Includes bibliographical references and index.

Preface

I Introduction

Introduction to Modeling Price Impact

The Handbook’s Scope

Introduction

What is Price Impact? Why do Traders Care About It?

The Causality Challenge for Price Impact Models

Four Core Modeling Principles

A Brief History of Price Impact Models

Trading Terminology

Trading Strategies

Trading Data: Fills, Orders, and Binned Data

Trading Signals, Alpha Signals

Intended, Predicted, and Realized Data

Basic Trading Parameters

Order Slippage, Arrival Price

Alpha Slippage, Slippage Due to Price Impact

Trading Experiments: A-B Tests and Back Tests

Outlining Applications

Transaction Cost Analysis (TCA) for Sell-Side Execu-
tion Teams


Portfolio Optimization for Buy-Side Statistical Arbi-
trage Teams


Liquidity Reports for Risk Management Teams

Portfolio Consolidation Analysis for Senior Manage-
ment


Roadmap

What to Expect from the Handbook

A Brief Summary of Each Chapter
II Acting on Price Impact

2 Mathematical Models of Price Impact

2.1 A Pedagogical Example

2.2 Mathematical Setup

2.2.1 Defining Price Impact and Instantaneous Transaction Costs

2.2.2 Establishing P & L in Discrete Time

2.2.3 Examples of Microstructure Assumptions

2.2.4 Reduced Form Models

2.3 The Obizhaeva and Wang (OW) Propagator Model

2.3.1 An Optimal Execution Problem

2.3.2 Closed- Form Optimal Trading Strategy

2.3.3 Intuition Behind the Optimal Trading Strategy

2.4 Extensions Related to the Objective Function

2.4.1 Alpha Signal

2.4.2 Two- Sided Trading

2.4.2.1 Bid- ask Spread as Regularization Term

2.5 Extensions Related to Time

2.5.1 Time Change

2.5.2 Stochastic Push

2.5.2.1 Sensitivity Analysis in Impact Space

2.5.3 Linear Propagator Models

2.6 Extensions Related to External Impact

2.6.1 Microstructure Assumptions

2.6.2 Optimal Trading Strategy with External Impact

2.6.3 Local Concavity

2.6.4 Global Concavity

2.7 Price Manipulation Paradoxes

2.7.1 Constraints on Price Impact Models

2.7.2 Extension to Locally Concave Models

2.7.3 Constraints on Volume Predictions

2.8 Summary of Results

2.8.1 Generalized OW Impact Model

2.8.2 Generalized OW Impact Model with External Impact

2.8.3 Control Problems

2.8.4 Price Manipulation Bounds

2.9 Exercises

3 Applications of Price Impact Models

3.1 A Pedagogical Example

3.2 Optimal Execution

3.2.1 Pre-Trade Cost Model

3.2.1.1 Idealized Optimal Execution Problem

3.2.1.2 Communication with the Portfolio Team

3.2.1.3 Implied Alpha

3.2.1.4 The Square- Root Law


Including Alpha Signals in the Execution Strategy

Alpha Latency

Reactive Execution Schedule
　


Allowing for Tactical Deviations at the Microstructure Level

Quantifying Deviations’ Impact

Block Trades and Auctions

Changing the Execution Strategy when New Orders Arrive

A Simulation Example

Summary

Transaction Cost Analysis (TCA)

