Stochastic Modelling of Big Data in Finance

305 Pages·2022·127.149 MB·English

by  Anatoliy Swishchuk

Stochastic Modelling
of Big Data in Finance
Stochastic Modelling of Big Data in Finance provides a rigorous overview and exploration of sto-
chastic modelling of big data in finance (BDF). The book describes various stochastic models, includ-
ing multivariate models, to deal with big data in finance. This includes data in high-frequency and
algorithmic trading, specifically in limit order books (LOB), and shows how those models can be
applied to different datasets to describe the dynamics of LOB, and to figure out which model is the
best with respect to a specific data set. The results of the book may be used to also solve acquisition,
liquidation and market making problems, and other optimization problems in finance.
Features
• Self-contained book suitable for graduate students and post-doctoral fellows in financial math-
ematics and data science, as well as for practitioners working in the financial industry who deal
with big data
• All results are presented visually to aid in understanding of concepts
Dr. Anatoliy Swishchuk is a Professor in Mathematical Finance at the Department of Mathematics
and Statistics, University of Calgary, Calgary, AB, Canada. He got his B.Sc. and M.Sc. degrees from
Kyiv State University, Kyiv, Ukraine. He earned two doctorate degrees in Mathematics and Physics
(PhD and DSc) from the prestigious National Academy of Sciences of Ukraine (NASU), Kiev, Ukraine,
and is a recipient of NASU award for young scientist with a gold medal for series of research publica-
tions in random evolutions and their applications.  
Dr. Swishchuk is a chair and organizer of finance and energy finance seminar ‘Lunch at the Lab’ at the
Department of Mathematics and Statistics. Dr. Swishchuk is a Director of Mathematical and Compu-
tational Finance Laboratory at the University of Calgary. He was a steering committee member of the
Professional Risk Managers International Association (PRMIA), Canada (2006-2015), and is a steer-
ing committee member of Global Association of Risk Professionals (GARP), Canada (since 2015).
Dr. Swishchuk is a creator of mathematical finance program at the Department of Mathematics & Sta-
tistics. He is also a proponent for a new specialization “Financial and Energy Markets Data Modelling”
in the Data Science and Analytics program. His research areas include financial mathematics, ran-
dom evolutions and their applications, biomathematics, stochastic calculus, and he serves on editorial
boards for four research journals. He is the author of more than 200 publications, including 15 books
and more than 150 articles in peer-reviewed journals. In 2018 he received a Peak Scholar award.
Chapman & Hall/CRC Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
appeal to both academics and practitioners. The inclusion of numerical code and concrete real-world
examples is highly encouraged.
Series Editors
M.A.H. Dempster
Centre for Financial Research
Department of Pure Mathematics and Statistics
University of Cambridge, UK
Dilip B. Madan
Robert H. Smith School of Business
University of Maryland, USA
Rama Cont
Department of Mathematics
Imperial College, UK
Robert A. Jarrow
Lynch Professor of Investment Management
Johnson Graduate School of Management
Cornell University, USA
Machine Learning for Factor Investing: R Version
Guillaume Coqueret, Tony Guida
Malliavin Calculus in Finance: Theory and Practice
Elisa Alos, David Garcia Lorite
Risk Measures and Insurance Solvency Benchmarks: Fixed-Probability Levels
in Renewal Risk Models
Vsevolod K. Malinovskii
Financial Mathematics: A Comprehensive Treatment in Discrete Time, Second Edition
Giuseppe Campolieti, Roman N. Makarov
Pricing Models of Volatility Products and Exotic Variance Derivatives
Yue Kuen Kwok, Wendong Zheng
Quantitative Finance with Python: A Practical Guide to Investment Management,
Trading, and Financial Engineering
Chris Kelliher
Stochastic Modelling of Big Data in Finance
Anatoliy Swishchuk
Introduction to Stochastic Finance with Market Examples, Second Edition
Nicolas Privault
Commodities: Fundamental Theory of Futures, Forwards, and Derivatives Pricing,
Second Edition
M.A.H. Dempster, Ke Tang
For more information about this series please visit: https://www.crcpress.
com/Chapman-and-HallCRC-Financial-Mathematics-Series/book series/
CHFINANCMTH
Stochastic Modelling
of Big Data in Finance
Anatoliy Swishchuk
Department of Mathematics and Statistics,
University of Calgary, Calgary, Canada
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2023 Anatoliy Swishchuk
CRC Press is an imprint of Taylor & Francis Group, LLC
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Library of Congress Cataloging-in-Publication Data
Names: Swishchuk, Anatoliy, author.
