Advanced Algorithmic Trading

Contents I

Intr Introdu oduct ction ion

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1 Introductio Introduction n To Advanced Advanced Algorithmic Algorithmic Trading Trading . . . . . . . . . . . . . . . . . 1.1 The Hunt Hunt for Alpha Alpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.2 Why Time Series Series Analysis, Analysis, Bayesian Bayesian Statist Statistics ics and Machine Machine Learning? Learning? . . . . . .

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1.2.1

Bayesian Bay esian Statistic Statisticss   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.2.2 1.2 .2

Time Time Series Series Analys Analysis is   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.2.3 1.2 .3

Machin Mac hinee Lea Learni rning ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.3 Ho How w Is The Book Lai Laid d Out? Out? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.4 Required Required Tech Technical nical Bac Backgrou kground nd   . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.4.1 1.4 .1

Mathem Mathemati atics cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.4.2 1.4 .2

Progra Programmi mming ng   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.5 How Does This Book Differ From From "Successful "Successful Algorith Algorithmic mic Tradin Trading"? g"? . . . . . . .

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1.6 Softwar Softwaree Installati Installation on   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.6.1 1.6 .1

Instal Installin ling g Python Python   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.6. 1.6.2 2

Inst Instal alli ling ng R   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.7 QSTrader QSTrader Backt Backtesti esting ng Simulatio Simulation n Software Software . . . . . . . . . . . . . . . . . . . . .

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1.7.1 1.7 .1 1.8 1.8

II

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Altern Alternati ative vess   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Wher Wheree to to Get Get Help Help   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bay Bayesian esian Stat Statis isti tics cs

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2 Introductio Introduction n to Bayes Bayesian ian Statistic Statisticss . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 What is Bay Bayesian esian Statistic Statistics? s? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1

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Frequentis requentistt vs Bay Bayesian esian Examples Examples . . . . . . . . . . . . . . . . . . . . . .

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2.2 Applying Applying Bayes Bayes’’ Rule for for Bayes Bayesian ian Inference Inference . . . . . . . . . . . . . . . . . . . . .

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2.3 Coin-Flipp Coin-Flipping ing Example Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Bayesian Bayesian Inferenc Inference e of a Binomial Binomial Proportion Proportion . . . . . . . . . . . . . . . . . . . 23 3.1 The Bay Bayesian esian Approach Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2 Assumption Assumptionss of of the the Approach Approach   . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Recalling Recalling Bay Bayes’ es’ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4 The Likelihood Likelihood Function unction   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4.1

Bernoulli Bernoulli Distributio Distribution n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4.2

Bernoulli Bernoulli Likelihood Likelihood Function unction   . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4.3 3.4 .3

Multip Multiple le Flip Flipss of the Coin Coin   . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.5 Quantifyi Quantifying ng our Prior Beliefs Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 3.5.1 3.5.1 3.5.2

Beta Beta Distri Distribut bution ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Is Is A Beta Prior Prior Conjug Conjugate ate to the the Bernoull Bernoullii Likelihood? Likelihood? . . . . . . .

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3.5.3

Multiple Multiple Ways Ways to Specify Specify a Beta Prior Prior   . . . . . . . . . . . . . . . . . . . .

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3.6 Using Bay Bayes’ es’ Rule Rule to Calculat Calculatee a Poster Posterior ior . . . . . . . . . . . . . . . . . . . . . .

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4 Marko Markov v Chain Chain Monte Monte Carlo Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 Bay Bayesian esian Inference Inference Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Why Markov Markov Chain Monte Monte Carlo? Carlo? . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2.1

Markov Markov Chain Monte Monte Carlo Carlo Algorithms Algorithms . . . . . . . . . . . . . . . . . . .

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4.3 The Metropo Metropolis lis Algo Algorit rithm hm   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.4 Introd Introduci ucing ng PyMC3 PyMC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.5 Inferring Inferring a Binomial Binomial Proportion Proportion with with Markov Markov Chain Monte Monte Carlo  Carlo   . . . . . . . . . 4.5.1 Inferring Inferring a Binonia Binoniall Proportion Proportion with with Conjugat Conjugatee Priors Priors Recap Recap   . . . . . . .

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4.5.2 Inferring Inferring a Binomial Binomial Proportion Proportion with PyMC3 PyMC3   . . . . . . . . . . . . . . . . 4.6 Biblio Bibliogra graphi phicc Note Note   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Bayesian Bayesian Linear Regression Regression   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Frequentis requentistt Linear Linear Regres Regression sion   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Bay Bayesian esian Linear Linear Regre Regression ssion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3 Bay Bayesian esian Linear Linear Regressio Regression n with PyMC3 PyMC3 . . . . . . . . . . . . . . . . . . . . . . .

