Model-free Hedging

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Author :
Publisher : CRC Press
ISBN 13 : 1351666231
Total Pages : 190 pages
Book Rating : 4.3/5 (516 download)

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Book Synopsis Model-free Hedging by : Pierre Henry-Labordere

Download or read book Model-free Hedging written by Pierre Henry-Labordere and published by CRC Press. This book was released on 2017-05-25 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Model-free Hedging: A Martingale Optimal Transport Viewpoint focuses on the computation of model-independent bounds for exotic options consistent with market prices of liquid instruments such as Vanilla options. The author gives an overview of Martingale Optimal Transport, highlighting the differences between the optimal transport and its martingale counterpart. This topic is then discussed in the context of mathematical finance.

Model-free Hedging

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Author :
Publisher : Chapman & Hall/CRC
ISBN 13 : 9781138062238
Total Pages : 0 pages
Book Rating : 4.0/5 (622 download)

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Book Synopsis Model-free Hedging by : Pierre Henry-Labordère

Download or read book Model-free Hedging written by Pierre Henry-Labordère and published by Chapman & Hall/CRC. This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Model-free Hedging: A Martingale Optimal Transport Viewpoint focuses on the computation of model-independent bounds for exotic options consistent with market prices of liquid instruments such as Vanilla options. The author gives an overview of Martingale Optimal Transport, highlighting the differences between the optimal transport and its martingale counterpart. This topic is then discussed in the context of mathematical finance.

Dynamic Hedging

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Author :
Publisher : John Wiley & Sons
ISBN 13 : 9780471152804
Total Pages : 536 pages
Book Rating : 4.1/5 (528 download)

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Book Synopsis Dynamic Hedging by : Nassim Nicholas Taleb

Download or read book Dynamic Hedging written by Nassim Nicholas Taleb and published by John Wiley & Sons. This book was released on 1997-01-14 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: Destined to become a market classic, Dynamic Hedging is the only practical reference in exotic options hedgingand arbitrage for professional traders and money managers Watch the professionals. From central banks to brokerages to multinationals, institutional investors are flocking to a new generation of exotic and complex options contracts and derivatives. But the promise of ever larger profits also creates the potential for catastrophic trading losses. Now more than ever, the key to trading derivatives lies in implementing preventive risk management techniques that plan for and avoid these appalling downturns. Unlike other books that offer risk management for corporate treasurers, Dynamic Hedging targets the real-world needs of professional traders and money managers. Written by a leading options trader and derivatives risk advisor to global banks and exchanges, this book provides a practical, real-world methodology for monitoring and managing all the risks associated with portfolio management. Nassim Nicholas Taleb is the founder of Empirica Capital LLC, a hedge fund operator, and a fellow at the Courant Institute of Mathematical Sciences of New York University. He has held a variety of senior derivative trading positions in New York and London and worked as an independent floor trader in Chicago. Dr. Taleb was inducted in February 2001 in the Derivatives Strategy Hall of Fame. He received an MBA from the Wharton School and a Ph.D. from University Paris-Dauphine.

Market Risk Analysis, Pricing, Hedging and Trading Financial Instruments

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Publisher : John Wiley & Sons
ISBN 13 : 0470997893
Total Pages : 427 pages
Book Rating : 4.4/5 (79 download)

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Book Synopsis Market Risk Analysis, Pricing, Hedging and Trading Financial Instruments by : Carol Alexander