Title: Stochastic modelling of big data in finance / Anatoliy Swishchuk,
Department of Mathematics and Statistics, University of Calgary,
Calgary, Canada.
Description: 1 Edition. | Boca Raton, FL : Chapman & Hall, CRC Press, 2023.
| Series: Chapman and Hall/CRC financial mathematics series | Includes
bibliographical references and index.
Identifiers: LCCN 2022022045 (print) | LCCN 2022022046 (ebook) | ISBN
9781032209265 (hardback) | ISBN 9781032209289 (paperback) | ISBN
9781003265986 (ebook)
Subjects: LCSH: Finance--Mathematical models. | Stochastic models. | Big
data.
Classification: LCC HG106 .S95 2023 (print) | LCC HG106 (ebook) | DDC
332.01/5195--dc23/eng/20220801
LC record available at https://lccn.loc.gov/2022022045
LC ebook record available at https://lccn.loc.gov/2022022046
ISBN: 978-1-032-20926-5 (hbk)
ISBN: 978-1-032-20928-9 (pbk)
ISBN: 978-1-003-26598-6 (ebk)
DOI: 10.1201/ 9781003265986
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
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Contents
Foreword xiii
Preface xv
Symbols xxi
Acknowledgements xxiii
1 A Brief Introduction: Stochastic Modelling of Big Data
in Finance 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Big Data in Finance: Limit Order Books . . . . . . . . . . . 3
1.2.1 Description of Limit Order Books Mechanism . . . . . 3
1.2.2 Big Data in Finance: Lobster Data . . . . . . . . . . . 4
1.2.3 MoreBigDatainFinance:XetraandFrankfurtMarkets
(Deutsche Boerse Group), on September 23, 2013 and
CISCO Data on November 3, 2014 . . . . . . . . . . . 5
1.3 Stochastic Modelling of Big Data in Finance: Limit Order
Books (LOB) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Semi-Markov Modelling of LOB . . . . . . . . . . . . 7
1.3.2 General Semi-Markov Modelling of LOB . . . . . . . . 9
1.3.3 Modelling of LOB with a Compound Hawkes Processes 10
1.3.4 Modelling of LOB with a General Compound Hawkes
Processes . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.5 ModellingofLOBwithaNon-linearGeneralCompound
Hawkes Processes. . . . . . . . . . . . . . . . . . . . . 12
1.3.6 Modelling of LOB with a Multivariable General
Compound Hawkes Processes . . . . . . . . . . . . . . 12
1.4 IllustrationandJustificationofOurMethodtoStudyBigData
in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Numerical Results: Lobster Data (Apple, Google and
Microsoft Stocks) . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 NumericalResults:XetraandFrankfurtMarketsstocks
(Deutsche Boerse Group), on September 23, 2013 . . . 14
1.4.3 Numerical Results: CISCO Data, November 3, 2014 . 15
1.5 Methodological Aspects of Using the Models . . . . . . . . . 15
vii
viii Contents
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
I Semi-Markovian Modelling of Big Data in Finance 21
2 A Semi-Markovian Modelling of Big Data in Finance 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 A Semi-Markovian Modelling of Limit Order Markets . . . . 26
2.2.1 Markov Renewal and Semi-Markov Processes . . . . . 26
2.2.2 Semi-Markovian Modelling of Limit Order Books . . . 27
2.3 Main Probabilistic Results . . . . . . . . . . . . . . . . . . . 32
2.3.1 Duration until the next price change . . . . . . . . . . 32
2.3.2 Probability of Price Increase . . . . . . . . . . . . . . 38
2.3.3 ThestockpriceseenasafunctionalofaMarkovrenewal
process . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Diffusion Limit of the Price Process . . . . . . . . . . . . . . 40
2.4.1 Balanced Order Flow case: Pa(1,1) = Pa(−1,−1) and
Pb(1,1)=Pb(−1,−1) . . . . . . . . . . . . . . . . . . 41
2.4.2 Other cases: either Pa(1,1)<Pa(−1,−1) or Pb(1,1)
