Algorithmic Trading Methods: Applications Using Advanced Statistics, Optimization, and Machine Learning Techniques

Front Cover
Algorithmic Trading Methods
Algorithmic Trading Methods: Applications using Advanced Statistics,
Optimization, and Machine Learning Techniques
Copyright
Contents
Preface
Acknowledgments
1 - Introduction
WHAT IS ELECTRONIC TRADING?
WHAT IS ALGORITHMIC TRADING?
TRADING ALGORITHM CLASSIFICATIONS
TRADING ALGORITHM STYLES
INVESTMENT CYCLE
INVESTMENT OBJECTIVE
INFORMATION CONTENT
INVESTMENT STYLES
INVESTMENT STRATEGIES
RESEARCH DATA
BROKER TRADING DESKS
RESEARCH FUNCTION
SALES FUNCTION
IMPLEMENTATION TYPES
ALGORITHMIC DECISION-MAKING PROCESS
2 - Algorithmic Trading
ADVANTAGES
DISADVANTAGES
GROWTH IN ALGORITHMIC TRADING
MARKET PARTICIPANTS
CLASSIFICATIONS OF ALGORITHMS
TYPES OF ALGORITHMS
ALGORITHMIC TRADING TRENDS
DAY OF WEEK EFFECT
INTRADAY TRADING PROFILES
TRADING VENUE CLASSIFICATION
Displayed Market
Dark Pool
Dark Pool Controversies
TYPES OF ORDERS
REVENUE PRICING MODELS
Order Priority
EXECUTION OPTIONS
ALGORITHMIC TRADING DECISIONS
Macro Level Strategies
Micro Level Decisions
Limit Order Models
Smart Order Routers
ALGORITHMIC ANALYSIS TOOLS
Pre-Trade Analysis
Intraday Analysis
Post-Trade Analysis
HIGH FREQUENCY TRADING
Auto Market Making
Quantitative Trading/Statistical Arbitrage
Rebate/Liquidity Trading
DIRECT MARKET ACCESS
3 - Transaction Costs
WHAT ARE TRANSACTION COSTS?
WHAT IS BEST EXECUTION?
WHAT IS THE GOAL OF IMPLEMENTATION?
UNBUNDLED TRANSACTION COST COMPONENTS
Commission
Fees
Taxes
Rebates
Spreads
Delay Cost
Price Appreciation
Market Impact
Timing Risk
Opportunity Cost
TRANSACTION COST CLASSIFICATION
TRANSACTION COST CATEGORIZATION
TRANSACTION COST ANALYSIS
Measuring/Forecasting
Cost vs. Profit and Loss
IMPLEMENTATION SHORTFALL
Complete Execution
Opportunity Cost (Andre Perold)
Expanded Implementation Shortfall (Wayne Wagner)
IMPLEMENTATION SHORTFALL FORMULATION
Trading Cost/Arrival Cost
EVALUATING PERFORMANCE
Trading Price Performance
Benchmark Price Performance
VWAP Benchmark
Participation-Weighted Price Benchmark
Relative Performance Measure
Pretrade Benchmark
Index-Adjusted Performance Metric
Z-Score Evaluation Metric
Market Cost-Adjusted Z-Score
Adaptation Tactic
COMPARING ALGORITHMS
Nonparametric Tests
Paired Samples
Sign Test
Wilcoxon Signed Rank Test
INDEPENDENT SAMPLES
Mann–Whitney U Test
MEDIAN TEST
DISTRIBUTION ANALYSIS
CHI-SQUARE GOODNESS OF FIT
KOLMOGOROV–SMIRNOV GOODNESS OF FIT
EXPERIMENTAL DESIGN
Proper Statistical Tests
Small Sample Size
Data Ties
Proper Categorization
Balanced Data Sets
FINAL NOTE ON POSTTRADE ANALYSIS
4 - Market Impact Models
INTRODUCTION
DEFINITION
Example 1: Temporary Market Impact
Example 2: Permanent Market Impact
Graphical Illustrations of Market Impact
Illustration #1: Price Trajectory
Illustration #2: Supply–Demand Equilibrium
After Shares Transact, We Face Some Uncertainty—What Happens Next?