Download or read book Market Risk Analysis, Pricing, Hedging and Trading Financial Instruments written by Carol Alexander and published by John Wiley & Sons. This book was released on 2008-06-09 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written by leading market risk academic, Professor Carol Alexander, Pricing, Hedging and Trading Financial Instruments forms part three of the Market Risk Analysis four volume set. This book is an in-depth, practical and accessible guide to the models that are used for pricing and the strategies that are used for hedging financial instruments, and to the markets in which they trade. It provides a comprehensive, rigorous and accessible introduction to bonds, swaps, futures and forwards and options, including variance swaps, volatility indices and their futures and options, to stochastic volatility models and to modelling the implied and local volatility surfaces. All together, the Market Risk Analysis four volume set illustrates virtually every concept or formula with a practical, numerical example or a longer, empirical case study. Across all four volumes there are approximately 300 numerical and empirical examples, 400 graphs and figures and 30 case studies many of which are contained in interactive Excel spreadsheets available from the the accompanying CD-ROM . Empirical examples and case studies specific to this volume include: Duration-Convexity approximation to bond portfolios, and portfolio immunization; Pricing floaters and vanilla, basis and variance swaps; Coupon stripping and yield curve fitting; Proxy hedging, and hedging international securities and energy futures portfolios; Pricing models for European exotics, including barriers, Asians, look-backs, choosers, capped, contingent, power, quanto, compo, exchange, ‘best-of’ and spread options; Libor model calibration; Dynamic models for implied volatility based on principal component analysis; Calibration of stochastic volatility models (Matlab code); Simulations from stochastic volatility and jump models; Duration, PV01 and volatility invariant cash flow mappings; Delta-gamma-theta-vega mappings for options portfolios; Volatility beta mapping to volatility indices.

Model Risk in Financial Markets

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Author :
Publisher : World Scientific
ISBN 13 : 9814663425
Total Pages : 384 pages
Book Rating : 4.8/5 (146 download)

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Book Synopsis Model Risk in Financial Markets by : Radu Tunaru

Download or read book Model Risk in Financial Markets written by Radu Tunaru and published by World Scientific. This book was released on 2015-06-08 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: The financial systems in most developed countries today build up a large amount of model risk on a daily basis. However, this is not particularly visible as the financial risk management agenda is still dominated by the subprime-liquidity crisis, the sovereign crises, and other major political events. Losses caused by model risk are hard to identify and even when they are internally identified, as such, they are most likely to be classified as normal losses due to market evolution. Model Risk in Financial Markets: From Financial Engineering to Risk Management seeks to change the current perspective on model innovation, implementation and validation. This book presents a wide perspective on model risk related to financial markets, running the gamut from financial engineering to risk management, from financial mathematics to financial statistics. It combines theory and practice, both the classical and modern concepts being introduced for financial modelling. Quantitative finance is a relatively new area of research and much has been written on various directions of research and industry applications. In this book the reader gradually learns to develop a critical view on the fundamental theories and new models being proposed. Contents:IntroductionFundamental RelationshipsModel Risk in Interest Rate ModellingArbitrage TheoryDerivatives Pricing Under UncertaintyPortfolio Selection Under UncertaintyProbability Pitfalls of Financial CalculusModel Risk in Risk Measures CalculationsParameter Estimation RiskComputational ProblemsPortfolio Selection Using Sharpe RatioBayesian Calibration for Low Frequency DataMCMC Estimation of Credit Risk MeasuresLast But Not Least. Can We Avoid the Next Big Systemic Financial Crisis?Notations for the Study of MLE for CIR Process Readership: Graduate students, researchers, practitioners, senior managers in financial institutions and hedge-funds, regulators and risk managers, who are keen to understand the pitfalls of financial modelling, and also those who are looking for a career in model validation, product control and risk management functions. Key Features:Some innovative results are presented for the first timeCovers a wide range of models, results and applications in financial markets to demonstrate that model risk is generally spreadKeywords:Model Risk;Risk Management;Financial Engineering;Financial Markets

Credit Risk: Modeling, Valuation and Hedging

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Author :
Publisher : Springer Science & Business Media
ISBN 13 : 3662048213
Total Pages : 517 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Credit Risk: Modeling, Valuation and Hedging by : Tomasz R. Bielecki

Download or read book Credit Risk: Modeling, Valuation and Hedging written by Tomasz R. Bielecki and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: The motivation for the mathematical modeling studied in this text on developments in credit risk research is the bridging of the gap between mathematical theory of credit risk and the financial practice. Mathematical developments are covered thoroughly and give the structural and reduced-form approaches to credit risk modeling. Included is a detailed study of various arbitrage-free models of default term structures with several rating grades.