<Pb(−1,−1) . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 More Big Data . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 More Data . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.2 Estimated Probabilities . . . . . . . . . . . . . . . . . 54
2.6.3 Assumption on Distributions f and f˜ . . . . . . . . . 59
2.6.4 Diffusion Limit (Not-Fixed Spread). . . . . . . . . . . 60
2.6.5 The Optimal Liquidation/Acquisition Problems . . . . 61
2.6.6 Market Making . . . . . . . . . . . . . . . . . . . . . . 62
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 General Semi-Markovian Modelling of Big Data in Finance 67
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.1 Motivation for Generalizing the Model . . . . . . . . . 68
3.1.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Reviewing the Assumptions with Our New Data Sets . . . . 69
3.2.1 Liquidity of Our Data . . . . . . . . . . . . . . . . . . 69
3.2.2 Empirical Distributions of Initial Queue Sizes and
Calculated Conditional Probabilities . . . . . . . . . . 70
3.2.3 Inter-arrival Times of Book Events . . . . . . . . . . . 71
3.2.4 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . 72
3.3 General Semi-Markov Model for the Limit Order Book with
Two States . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.1 Diffusion Limits . . . . . . . . . . . . . . . . . . . . . 73
Contents ix
3.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . 77
3.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . 78
3.3.4 Application of the Model . . . . . . . . . . . . . . . . 79
3.3.4.1 Examination of the Data . . . . . . . . . . . 80
3.3.4.2 Model Implementation . . . . . . . . . . . . 82
3.3.4.3 Results for Constructed Sample Day . . . . . 83
3.4 General Semi-Markov Model for the Limit Order Book with
arbitrary number of states . . . . . . . . . . . . . . . . . . . 85
3.4.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4.2 Diffusion Limits . . . . . . . . . . . . . . . . . . . . . 86
3.4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . 89
3.4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . 90
3.5 Discussion on Price Spreads . . . . . . . . . . . . . . . . . . 91
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
II Modelling of Big Data in Finance with Hawkes
Processes 97
4 A Brief Introduction to Hawkes Processes 99
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Definition of Hawkes Processes (HPs) . . . . . . . . . . . . . 101
4.3 Compound Hawkes Processes . . . . . . . . . . . . . . . . . . 104
4.3.1 Special Cases of Compound Hawkes Processes in Limit
Order Books . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 Limit Theorems for Hawkes Processes: LLN and FCLT . . . 106
4.4.1 Law of Large Numbers (LLN) for Hawkes Processes . 106
4.4.2 FunctionalCentralLimitTheorems(FCLT)forHawkes
Processes . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Limit Theorems for Poisson Processes: LLN and FCLT . . . 107
4.5.1 Law of Large Numbers (LLN) for Poisson Processes . 107
4.5.2 FunctionalCentralLimitTheorems(FCLT)forHawkes
Processes . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.6 Stylized Properties of Hawkes Process . . . . . . . . . . . . . 107
4.6.1 Non-exponential Inter-arrival Times . . . . . . . . . . 108
4.6.2 Clustering Effect of Trades . . . . . . . . . . . . . . . 110
4.6.3 Non-independency of Mid-price Changes . . . . . . . . 112
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Stochastic Modelling of Big Data in Finance with CHP 121
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Definitions of HP, CHP and RSCHP . . . . . . . . . . . . . . 123
5.2.1 One-dimensional Hawkes Process . . . . . . . . . . . . 123
5.2.2 Compound Hawkes Process (CHP) . . . . . . . . . . . 125