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Author : Chloe Alice Lacombe
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (122 download)
Download or read book Large Deviations for Rough and Complete Stochastic Volatility Models written by Chloe Alice Lacombe and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Archil Gulisashvili
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)
Download or read book Multivariate Stochastic Volatility Models and Large Deviation Principles written by Archil Gulisashvili and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We establish a comprehensive sample path large deviation principle (LDP) for log-price processes associated with multivariate time-inhomogeneous stochastic volatility models. Examples of models for which the new LDP holds include Gaussian models, non-Gaussian fractional models, mixed models, models with reflection, and models in which the volatility process is a solution to a Volterra type stochastic integral equation. The sample path and small-noise LDPs for log-price processes are used to obtain large deviation style asymptotic formulas for the distribution function of the first exit time of a log-price process from an open set, multidimensional binary barrier options, call options, Asian options, and the implied volatility. Such formulas capture leading order asymptotics of the above-mentioned important quantities arising in the theory of stochastic volatility models. We also prove a sample path LDP for solutions to Volterra type stochastic integral equations with predictable coefficients depending on auxiliary stochastic processes.
Author : Archil Gulisashvili
Publisher :
ISBN 13 :
Total Pages : 40 pages
Book Rating : 4.:/5 (13 download)
Download or read book Gaussian Stochastic Volatility Models written by Archil Gulisashvili and published by . This book was released on 2019 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.
Author : Christian Bayer
Publisher : SIAM
ISBN 13 : 1611977789
Total Pages : 292 pages
Book Rating : 4.6/5 (119 download)
Download or read book Rough Volatility written by Christian Bayer and published by SIAM. This book was released on 2023-12-18 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Jin Feng
Publisher : American Mathematical Soc.
ISBN 13 : 0821841459
Total Pages : 426 pages
Book Rating : 4.8/5 (218 download)
Download or read book Large Deviations for Stochastic Processes written by Jin Feng and published by American Mathematical Soc.. This book was released on 2006 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to the results on large deviations for a class of stochastic processes. Following an introduction and overview, the material is presented in three parts. Part 1 gives necessary and sufficient conditions for exponential tightness that are analogous to conditions for tightness in the theory of weak convergence. Part 2 focuses on Markov processes in metric spaces. For a sequence of such processes, convergence of Fleming's logarithmically transformed nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence of linear semigroups in weak convergence. Viscosity solution methods provide applicable conditions for the necessary convergence. Part 3 discusses methods for verifying the comparison principle for viscosity solutions and applies the general theory to obtain a variety of new and known results on large deviations for Markov processes. In examples concerning infinite dimensional state spaces, new comparison principles are de
Author : Chloe Lacombe
Publisher :
ISBN 13 :
Total Pages : 28 pages
Book Rating : 4.:/5 (13 download)
Download or read book Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models written by Chloe Lacombe and published by . This book was released on 2019 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around-the-money.
Author : Peter K. Friz
Publisher : Springer
ISBN 13 : 3319116053
Total Pages : 590 pages
Book Rating : 4.3/5 (191 download)
Download or read book Large Deviations and Asymptotic Methods in Finance written by Peter K. Friz and published by Springer. This book was released on 2015-06-16 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : S. R. S. Varadhan
Publisher : American Mathematical Soc.
ISBN 13 : 082184086X
Total Pages : 114 pages
Book Rating : 4.8/5 (218 download)
Download or read book Large Deviations written by S. R. S. Varadhan and published by American Mathematical Soc.. This book was released on 2016-12-08 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. This book, which is based on a graduate course on large deviations at the Courant Institute, focuses on three concrete sets of examples: (i) diffusions with small noise and the exit problem, (ii) large time behavior of Markov processes and their connection to the Feynman-Kac formula and the related large deviation behavior of the number of distinct sites visited by a random walk, and (iii) interacting particle systems, their scaling limits, and large deviations from their expected limits. For the most part the examples are worked out in detail, and in the process the subject of large deviations is developed. The book will give the reader a flavor of how large deviation theory can help in problems that are not posed directly in terms of large deviations. The reader is assumed to have some familiarity with probability, Markov processes, and interacting particle systems.
