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Rough Volatility

Author: Christian Bayer

Publisher: SIAM

Published: 2023-12-18

Total Pages: 292

ISBN-13: 1611977789

Volatility underpins financial markets by encapsulating uncertainty about prices, individual behaviors, and decisions and has traditionally been modeled as a semimartingale, with consequent scaling properties. The mathematical description of the volatility process has been an active topic of research for decades; however, driven by empirical estimates of the scaling behavior of volatility, a new paradigm has emerged, whereby paths of volatility are rougher than those of semimartingales. According to this perspective, volatility behaves essentially as a fractional Brownian motion with a small Hurst parameter. The first book to offer a comprehensive exploration of the subject, Rough Volatility contributes to the understanding and application of rough volatility models by equipping readers with the tools and insights needed to delve into the topic, exploring the motivation for rough volatility modeling, providing a toolbox for computation and practical implementation, and organizing the material to reflect the subject’s development and progression. This book is designed for researchers and graduate students in quantitative finance as well as quantitative analysts and finance professionals.

The Volatility Surface

Author: Jim Gatheral

Publisher: Wiley

Published: 2006-09-11

Total Pages: 0

ISBN-13: 9780471792512

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Praise for The Volatility Surface "I'm thrilled by the appearance of Jim Gatheral's new book The Volatility Surface. The literature on stochastic volatility is vast, but difficult to penetrate and use. Gatheral's book, by contrast, is accessible and practical. It successfully charts a middle ground between specific examples and general models--achieving remarkable clarity without giving up sophistication, depth, or breadth." --Robert V. Kohn, Professor of Mathematics and Chair, Mathematical Finance Committee, Courant Institute of Mathematical Sciences, New York University "Concise yet comprehensive, equally attentive to both theory and phenomena, this book provides an unsurpassed account of the peculiarities of the implied volatility surface, its consequences for pricing and hedging, and the theories that struggle to explain it." --Emanuel Derman, author of My Life as a Quant "Jim Gatheral is the wiliest practitioner in the business. This very fine book is an outgrowth of the lecture notes prepared for one of the most popular classes at NYU's esteemed Courant Institute. The topics covered are at the forefront of research in mathematical finance and the author's treatment of them is simply the best available in this form." --Peter Carr, PhD, head of Quantitative Financial Research, Bloomberg LP Director of the Masters Program in Mathematical Finance, New York University "Jim Gatheral is an acknowledged master of advanced modeling for derivatives. In The Volatility Surface he reveals the secrets of dealing with the most important but most elusive of financial quantities, volatility." --Paul Wilmott, author and mathematician "As a teacher in the field of mathematical finance, I welcome Jim Gatheral's book as a significant development. Written by a Wall Street practitioner with extensive market and teaching experience, The Volatility Surface gives students access to a level of knowledge on derivatives which was not previously available. I strongly recommend it." --Marco Avellaneda, Director, Division of Mathematical Finance Courant Institute, New York University "Jim Gatheral could not have written a better book." --Bruno Dupire, winner of the 2006 Wilmott Cutting Edge Research Award Quantitative Research, Bloomberg LP

A Course on Rough Paths

Author: Peter K. Friz

Publisher: Springer Nature

Published: 2020-05-27

Total Pages: 346

ISBN-13: 3030415562

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With many updates and additional exercises, the second edition of this book continues to provide readers with a gentle introduction to rough path analysis and regularity structures, theories that have yielded many new insights into the analysis of stochastic differential equations, and, most recently, stochastic partial differential equations. Rough path analysis provides the means for constructing a pathwise solution theory for stochastic differential equations which, in many respects, behaves like the theory of deterministic differential equations and permits a clean break between analytical and probabilistic arguments. Together with the theory of regularity structures, it forms a robust toolbox, allowing the recovery of many classical results without having to rely on specific probabilistic properties such as adaptedness or the martingale property. Essentially self-contained, this textbook puts the emphasis on ideas and short arguments, rather than aiming for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis and probability courses, with little more than Itô-integration against Brownian motion required for most of the text. From the reviews of the first edition: "Can easily be used as a support for a graduate course ... Presents in an accessible way the unique point of view of two experts who themselves have largely contributed to the theory" - Fabrice Baudouin in the Mathematical Reviews "It is easy to base a graduate course on rough paths on this ... A researcher who carefully works her way through all of the exercises will have a very good impression of the current state of the art" - Nicolas Perkowski in Zentralblatt MATH

Brazilian Derivatives and Securities

Author: Marcos C. S. Carreira

Publisher: Springer

Published: 2016-07-11

Total Pages: 303

ISBN-13: 113747727X

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The Brazilian financial markets operate in a very different way to G7 markets. Key differences include onshore and offshore markets, exponential rates, business days day-counts, and price formation from the futures markets (instead of the cash markets). This book provides a quantitative, applied guide to the offshore and onshore Brazilian markets, with a focus on the financial instruments unique to the region. It offers a comprehensive introduction to the key financial 'archaeology' in the Brazil context, exploring interest rates, FX and inflation and key differences from G7 market finance. It explores the core industry investment banking business in detail, from FX to interest rates and cash and inflation. Finally it introduces the region's unique financial instruments, as well as their pricing and risk management needs. Covering both introductory and complex topics, this book provides existing practitioners in Brazil, as well as those interested in becoming involved in these markets, everything they need to understand the market dynamics, risks, pricing and calibration of curves for all products currently available.