Innovations in Quantitative Risk Management

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Author :
Publisher : Springer
ISBN 13 : 331909114X
Total Pages : 434 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Innovations in Quantitative Risk Management by : Kathrin Glau

Download or read book Innovations in Quantitative Risk Management written by Kathrin Glau and published by Springer. This book was released on 2015-01-09 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantitative models are omnipresent –but often controversially discussed– in todays risk management practice. New regulations, innovative financial products, and advances in valuation techniques provide a continuous flow of challenging problems for financial engineers and risk managers alike. Designing a sound stochastic model requires finding a careful balance between parsimonious model assumptions, mathematical viability, and interpretability of the output. Moreover, data requirements and the end-user training are to be considered as well. The KPMG Center of Excellence in Risk Management conference Risk Management Reloaded and this proceedings volume contribute to bridging the gap between academia –providing methodological advances– and practice –having a firm understanding of the economic conditions in which a given model is used. Discussed fields of application range from asset management, credit risk, and energy to risk management issues in insurance. Methodologically, dependence modeling, multiple-curve interest rate-models, and model risk are addressed. Finally, regulatory developments and possible limits of mathematical modeling are discussed.

Credit Risk: Modeling, Valuation and Hedging

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Author :
Publisher : Springer Science & Business Media
ISBN 13 : 9783540675938
Total Pages : 524 pages
Book Rating : 4.6/5 (759 download)

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Book Synopsis Credit Risk: Modeling, Valuation and Hedging by : Tomasz R. Bielecki

Download or read book Credit Risk: Modeling, Valuation and Hedging written by Tomasz R. Bielecki and published by Springer Science & Business Media. This book was released on 2004-01-22 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: The motivation for the mathematical modeling studied in this text on developments in credit risk research is the bridging of the gap between mathematical theory of credit risk and the financial practice. Mathematical developments are covered thoroughly and give the structural and reduced-form approaches to credit risk modeling. Included is a detailed study of various arbitrage-free models of default term structures with several rating grades.

The SABR/LIBOR Market Model

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Author :
Publisher : John Wiley & Sons
ISBN 13 : 1119995639
Total Pages : 308 pages
Book Rating : 4.1/5 (199 download)

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Book Synopsis The SABR/LIBOR Market Model by : Riccardo Rebonato

Download or read book The SABR/LIBOR Market Model written by Riccardo Rebonato and published by John Wiley & Sons. This book was released on 2011-03-01 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a major innovation in the interest rate space. It explains a financially motivated extension of the LIBOR Market model which accurately reproduces the prices for plain vanilla hedging instruments (swaptions and caplets) of all strikes and maturities produced by the SABR model. The authors show how to accurately recover the whole of the SABR smile surface using their extension of the LIBOR market model. This is not just a new model, this is a new way of option pricing that takes into account the need to calibrate as accurately as possible to the plain vanilla reference hedging instruments and the need to obtain prices and hedges in reasonable time whilst reproducing a realistic future evolution of the smile surface. It removes the hard choice between accuracy and time because the framework that the authors provide reproduces today's market prices of plain vanilla options almost exactly and simultaneously gives a reasonable future evolution for the smile surface. The authors take the SABR model as the starting point for their extension of the LMM because it is a good model for European options. The problem, however with SABR is that it treats each European option in isolation and the processes for the various underlyings (forward and swap rates) do not talk to each other so it isn't obvious how to relate these processes into the dynamics of the whole yield curve. With this new model, the authors bring the dynamics of the various forward rates and stochastic volatilities under a single umbrella. To ensure the absence of arbitrage they derive drift adjustments to be applied to both the forward rates and their volatilities. When this is completed, complex derivatives that depend on the joint realisation of all relevant forward rates can now be priced. Contents THE THEORETICAL SET-UP The Libor Market model The SABR Model The LMM-SABR Model IMPLEMENTATION AND CALIBRATION Calibrating the LMM-SABR model to Market Caplet prices Calibrating the LMM/SABR model to Market Swaption Prices Calibrating the Correlation Structure EMPIRICAL EVIDENCE The Empirical problem Estimating the volatility of the forward rates Estimating the correlation structure Estimating the volatility of the volatility HEDGING Hedging the Volatility Structure Hedging the Correlation Structure Hedging in conditions of market stress