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5.3.1

What are Generalise Generalised d Linear Linear Models? Models? . . . . . . . . . . . . . . . . . . . .

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5.3.2

Simulati Simulating ng Data Data and Fitti Fitting ng the Model with with PyMC3 PyMC3 . . . . . . . . . . . .

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5.4 Biblio Bibliogra graphi phicc Note Note   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5 5.5

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Full ull Code Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Bayesian Bayesian Stochastic Stochastic Volatil Volatility ity Model Model . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1 Stochasti Stochasticc Volatilit olatility y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.2 Bay Bayesian esian Stochastic Stochastic Volatilit olatility y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 PyMC3 PyMC3 Implement Implementation ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.4 6.4

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6.3.1 6.3 .1

Obtain Obtaining ing the the Pric Pricee Histor History y . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.3.2 6.3.2 6.3.3 6.3 .3

Model Model Specifi Specificat cation ion in PyMC PyMC33 . . . . . . . . . . . . . . . . . . . . . . . . Fittin Fitting g the the Model Model with with NUTS NUTS . . . . . . . . . . . . . . . . . . . . . . . . .

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Full ull Code Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Time Time Seri Series es Analy Analysi siss

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7 Introd Introduct uction ion to Time Time Series Series Analysis Analysis   . . . . . . . . . . . . . . . . . . . . . . . . 73 7.1 What What is Time Time Serie Seriess Analys Analysis? is? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 How Can Can We Apply Apply Time Series Series Analysis Analysis in Quantita Quantitativ tivee Finance? Finance? . . . . . . . .

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7.3 Time Series Series Analys Analysis is Soft Softwar waree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Time Series Series Analys Analysis is Roadmap Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.5 How Does Does This Relat Relatee to Other Other Statisti Statistical cal Tools? Tools? . . . . . . . . . . . . . . . . . .

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8 Serial Serial Correl Correlati ation on   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1 Expectation Expectation,, Varian Variance ce and Cov Covarian ariance ce . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1.1

Example: Example: Sample Sample Covarian Covariance ce in R . . . . . . . . . . . . . . . . . . . . . .

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3 8.2 Correl Correlati ation on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 8.2 .1

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Exampl Example: e: Sample Sample Corre Correlat lation ion in in R   . . . . . . . . . . . . . . . . . . . . . .

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8.3 Stationari Stationarity ty in Time Series Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.4 Serial Serial Correl Correlati ation on   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.5 The Correl Correlogr ogram am   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.6 8.6

8.5.1 8.5 .1

Exampl Examplee 1 - Fixed Fixed Line Linear ar Tren Trend d . . . . . . . . . . . . . . . . . . . . . . .

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8.5.2 8.5 .2

Exampl Examplee 2 - Repea Repeated ted Sequen Sequence ce . . . . . . . . . . . . . . . . . . . . . . .

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Next Next Step Stepss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Random Random Walk Walkss and White White Noise Models Models . . . . . . . . . . . . . . . . . . . . . . 87 9.1 Time Time Series Series Model Modellin ling g Process Process . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.2 Bac Backwa kward rd Shift Shift and Differe Difference nce Operator Operatorss . . . . . . . . . . . . . . . . . . . . . .

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9.3 9.3

Whit Whitee No Nois isee   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.3.1

Second-Ord Second-Order er Properties Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.3.2 9.3 .2

Correl Correlogr ogram am . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.4 Random Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.4.1

Second-Ord Second-Order er Properties Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.4.2 9.4 .2

Correl Correlogr ogram am . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9.4.3

Fitting Fitting Random Random Walk Walk Models Models to Financial Financial Data Data . . . . . . . . . . . . . .

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10 Autoregressive Moving Average Models . . . . . . . . . . . . . . . . . . . . . . 97 10.1 How Will We We Proceed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.2 Strictly Stationary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.3 Aka Akaike ike Information Criterion Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.4 Autoregre Autoregressiv ssivee (AR) Models of order p   . . . . . . . . . . . . . . . . . . . . . . . .