Illustration #3: Temporary Impact Decay Function
Example #3: Temporary Decay Formulation
Illustration #4: Various Market Impact Price Trajectories
Developing a Market Impact Model
Essential Properties of a Market Impact Model
The Shape of the Market Impact Function
Example: Convex Shape
Example: Linear Shape
Example: Concave Shape
DERIVATION OF MODELS
Almgren and Chriss Market Impact Model
Random Walk With Price Drift—Discrete Time Periods
Random Walk With Market Impact (No Price Drift)
I-STAR MARKET IMPACT MODEL
MODEL FORMULATION
I-Star: Instantaneous Impact Equation
The Market Impact Equation
Derivation of the Model
Cost Allocation Method
I∗ Formulation
Comparison of Approaches
5 - Probability and Statistics
INTRODUCTION
RANDOM VARIABLES
PROBABILITY DISTRIBUTIONS
Example: Discrete Probability Distribution Function
Example: Continuous Probability Distribution Function
Descriptive Statistics
PROBABILITY DISTRIBUTION FUNCTIONS
CONTINUOUS DISTRIBUTION FUNCTIONS
Normal Distribution
Standard Normal Distribution
Student's t-Distribution
Log-Normal Distribution
Uniform Distribution
Exponential Distribution
Chi-Square Distribution
Logistic Distribution
Triangular Distribution
DISCRETE DISTRIBUTIONS
Binomial Distribution
Poisson Distribution
END NOTES
6 - Linear Regression Models
INTRODUCTION
Linear Regression Requirements
Regression Metrics
LINEAR REGRESSION
True Linear Regression Model
Simple Linear Regression Model
Solving the Simple Linear Regression Model
Step 1: Estimate Model Parameters
Step 2: Evaluate Model Performance Statistics
Standard Error of the Regression Model
R2 Goodness of Fit
Step 3: Test for Statistical Significance of Factors
T-test: Hypothesis Test:
F-test: Hypothesis Test:
Example: Simple Linear Regression
Multiple Linear Regression Model
Solving the Multiple Linear Regression Model
Step 1: Estimate Model Parameters
Step 2: Calculate Model Performance Statistics
Standard Error of the Regression Model
R2 Goodness of Fit
Step 3: Test for Statistical Significance of Factors
T-test: Hypothesis Test:
F-test: Hypothesis Test:
Example: Multiple Linear Regression
MATRIX TECHNIQUES
Estimate Parameters
Compute Standard Errors of b
R2 Statistic
F-Statistic
LOG REGRESSION MODEL
Example: Log-Transformation
Example: Log-Linear Transformation
POLYNOMIAL REGRESSION MODEL
FRACTIONAL REGRESSION MODEL
7 - Probability Models
INTRODUCTION
DEVELOPING A PROBABILITY MODEL
Comparison of Linear Regression Model to Probability Model
Power Function Model
Logit Model
Probit Model
Comparison of Logit and Probit Models
Outcome Data
Model Formulation
Mean
Variance
Grouping Data
Solving Binary Output Models
Step 1: Specify Probability Function
Step 2: Set Up a Likelihood Function Based on Actual Outcome Results for all Observations. For Example, If We Have n Observ ...