Modeling, Measuring and Hedging Operational Risk

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Author :
Publisher : John Wiley & Sons
ISBN 13 :
Total Pages : 360 pages
Book Rating : 4.F/5 ( download)

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Book Synopsis Modeling, Measuring and Hedging Operational Risk by : Marcelo G. Cruz

Download or read book Modeling, Measuring and Hedging Operational Risk written by Marcelo G. Cruz and published by John Wiley & Sons. This book was released on 2002-03-12 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: Worldwide banks are keen to find ways of effectively measuring and managing operational risk , yet many find themselves poorly equipped to do this. Operational risk includes concerns about such issues as transaction processing errors, liability situations, and back-office failure. Measuring and Modelling Operational Risk focuses on the measuring and modelling techniques banks and investment companies need to quantify operational risk and provides practical, sensible solutions for doing so. * Author is one of the leading experts in the field of operational risk. * Interest in the field is growing rapidly and this is the only book that focuses on the quantitative measuring and modelling of operational risk. * Includes case vignettes and real-world examples based on the author's extensive experience.

The Concepts and Practice of Mathematical Finance

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Publisher : Cambridge University Press
ISBN 13 : 9780521823555
Total Pages : 496 pages
Book Rating : 4.8/5 (235 download)

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Book Synopsis The Concepts and Practice of Mathematical Finance by : Mark Suresh Joshi

Download or read book The Concepts and Practice of Mathematical Finance written by Mark Suresh Joshi and published by Cambridge University Press. This book was released on 2003-12-24 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: For those starting out as practitioners of mathematical finance, this is an ideal introduction. It provides the reader with a clear understanding of the intuition behind derivatives pricing, how models are implemented, and how they are used and adapted in practice. Strengths and weaknesses of different models, e.g. Black-Scholes, stochastic volatility, jump-diffusion and variance gamma, are examined. Both the theory and the implementation of the industry-standard LIBOR market model are considered in detail. Uniquely, the book includes extensive discussion of the ideas behind the models, and is even-handed in examining various approaches to the subject. Thus each pricing problem is solved using several methods. Worked examples and exercises, with answers, are provided in plenty, and computer projects are given for many problems. The author brings to this book a blend of practical experience and rigorous mathematical background, and supplies here the working knowledge needed to become a good quantitative analyst.

Pricing Models of Volatility Products and Exotic Variance Derivatives

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Author :
Publisher : CRC Press
ISBN 13 : 1000584259
Total Pages : 283 pages
Book Rating : 4.0/5 (5 download)

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Book Synopsis Pricing Models of Volatility Products and Exotic Variance Derivatives by : Yue Kuen Kwok

Download or read book Pricing Models of Volatility Products and Exotic Variance Derivatives written by Yue Kuen Kwok and published by CRC Press. This book was released on 2022-05-08 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pricing Models of Volatility Products and Exotic Variance Derivatives summarizes most of the recent research results in pricing models of derivatives on discrete realized variance and VIX. The book begins with the presentation of volatility trading and uses of variance derivatives. It then moves on to discuss the robust replication strategy of variance swaps using portfolio of options, which is one of the major milestones in pricing theory of variance derivatives. The replication procedure provides the theoretical foundation of the construction of VIX. This book provides sound arguments for formulating the pricing models of variance derivatives and establishes formal proofs of various technical results. Illustrative numerical examples are included to show accuracy and effectiveness of analytic and approximation methods. Features Useful for practitioners and quants in the financial industry who need to make choices between various pricing models of variance derivatives Fabulous resource for researchers interested in pricing and hedging issues of variance derivatives and VIX products Can be used as a university textbook in a topic course on pricing variance derivatives