TCA Best Practices

Control for Basic Trading Parameters

TCA Predictions

An Experiment to Size Orders Correctly

Clean-Up Costs for Partial Executions

An Experiment for Consecutive Orders

A Simulation to Improve High-Touch Trading

Summary of Results

Optimal Execution Without Intraday Alpha

Pre-Trade Cost Model

Implied Alpha

The Case of Sizable Orders

Implied Alpha’s TCA Implication

Clean-up Costs

Intraday and Low-Latency Alphas

Intraday Alpha

The Cost of Tactical Algorithms

Optimal Execution for Multiple Orders

Combining Order Executions

TCA for Consecutive Orders

Exercises
Further Applications of Price Impact Models

A Pedagogical Example

Statistical Arbitrage

Using External Impact as an Alpha Signal

Cont, Cucuringu, and Zhang’s Alpha Signal

Model Architecture

Extensions

Adjusting Regression Techniques for Liquidity

Using Price Impact for Simulation

Waelbroeck’s Simulation Environment

Business Applications of a Market Simulator

Portfolio and Risk Management

How Price Impact Distorts Accounting P&L and Perceived Risk

Expected Closing P&L

P&L Bias Examples

P&L Bias in Steady State

General Implications and Actions

Portfolio Management Implications

Liquidity Risk Implications

Senior Management Implications

Simulating Fire Sales

Liquidation Without Fire Sale

Liquidation With Fire Sale

Combining Two Portfolios’ Trading

Theory in the Case Without Mutual Information

Theory in the Case With Mutual Information

Empirical Simulation Approach

Summary of Results

Alpha Research

Market Simulator

Liquidity Risk Management

Combining Two Portfolios’ Trading

Exercises
III Measuring Price Impact

An Introduction to the Mathematics of Causal Inference

A Pedagogical Example

A Technical Primer on Causal Inference

Causal Structures

Do-Calculus

Simpson’s Paradox

Identifiability of Causal Formulas

Methods to Reduce Causal Biases

Standard A-B testing

Causal Regularization

Regularization in the Predictive Case

Regularization in the Causal Case

Summary of Results

Causal Structures and Models

Do-Calculus

A-B Testing

Interventional Data

Causal Regularization

Exercises
Dealing with Biases when Fitting Price Impact Models

A Pedagogical Example

Chapter Roadmap

A Non-Technical Primer on Causal Inference

Applying Causal Inference to Trading

A Template for Dealing with Causal Biases

Prediction Bias

Definitions

Actions and Counterfactuals

Why Impact Research is Complex

Experiments and Regularization

A Simulation Example

Synchronization Bias

Definitions

Actions and Counterfactuals

Experiments

Implementation Bias

Definitions

Actions and Counterfactuals

Experiments and Regularization

Issuer Bias

Definitions

Actions and Counterfactuals

Experiments and Regularization

Concluding Thoughts

Summary of Results

Prediction Bias

Synchronization Bias

Implementation Bias

Issuer Bias

Exercises
Empirical Analysis of Price Impact Models

A Pedagogical Example

Methodology

Pre-Processing the Event-Based Data

Definition of the Base Features and Binned Data

Definition of the Time Kernel and Price Impact Computation

Definition of the Prediction Horizon and Training Samples

Definition of the Testing and Validation Samples

Review of the Models in the Literature
　


The Order Flow Imbalance (OFI) Model

The Original OW Model

The Locally Concave Bouchaud Model

The Reduced-Form Model

The Globally Concave AFS Model

Empirical Model Comparisons

Across Timescales

Across Time of Day

Across Clocks

Across Stocks

The Magnitude of Price Impact

Cross-Impact

Causal Bias for Cross-Impact

Price Impact for Factor Trading

Causal Graph for the EigenLiquidity Model

Distinction Between do(Qi, Qj) and do(Q¯)

Counterfactuals Under

Cross-Impact for Risk-Management and Fac-
tor Research


Price Impact for Pairs Trading

Bonds in Schneider and Lillo (2019)[200]

Options in Said et al. (2021)[197]

Commodity Futures in Tomas, Mastromatteo,
and Benzaquen (2022)[217]


Rosenbaum and Tomas (2021)[193]

Sparse Equities Cross-Impact in Cont, Cu- curingu, and Zhang (2021)[71

Summary of Results

Discrete Formulas for Price Impact Models

The Original OW Model

The Locally Concave Bouchaud Model

The Reduced-Form Model

The Globally Concave AFS Model

Summary Table

The EigenLiquidity Model

Exercises
IV Appendix

A Using Kdb+ for Trading Models


A Gentle Introduction to Kdb+

What is Kdb+ and Why Does It Matter to Quants?

First Steps in Kdb+

Basic Operations in Q
　


Q does not Follow the Traditional Order of Operations

Assignments and Other Basic Operators

Atoms, Lists, and Dictionaries

Strings and Symbols

Functions and Loops

Tables are Flipped Dictionaries of Lists

Setting Up a Small Database

A Cheat-Sheet for Quantitative Trading

Data Wrangling in Kdb+

Qsql Queries

Joins

Generalizing Qsql

Long or Wide Format?

Vectorized Operations and Parallelism in Kdb+

An Efficient Implementation of the Generalized OW Model

Key Mathematical Idea

Key Algorithmic Idea

Computing Impact States

An Efficient Implementation of TCA

Key Algorithmic Idea

Computing TCA Returns
Functional Convergence Theorems for Microstructure

Or How to Deal with Local Non-Linearities in Microstructure

Functional Law of Large Numbers

Functional Central Limit Theorem

Further ReadingsSolutions to Exercises
C. 1 Solutions to Chapter 2

C. 2 Solutions to Chapter 3

C.3 Solutions to Chapter 4

C.4 Solutions to Chapter 5

C.5 Solutions to Chapter 6

C.6 Solutions to Chapter 7

Bibliography

Index

Handbook of Price Impact Modeling provides practitioners and students with a mathematical framework grounded in academic references to apply price impact models to quantitative trading and portfolio management. Automated trading is now the dominant form of trading across all frequencies. Furthermore, trading algorithm rise introduces new questions professionals must answer, for instance:


How do stock prices react to a trading strategy?

How to scale a portfolio considering its trading costs and liquidity risk?

How to measure and improve trading algorithms while avoiding biases?
Price impact models answer these novel questions at the forefront of quantitative finance. Hence, practitioners and students can use this Handbook as a comprehensive, modern view of systematic trading.

For financial institutions, the Handbook’s framework aims to minimize the firm’s price impact, measure market liquidity risk, and provide a unified, succinct view of the firm’s trading activity to the C-suite via analytics and tactical research.

The Handbook’s focus on applications and everyday skillsets makes it an ideal textbook for a master’s in finance class and students joining quantitative trading desks. Using price impact models, the reader learns how to:


Build a market simulator to back test trading algorithms

Implement closed-form strategies that optimize trading signals

Measure liquidity risk and stress test portfolios for fire sales

Analyze algorithm performance controlling for common trading biases

Estimate price impact models using public trading tape
Finally, the reader finds a primer on the database kdb+ and its programming language q, which are standard tools for analyzing high-frequency trading data at banks and hedge funds.

Authored by a finance professional, this book is a valuable resource for quantitative researchers and traders.