SOLVING PROBABILITY OUTPUT MODELS
EXAMPLES
Example 7.1 Power Function
Example 7.2 Logit Model
COMPARISON OF POWER FUNCTION TO LOGIT MODEL
Example 7.3 Logistic Regression
CONCLUSIONS
8 - Nonlinear Regression Models
INTRODUCTION
REGRESSION MODELS
Linear Regression Model
Polynomial Regression Model
Fractional Regression Model
Log-linear Regression Model
Logistic Regression Model
Nonlinear Model
NONLINEAR FORMULATION
SOLVING NONLINEAR REGRESSION MODEL
ESTIMATING PARAMETERS
Maximum Likelihood Estimation (MLE)
Step I: Define the Model
Step II: Define the Likelihood Function
Step III: Maximize the Log-Likelihood Function
NONLINEAR LEAST SQUARES (NON-OLS)
Step I: Define the Model
Step II: Define the Error Term
Step III: Define a Loss Function—Sum of Square Errors
Step IV: Minimize the Sum of Square Error
HYPOTHESIS TESTING
EVALUATE MODEL PERFORMANCE
SAMPLING TECHNIQUES
RANDOM SAMPLING
SAMPLING WITH REPLACEMENT
SAMPLING WITHOUT REPLACEMENT
MONTE CARLO SIMULATION
BOOTSTRAPPING TECHNIQUES
JACKKNIFE SAMPLING TECHNIQUES
Important Notes on Sampling in Nonlinear Regression Models
9 - Machine Learning Techniques
INTRODUCTION
TYPES OF MACHINE LEARNING
EXAMPLES
Cluster Analysis
CLASSIFICATION
REGRESSION
NEURAL NETWORKS
10 - Estimating I-Star Market Impact Model Parameters
INTRODUCTION
I-STAR MARKET IMPACT MODEL
SCIENTIFIC METHOD
Step 1: Ask a Question
Step 2: Research the Problem
Step 3: Construct a Hypothesis
Step 4: Test the Hypothesis
Step 6: Conclusions Communicate
Solution Technique
The Question
Research the Problem
Construct a Hypothesis
Test the Hypothesis
Underlying Data Set
Data Definitions
Imbalance/Order Size
Average daily volume
Actual market volume
Stock volatility
POV Rate
Arrival Cost
Imbalance Size Issues
Model Verification
Model Verification #1: Graphical Illustration
Model Verification #2: Regression Analysis
Model Verification #3: z-Score Analysis
Model Verification #4: Error Analysis
Stock Universe
Analysis Period
Time Period
Number of Data Points
Imbalance
Side
Volume
Turnover
VWAP
First Price
Average Daily Volume
Annualized Volatility
Size
POV Rate
Cost
Estimating Model Parameters
Sensitivity Analysis
Cost Curves
Statistical Analysis
Error Analysis
Stock-Specific Error Analysis
11 - Risk, Volatility, and Factor Models
INTRODUCTION
VOLATILITY MEASURES
Log-Returns
Average Return
Variance
Volatility
Covariance
Correlation
Dispersion
Value-at-Risk
IMPLIED VOLATILITY
Beta
Range
FORECASTING STOCK VOLATILITY
Volatility Models
Returns
Historical Moving Average (HMA)
Exponential Weighted Moving Average (EWMA)
ARCH Volatility Model
GARCH Volatility Model
HMA-VIX Adjustment Model
Determining Parameters via Maximum Likelihood Estimation
Likelihood Function
Measuring Model Performance
Root Mean Square Error (RMSE)
Root Mean Z-Score Squared Error (RMZSE)
Outlier Analysis
HISTORICAL DATA AND COVARIANCE
False Relationships
Example #1: False Negative Signal Calculations
Example #2: False Positive Signal Calculation
Degrees of Freedom
FACTOR MODELS
Matrix Notation
Factor Model in Matrix Notation
TYPES OF FACTOR MODELS
Index Model
Single-Index Model
Multi-Index Models
Macroeconomic Factor Models
Cross Sectional Multi-Factor Model
Statistical Factor Models
How Many Factors Should be Selected?