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10.4.1 10.4.1 Rationale Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10.4.2 Stationarity for Autoregressive Autoregressive Processes Processes

. . . . . . . . . . . . . . . . . . 100

10.4.3 10.4.3 Second Second Order Propertie Propertiess   . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.4.4 10.4.4 Simulatio Simulations ns and Correlogr Correlograms ams . . . . . . . . . . . . . . . . . . . . . . . . . 102 10.4.5 10.4.5 Financial Financial Data Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 10.5 Mo Moving ving Aver Average age (MA) Models of order q   . . . . . . . . . . . . . . . . . . . . . . . 111 10.5.1 10.5.1 Rationale Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.5.2 10.5.2 Definition Definition   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.5.3 10.5.3 Second Second Order Propertie Propertiess   . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.5.4 10.5.4 Simulatio Simulations ns and Correlogr Correlograms ams . . . . . . . . . . . . . . . . . . . . . . . . . 113 10.5.5 10.5.5 Financial Financial Data Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.5.6 10.5.6 Next Steps Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.6 Autogressive Autogressive Moving Average Average (ARMA) Models of order p, q   . . . . . . . . . . . . 124 10.6.1 Bayesian Bayesian Information Criterion   . . . . . . . . . . . . . . . . . . . . . . . . 125 10.6.2 10.6.2 Ljung-Bo Ljung-Box x Test Test   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.6.3 10.6.3 Rationale Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.6.4 10.6.4 Definition Definition   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.6.5 10.6.5 Simulatio Simulations ns and Correlogr Correlograms ams . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.6.6 10.6.6 Choosing Choosing the Best ARMA(p,q) ARMA(p,q) Model   . . . . . . . . . . . . . . . . . . . . 130 10.6.7 10.6.7 Financial Financial Data Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4 10.7 Next Steps Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 11 Autoregressive Integrated Moving Average and Conditional Heteroskedastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 135 11.1 Quick Quick Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 11.2 Autoregressive Autoregressive Integrated Moving Average Average (ARIMA) Models of order p, d, q . . 136 11.2.1 11.2.1 Rationale Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.2.2 11.2.2 Definitions Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.2.3 11.2.3 Simulati Simulation, on, Correlogram Correlogram and Model Fitting   . . . . . . . . . . . . . . . . . 137 11.2.4 11.2.4 Financial Financial Data and Predicti Prediction on . . . . . . . . . . . . . . . . . . . . . . . . 139 11.2.5 11.2.5 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.3 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.4 Conditional Heteroskedasticity Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.5 Autoregressive Autoregressive Conditional Heteroskedastic Heteroskedastic Models   . . . . . . . . . . . . . . . . . 145 11.5.1 11.5.1 ARCH ARCH Definition Definition   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 11.5.2 11.5.2 Why Does This Model Volati Volatilit lity? y? . . . . . . . . . . . . . . . . . . . . . . . 146 11.5.3 11.5.3 When Is It Appropriat Appropriatee To Apply ARCH(1 ARCH(1)? )?   . . . . . . . . . . . . . . . . 146 11.5.4 11.5.4 ARCH(p) ARCH(p) Models Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11.6 Generalised Autoregressive Autoregressive Conditional Heteroskedastic Heteroskedastic Models   . . . . . . . . . . 147 11.6.1 11.6.1 GARCH GARCH Definition Definition   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.6.2 11.6.2 Simulati Simulations, ons, Correlograms Correlograms and Model Fittings Fittings . . . . . . . . . . . . . . . 147 11.6.3 11.6.3 Financial Financial Data Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11.7 Next Steps Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 12 Cointegrated Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 12.1 Mean Reversion Trading Trading Strategies   . . . . . . . . . . . . . . . . . . . . . . . . . . 155 12.2 Cointegration Cointegration   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 12.3 Unit Root Tests Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 12.3.1 Augmented Dickey-F Dickey-Fuller uller Test Test . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.2 Phillips-Perron Phillips-Perron Test Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.3 Phillips-Ouliaris Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.4 12.3.4 Difficultie Difficultiess with Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . 157 12.4 Simulate Simulated d Cointegra Cointegrated ted Time Series with R   . . . . . . . . . . . . . . . . . . . . . 157 12.5 Cointegrated Cointegrated Augmented Dickey Fuller Fuller Test   . . . . . . . . . . . . . . . . . . . . . 162 12.6 CADF on Simulate Simulated d Data   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 12.7 CADF on Financial Financial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 12.7.1 12.7.1 EWA EWA and EWC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 12.7.2 12.7.2 RDS-A and RDS-B RDS-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 12.8 Full Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 12.9 Johansen Test Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12.9.1 12.9.1 Johansen Johansen Test Test on Simulated Simulated Data . . . . . . . . . . . . . . . . . . . . . . 175 12.9.2 12.9.2 Johansen Johansen Test Test on Financial Financial Data   . . . . . . . . . . . . . . . . . . . . . . . 178 12.9.3 12.9.3 Full Code Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13 State Space Models and the Kalman Filter . . . . . . . . . . . . . . . . . . . . 18 185 13.1 Linear Linear State-Space State-Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186