Large Deviations and Asymptotic Methods in Finance

Author: Peter K. Friz

Publisher: Springer

Published: 2015-06-16

Total Pages: 590

ISBN-13: 3319116053

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Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts. Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour. Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.

Rough Volatility

Author: Giulia Livieri

Publisher:

Published: 2017

Total Pages: 18

ISBN-13:

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It has been recently shown that spot volatilities can be very well modeled by rough stochastic volatility type dynamics. In such models, the log-volatility follows a fractional Brownian motion with Hurst parameter smaller than 1/2. This result has been established using high frequency volatility estimations from historical price data. We revisit this finding by studying implied volatility based approximations of the spot volatility. Using at-the-money options on the S&P500 index with short maturity, we are able to confirm that volatility is rough. The Hurst parameter found here, of order 0.3, is slightly larger than that usually obtained from historical data. This is easily explained from a smoothing effect due to the remaining time to maturity of the considered options.

Malliavin Calculus in Finance

Author: Elisa Alos

Publisher: CRC Press

Published: 2021-07-14

Total Pages: 350

ISBN-13: 1000403513

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Malliavin Calculus in Finance: Theory and Practice aims to introduce the study of stochastic volatility (SV) models via Malliavin Calculus. Malliavin calculus has had a profound impact on stochastic analysis. Originally motivated by the study of the existence of smooth densities of certain random variables, it has proved to be a useful tool in many other problems. In particular, it has found applications in quantitative finance, as in the computation of hedging strategies or the efficient estimation of the Greeks. The objective of this book is to offer a bridge between theory and practice. It shows that Malliavin calculus is an easy-to-apply tool that allows us to recover, unify, and generalize several previous results in the literature on stochastic volatility modeling related to the vanilla, the forward, and the VIX implied volatility surfaces. It can be applied to local, stochastic, and also to rough volatilities (driven by a fractional Brownian motion) leading to simple and explicit results. Features Intermediate-advanced level text on quantitative finance, oriented to practitioners with a basic background in stochastic analysis, which could also be useful for researchers and students in quantitative finance Includes examples on concrete models such as the Heston, the SABR and rough volatilities, as well as several numerical experiments and the corresponding Python scripts Covers applications on vanillas, forward start options, and options on the VIX. The book also has a Github repository with the Python library corresponding to the numerical examples in the text. The library has been implemented so that the users can re-use the numerical code for building their examples. The repository can be accessed here: https://bit.ly/2KNex2Y.

Options - 45 Years Since The Publication Of The Black-scholes-merton Model: The Gershon Fintech Center Conference

Author: David Gershon

Publisher: World Scientific

Published: 2022-12-21

Total Pages: 554

ISBN-13: 9811259151

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This book contains contributions by the best-known and consequential researchers who, over several decades, shaped the field of financial engineering. It presents a comprehensive and unique perspective on the historical development and the current state of derivatives research. The book covers classical and modern approaches to option pricing, realized and implied volatilities, classical and rough stochastic processes, and contingent claims analysis in corporate finance. The book is invaluable for students, academic researchers, and practitioners working with financial derivatives, market regulation, trading, risk management, and corporate decision-making.

Machine Learning for Risk Calculations

Author: Ignacio Ruiz

Publisher: John Wiley & Sons

Published: 2021-12-28

Total Pages: 471

ISBN-13: 1119791383

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State-of-the-art algorithmic deep learning and tensoring techniques for financial institutions The computational demand of risk calculations in financial institutions has ballooned and shows no sign of stopping. It is no longer viable to simply add more computing power to deal with this increased demand. The solution? Algorithmic solutions based on deep learning and Chebyshev tensors represent a practical way to reduce costs while simultaneously increasing risk calculation capabilities. Machine Learning for Risk Calculations: A Practitioner’s View provides an in-depth review of a number of algorithmic solutions and demonstrates how they can be used to overcome the massive computational burden of risk calculations in financial institutions. This book will get you started by reviewing fundamental techniques, including deep learning and Chebyshev tensors. You’ll then discover algorithmic tools that, in combination with the fundamentals, deliver actual solutions to the real problems financial institutions encounter on a regular basis. Numerical tests and examples demonstrate how these solutions can be applied to practical problems, including XVA and Counterparty Credit Risk, IMM capital, PFE, VaR, FRTB, Dynamic Initial Margin, pricing function calibration, volatility surface parametrisation, portfolio optimisation and others. Finally, you’ll uncover the benefits these techniques provide, the practicalities of implementing them, and the software which can be used. Review the fundamentals of deep learning and Chebyshev tensors Discover pioneering algorithmic techniques that can create new opportunities in complex risk calculation Learn how to apply the solutions to a wide range of real-life risk calculations. Download sample code used in the book, so you can follow along and experiment with your own calculations Realize improved risk management whilst overcoming the burden of limited computational power Quants, IT professionals, and financial risk managers will benefit from this practitioner-oriented approach to state-of-the-art risk calculation.

Derivatives in Financial Markets with Stochastic Volatility

Author: Jean-Pierre Fouque

Publisher: Cambridge University Press

Published: 2000-07-03

Total Pages: 222

ISBN-13: 9780521791632

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This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.