12 - Volume Forecasting Techniques
INTRODUCTION
MARKET IMPACT MODEL
AVERAGE DAILY VOLUME
Methodology
Definitions
Monthly Volume Forecasting Model
Analysis
Regression Results
OBSERVATIONS OVER THE 19-YEAR PERIOD: 2000–18
OBSERVATIONS OVER THE MOST RECENT 3-YEAR PERIOD: 2016–18
Volumes and Stock Price Correlation
FORECASTING DAILY VOLUMES
Our Daily Volume Forecasting Analysis is as Follows
Definitions
Daily Forecasting Analysis—Methodology
Variable Notation
ARMA Daily Forecasting Model
Analysis Goal
Step 1. Determine Which is More Appropriate: ADV or MDV and the Historical Look-Back Number of Days
Conclusion #1
Step 2. Estimate the DayOfWeek(t) Parameter
Conclusion #2
Step 3. Estimate the Autoregressive Parameter β^
Forecast Improvements
Daily Volume Forecasting Model
Conclusion #3
Forecasting Intraday Volumes Profiles
Forecasting Intraday Volume Profiles
Predicting Remaining Daily Volume
13 - Algorithmic Decision-Making Framework
INTRODUCTION
EQUATIONS
Variables
Important Equations
ALGORITHMIC DECISION-MAKING FRAMEWORK
Select Benchmark Price
Arrival Price Benchmark
Historical Price Benchmark
Future Price Benchmark
COMPARISON OF BENCHMARK PRICES
Specify Trading Goal
Further Insight
Specify Adaptation Tactic
Projected Cost
Target Cost Tactic
Aggressive in the Money
Passive-in-the-Money
COMPARISON ACROSS ADAPTATION TACTICS
MODIFIED ADAPTATION TACTICS
How Often Should we Reoptimization Our Tactic?
14 - Portfolio Algorithms and Trade Schedule Optimization
INTRODUCTION
TRADER'S DILEMMA
Variables
TRANSACTION COST EQUATIONS
Market Impact
Price Appreciation
Timing Risk
One-Sided Optimization Problem
OPTIMIZATION FORMULATION
Constraint Description
Objective Function Difficulty
Optimization Objective Function Simplification
PORTFOLIO OPTIMIZATION TECHNIQUES
Quadratic Programming Approach
Trade Schedule Exponential
Residual Schedule Exponential
Trading Rate Parameter
Market Impact Expression
Timing Risk Expression
Comparison of Optimization Techniques
How Long did it Take to Solve the Portfolio Objective Problem?
How Accurate Was the Solution for Each Optimization Technique?
PORTFOLIO ADAPTATION TACTICS
Description of AIM and PIM for Portfolio Trading
How Often Should we Reoptimize?
Appendix
15 - Advanced Algorithmic Modeling Techniques
INTRODUCTION
TRADING COST EQUATIONS
Model Inputs
TRADING STRATEGY
Percentage of Volume
Trading Rate
Trade Schedule
Comparison of POV Rate to Trade Rate
TRADING TIME
TRADING RISK COMPONENTS
TRADING COST MODELS—REFORMULATED
Market Impact Expression
I-Star
Market Impact for a Single Stock Order
Important note
Market Impact for a Basket of Stock
TIMING RISK EQUATION
Derivation of the 1/3 Factor
Timing Risk For a Basket of Stock
COMPARISON OF MARKET IMPACT ESTIMATES
Forecasting Covariance
Efficient Trading Frontier
Single Stock Trade Cost Objective Function
Portfolio Trade Cost Objective Function
MANAGING PORTFOLIO RISK
Residual Risk Curve
Minimum Trading Risk Quantity
Maximum Trading Opportunity
When to Use These Criteria?
Program-Block Decomposition
16 - Decoding and Reverse Engineering Broker Models with Machine Learning Techniques
INTRODUCTION
PRE-TRADE OF PRE-TRADES
I-Star Model Approach
Neural Network Model Approach
PORTFOLIO OPTIMIZATION
What Should the Portfolio Manager Do?