Proceedings of the International Workshop on Finance 2011. Doshisha University, Kyoto, Japan. 3-4 August 2011

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Author :
Publisher : World Scientific
ISBN 13 : 981440733X
Total Pages : 231 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Proceedings of the International Workshop on Finance 2011. Doshisha University, Kyoto, Japan. 3-4 August 2011 by : Akihiko Takahashi

Download or read book Proceedings of the International Workshop on Finance 2011. Doshisha University, Kyoto, Japan. 3-4 August 2011 written by Akihiko Takahashi and published by World Scientific. This book was released on 2012-05-21 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the Proceedings of the International Workshop on Finance 2011, held in Kyoto in the summer of 2011 with the aim of exchanging new ideas in financial engineering among researchers from various countries from both academia and industry. The workshop was held as a successor to the Daiwa International Workshop (2004OCo2008), and the KIER-TMU International Workshop (2009OCo2010). This workshop was organized by the Center for Advanced Research in Finance (CARF), Graduate School of Economics, the University of Tokyo, and Graduate School of Social Sciences, Tokyo Metropolitan University OCo and co-organized by Life Risk Research Center, Doshisha University. The workshop serves as a bridge between academic researchers and practitioners. This book contains about fifteen papers, all refereed, representing the presentations at the workshop. The papers address state-of-the-art techniques in financial engineering."

Recent Advances in Financial Engineering

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Author :
Publisher : World Scientific
ISBN 13 : 9814407321
Total Pages : 231 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Recent Advances in Financial Engineering by : Akihiko Takahashi

Download or read book Recent Advances in Financial Engineering written by Akihiko Takahashi and published by World Scientific. This book was released on 2012 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the Proceedings of the International Workshop on Finance 2011, held in Kyoto in the summer of 2011 with the aim of exchanging new ideas in financial engineering among researchers from various countries from both academia and industry. the workshop was held as a successor to the Daiwa International Workshop (2004–2008), and the KIER-TMU International Workshop (2009–2010). This workshop was organized by the Center for Advanced Research in Finance (CARF), Graduate School of Economics, the University of Tokyo, and Graduate School of Social Sciences, Tokyo Metropolitan University — and co-organized by Life Risk Research Center, Doshisha University.The workshop serves as a bridge between academic researchers and practitioners.This book contains about fifteen papers, all refereed, representing the presentations at the workshop. the papers address state-of-the-art techniques in financial engineering.

Recent Advances in Financial Engineering 2011

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Author :
Publisher : World Scientific
ISBN 13 : 9814407348
Total Pages : 232 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Recent Advances in Financial Engineering 2011 by : Akihiko Takahashi

Download or read book Recent Advances in Financial Engineering 2011 written by Akihiko Takahashi and published by World Scientific. This book was released on 2012-05-21 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the Proceedings of the International Workshop on Finance 2011, held in Kyoto in the summer of 2011 with the aim of exchanging new ideas in financial engineering among researchers from various countries from both academia and industry. The workshop was held as a successor to the Daiwa International Workshop (2004–2008), and the KIER-TMU International Workshop (2009–2010). This workshop was organized by the Center for Advanced Research in Finance (CARF), Graduate School of Economics, the University of Tokyo, and Graduate School of Social Sciences, Tokyo Metropolitan University — and co-organized by Life Risk Research Center, Doshisha University. The workshop serves as a bridge between academic researchers and practitioners. This book contains about fifteen papers, all refereed, representing the presentations at the workshop. The papers address state-of-the-art techniques in financial engineering. Contents:On the Representation of General Interest Rate Models as Square-Integrable Wiener Functionals (L P Hughston and F Mina)On Pricing Contingent Capital Notes (D B Madan)A Survey on Modeling and Analysis of Basis Spreads (M Fujii and A Takahashi)Conservative Delta Hedging Under Transaction Costs (M Fukasawa)The Theory of Optimal Investment in Information Security and Adjustment Costs: An Impulse Control Approach (M Goto and K Tatsumi)Strategic Investment with Three Asymmetric Firms (S Ko and T Shibata)An Empirical Analysis of Japanese Interest Rate Swap Spread (J Shimada, T Takahashi, T Miyakoshi, and Y Tsukuda)A Remark on Approximation of the Solutions to Partial Differential Equations in Finance (A Takahashi and T Yamada)Optimal Trading with Cointegrated Pairs of Stocks (Y Yamada and J A Primbs)Analytical Approximation of Pricing Average Options under the Heston Model (A Yamazaki) Readership: Graduate and postgraduate students of financial engineering and mathematical finance; academics and practitioners; quantitative researchers on financial markets. Keywords:Financial Engineering;Mathematical Finance;Money & Banking;Risk Management;Real Option;Corporate Finance;Computational FinanceKey Features:Wide coverage of the research themesMost up-to-date research resultsResearch straddling academia and industryAddresses state-of-the-art techniques in financial engineering

