195
11.3.5 Threshold Processes..............................................................................................................196
11.3.6 Chaotic Processes..................................................................................................................197
11.4. Performance criteria and statistical tests.............................................................198
11.4.1 Winning Probability.................................................................................................................199
11.4.2 Sharpe Ratio............................................................................................................................199
11.4.3 Luck Coefficient.......................................................................................................................201
11.5. Monte carlo simulation..............................................................................................202
11.6. Test results...................................................................................................................203
11.6.1 ARMA Processes....................................................................................................................203
11.6.2 Bilinear Processes...................................................................................................................205
11.6.3 ARCH and GARCH Processes.............................................................................................206
11.6.4 Threshold Processes..............................................................................................................208
11.6.5 Chaotic Processes..................................................................................................................209
11.6.6 Summary..................................................................................................................................210
11.7. Empirical analysis.......................................................................................................210
11.7.1 Data Description and Analysis..............................................................................................210
11.7.2 Experimental Design...............................................................................................................215
11.7.3 Results of the Experiments....................................................................................................216
11.8. Concluding remarks...................................................................................................219
12. Using GA's + ANN's and SVM for financial time series forecasting – 2
applications.........................................................................................................................220
12.1. GAs for financial time series forecasting.............................................................221
12.1.1 ANNs for financial time series forecasting...........................................................................221
12.1.2 Multiple experts for financial time series forecasting.........................................................222
12.2. A hybrid approach for dealing with stock market forecasting........................222
12.2.1 Context-based identification of multistationary models.....................................................223
12.2.2 The guarded experts framework...........................................................................................223
12.2.3 Neural XCS..............................................................................................................................225
12.2.4 Handling a population of NXCS experts..............................................................................225
12.3. Generating and maintaining NXCS experts..........................................................226
12.3.1 NXCS mechanisms for experts selection and outputs blending......................................227
12.4. Customizing NXCS experts for stock market forecasting................................227
12.4.1 Embodying technical-analysis domain knowledge into NXCS guards............................228
12.4.2 Devising a feedforward ANN for stock market forecasting...............................................230
12.5. Experimental results...................................................................................................233
12.6. Forecasting stock market direction with Support Vector Machines.............234
12.6.1 Experiment design...................................................................................................................234
12.6.2 Model inputs selection............................................................................................................235
12.6.3 Comparisons with other forecasting methods.....................................................................235
12.6.4 Experiment results...................................................................................................................237
13. Other non-linear Techniques...............................................................................237
13.1. Takens' theorem - delay embedding theorem – State space...........................237
13.1.1 FX application - Takens' theorem - delay embedding theorem........................................239
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13.2. Macroeconomics Forecasting – From VARMA to Reduced form VAR.........239
13.3. Markov switching SP500 / VIX 3 states..................................................................242
14. Asymmetric Algorithmic trading strategies....................................................245
14.1. Commodity Algo..........................................................................................................245
14.1.1 Commodity market characteristics........................................................................................245
15. Integrated Algorithmic Trading Strategy Theory - Putting it all together247
15.1. Integrated Algorithmic Trading Strategy Theory – Definitions and techniques
247
15.2. Modelling Framework.................................................................................................247
15.3. Techniques....................................................................................................................247
15.4. Perfomance measurement........................................................................................248
15.5. GOLD IATST..................................................................................................................248
15.6. Multi Strategy IATST...................................................................................................250
16. Conclusions.............................................................................................................251
17. Appendix...................................................................................................................253
17.1. Forecast error...............................................................................................................253
17.2. Analytical proof of the law of reversion................................................................254
17.3. The Ornstein-Uhlenbeck process...........................................................................255
17.4. Relation between CIR and OU processes.............................................................256
17.5. The Levy-Khintchine representation of a Levy process...................................257
17.6. Representation of compound Poisson process.................................................258
17.7. Deterministic linear dynamical system.................................................................259
17.8. OU (auto)covariance: general expression............................................................260
17.9. OU (auto)covariance: explicit solution..................................................................261
17.10. The sup-Wald test....................................................................................................264
17.11. Power of the stability test......................................................................................265
17.12. Hedge fund returns and market neutrality........................................................266
17.13. Akaike information criterion.................................................................................267
17.14. Genetic algorithms and technical analysis......................................................269
17.14.1 Coding Trading Strategies.................................................................................................269
17.14.2 Ordinary Genetic Algorithms.............................................................................................271
17.15. Notes on Advanced Maths for Quantitative Investment...............................272
17.15.1 Stochastic processes......................................................................................................272
17.15.2 Stochastic differential equation....................................................................................274
17.15.3 Stochastic integral............................................................................................................275
17.15.4 Martingales.........................................................................................................................275
17.15.5 Black-Scholes theory – Riskless portfolio.................................................................276
17.15.6 Optimization in finance....................................................................................................277
17.15.7 Black-Litterman.................................................................................................................279
17.15.8 Statistical Physics and Finance....................................................................................280
17.15.9 Probability distributions..................................................................................................281
17.15.10 Copulas................................................................................................................................284
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17.15.11 Important distributions....................................................................................................285
18. Abbreviations and Acronyms..............................................................................289
19. Programming code.................................................................................................291
20. References................................................................................................................291

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