Deriving Portfolio Optimization Market Impact Models
Example: Share Quantity Regression Model
Example: Trade Value Regression Model
17 - Portfolio Construction with Transaction Cost Analysis
INTRODUCTION
PORTFOLIO OPTIMIZATION AND CONSTRAINTS
TRANSACTION COSTS IN PORTFOLIO OPTIMIZATION
PORTFOLIO MANAGEMENT PROCESS
Example: Efficient Trading Frontier With and Without Short Positions
Example: Maximizing Investor Utility
TRADING DECISION PROCESS
What is the Appropriate Optimal Strategy to Use?
UNIFYING THE INVESTMENT AND TRADING THEORIES
Which Execution Strategy Should the Trader Use?
COST-ADJUSTED FRONTIER
DETERMINING THE APPROPRIATE LEVEL OF RISK AVERSION
BEST EXECUTION FRONTIER
PORTFOLIO CONSTRUCTION WITH TRANSACTION COSTS
Quest for Best Execution Frontier
Return
Risk
EXAMPLE
Important Findings
CONCLUSION
18 - Quantitative Analysis with TCA
INTRODUCTION
Quantitative Overlays
Market Impact Factor Scores
Cost Curves
Alpha Capture
Investment Capacity
Portfolio Optimization
Backtesting
Liquidation Cost
Sensitivity Analysis
ARE THE EXISTING MODELS USEFUL ENOUGH FOR PORTFOLIO CONSTRUCTION?
Current State of Vendor Market Impact Models
PRETRADE OF PRETRADES
Applications
Example #1
Example #2
Example #3
Example #4
HOW EXPENSIVE IS IT TO TRADE?
Acquisition and Liquidation Costs
Portfolio Management—Screening Techniques
BACKTESTING STRATEGIES
MARKET IMPACT SIMULATION
Simulation Scenario
MULTI-ASSET CLASS INVESTING
Investing in Beta Exposure and Other Factors
Example #5
Equities
Exchange-Traded Funds
Futures
Beta Investment Allocation
MULTI-ASSET TRADING COSTS
Global Equity Markets
Multi-Asset Classes
Why do Trading Costs Vary Across Asset Classes?
Definitions
Observations
Equities
Exchange Traded Funds
Futures
Bonds
Commodities
Currency
Room for Improvement
MARKET IMPACT FACTOR SCORES
Current State of Market Impact Factor Scores
MARKET IMPACT FACTOR SCORE ANALYSIS
ALPHA CAPTURE PROGRAM
Example #6
Example #7
Alpha Capture Curves
Important Note
19 - Machine Learning and Trade Schedule Optimization
INTRODUCTION
MULTIPERIOD TRADE SCHEDULE OPTIMIZATION PROBLEM
Setting up the Problem
Trader's Dilemma Objective Function
NONLINEAR OPTIMIZATION CONVERGENCE
Newton's Method
Example #1
Example #2
MACHINE LEARNING
Neural Networks
Neural Network Errors
MACHINE LEARNING TRAINING EXPERIMENT
Step I: Generating Simulated Trade Baskets
Step II: Compile Stock and Basket Data Statistics
X-Input Variables
Y-Output Variable
Step III: Solve the Multiperiod Trade Schedule Optimization Problem
Step IV: Train the NNET
Step V. Calculate the Initial Parameter Values for the NNET
Principal Component Analysis
Stepwise Regression Analysis
Neural Network Structure
Neural Network Error
PERFORMANCE RESULTS
CONCLUSIONS
20 - TCA Analysis Using MATLAB, Excel, and Python
INTRODUCTION
TRANSACTION COST ANALYSIS FUNCTIONS
TRANSACTION COST MODEL
MATLAB FUNCTIONS
EXCEL AND PYTHON FUNCTIONS
TCA REPORT EXAMPLES
CONCLUSION
21 - Transaction Cost Analysis (TCA) Library
INTRODUCTION
TCA Library
TRANSACTION COST ANALYSIS USING THE TCA LIBRARY
List of TCA Functions
Pretrade Analysis
Posttrade Analysis
Portfolio Management
Optimization
Calculations
Conversions
REFERENCES
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z
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