Applications of Computational Intelligence in Data-Driven Trading

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Author :
Publisher : John Wiley & Sons
ISBN 13 : 1119550505
Total Pages : 304 pages
Book Rating : 4.1/5 (195 download)

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Book Synopsis Applications of Computational Intelligence in Data-Driven Trading by : Cris Doloc

Download or read book Applications of Computational Intelligence in Data-Driven Trading written by Cris Doloc and published by John Wiley & Sons. This book was released on 2019-10-29 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: “Life on earth is filled with many mysteries, but perhaps the most challenging of these is the nature of Intelligence.” – Prof. Terrence J. Sejnowski, Computational Neurobiologist The main objective of this book is to create awareness about both the promises and the formidable challenges that the era of Data-Driven Decision-Making and Machine Learning are confronted with, and especially about how these new developments may influence the future of the financial industry. The subject of Financial Machine Learning has attracted a lot of interest recently, specifically because it represents one of the most challenging problem spaces for the applicability of Machine Learning. The author has used a novel approach to introduce the reader to this topic: The first half of the book is a readable and coherent introduction to two modern topics that are not generally considered together: the data-driven paradigm and Computational Intelligence. The second half of the book illustrates a set of Case Studies that are contemporarily relevant to quantitative trading practitioners who are dealing with problems such as trade execution optimization, price dynamics forecast, portfolio management, market making, derivatives valuation, risk, and compliance. The main purpose of this book is pedagogical in nature, and it is specifically aimed at defining an adequate level of engineering and scientific clarity when it comes to the usage of the term “Artificial Intelligence,” especially as it relates to the financial industry. The message conveyed by this book is one of confidence in the possibilities offered by this new era of Data-Intensive Computation. This message is not grounded on the current hype surrounding the latest technologies, but on a deep analysis of their effectiveness and also on the author’s two decades of professional experience as a technologist, quant and academic.

Machine Learning and Data Sciences for Financial Markets

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Author :
Publisher : Cambridge University Press
ISBN 13 : 1316516199
Total Pages : 742 pages
Book Rating : 4.3/5 (165 download)

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Book Synopsis Machine Learning and Data Sciences for Financial Markets by : Agostino Capponi

Download or read book Machine Learning and Data Sciences for Financial Markets written by Agostino Capponi and published by Cambridge University Press. This book was released on 2023-04-30 with total page 742 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leveraging the research efforts of more than sixty experts in the area, this book reviews cutting-edge practices in machine learning for financial markets. Instead of seeing machine learning as a new field, the authors explore the connection between knowledge developed by quantitative finance over the past forty years and techniques generated by the current revolution driven by data sciences and artificial intelligence. The text is structured around three main areas: 'Interactions with investors and asset owners,' which covers robo-advisors and price formation; 'Risk intermediation,' which discusses derivative hedging, portfolio construction, and machine learning for dynamic optimization; and 'Connections with the real economy,' which explores nowcasting, alternative data, and ethics of algorithms. Accessible to a wide audience, this invaluable resource will allow practitioners to include machine learning driven techniques in their day-to-day quantitative practices, while students will build intuition and come to appreciate the technical tools and motivation for the theory.