Author : Peter K. Friz
Publisher : Springer Nature
ISBN 13 : 3030415562
Total Pages : 346 pages
Book Rating : 4.0/5 (34 download)
Download or read book A Course on Rough Paths written by Peter K. Friz and published by Springer Nature. This book was released on 2020-05-27 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Author : Antoine Jacquier
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (93 download)
Download or read book Implied Volatility Asymptotics Under Affine Stochastic Volatility Models written by Antoine Jacquier and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :
Publisher : Academic Press
ISBN 13 : 0080874576
Total Pages : 329 pages
Book Rating : 4.0/5 (88 download)
Download or read book Large Deviations written by and published by Academic Press. This book was released on 1989-06-21 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first four chapters of this volume are based on lectures given by Stroock at MIT in 1987. They form an introduction to the basic ideas of the theory of large deviations and make a suitable package on which to base a semester-length course for advanced graduate students with a strong background in analysis and some probability theory. A large selection of exercises presents important material and many applications. The last two chapters present various non-uniform results (Chapter 5) and outline the analytic approach that allows one to test and compare techniques used in previous chapters (Chapter 6).
Author : X Y Wang
Publisher : Independently Published
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.3/5 (938 download)
Download or read book Stochastic Calculus for Quants written by X Y Wang and published by Independently Published. This book was released on 2023-05-07 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Stochastic Calculus for Quants: Questions and Answers" is a comprehensive guide designed for aspiring and experienced quantitative analysts, offering an in-depth understanding of the key concepts and applications of stochastic calculus in the field of quantitative finance. This book is a valuable resource for anyone preparing for interviews, seeking to enhance their knowledge, or looking to excel in the quantitative finance industry. The book is organized into five main sections, ranging from basic to guru levels, addressing various stochastic calculus concepts and their relevance to quantitative finance. The author explains foundational concepts such as stochastic processes, Brownian motion, random walks, Ito's lemma, and martingales, providing clear definitions and real-life financial examples. Intermediate topics include European and American option pricing, the Black-Scholes-Merton model, Greeks in options trading, jump diffusion models, path dependency, stochastic volatility models, and mean reversion. The book also delves into advanced subjects like the Longstaff-Schwartz algorithm, local volatility, SABR model, LIBOR Market Model, copulas, principal component analysis, and the Heath-Jarrow-Morton framework. The expert section covers cutting-edge topics such as rough volatility, the Bergomi model, stochastic optimal control, affine processes, the multiple curve framework, large deviations theory, dynamic programming, and stochastic PDEs. Finally, the guru section explores the rough Bergomi model, signature-based methods, large-scale optimization, stochastic portfolio theory, functional Ito calculus, and stochastic optimal transport methods. "Stochastic Calculus for Quants: Questions and Answers" is a must-read for anyone seeking a deep understanding of stochastic calculus and its applications in the world of quantitative finance. With its progressive structure, practical examples, and extensive coverage of essential topics, this book will undoubtedly become an indispensable resource for professionals, academics, and students alike.
Author : Christian Bayer
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (13 download)
Download or read book Log-modulated Rough Stochastic Volatility Models written by Christian Bayer and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Ozan Akdogan
Publisher :
ISBN 13 :
Total Pages : 19 pages
Book Rating : 4.:/5 (13 download)
Download or read book Vol-of-Vol Expansion for (Rough) Stochastic Volatility Models written by Ozan Akdogan and published by . This book was released on 2019 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby we extend the scope of existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, explicit (in the sense of non-recursive) representations of the so-called push-down Malliavin weights that utilizes a precise understanding of the terms of this expansion.
Author : Christian Bayer
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (13 download)
Download or read book Deep Calibration of Rough Stochastic Volatility Models written by Christian Bayer and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparked by Alòs, León und Vives (2007); Fukasawa (2011, 2017); Gatheral, Jaisson und Rosenbaum (2018), so-called rough stochastic volatility models such as the rough Bergomi model by Bayer, Friz und Gatheral (2016) constitute the latest evolution in option price modeling. Unlike standard bivariate diffusion models such as Heston (1993), these non-Markovian models with fractional volatility drivers allow to parsimoniously recover key stylized facts of market implied volatility surfaces such as the exploding power-law behaviour of the at-the-money volatility skew as time to maturity goes to zero. Standard model calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, rendering calibration of many (rough) stochastic volatility models prohibitively expensive since there the map can often only be approximated by costly Monte Carlo (MC) simulations (Bennedsen, Lunde & Pakkanen, 2017; McCrickerd & Pakkanen, 2018; Bayer et al., 2016; Horvath, Jacquier & Muguruza, 2017). As a remedy, we propose to combine a standard Levenberg-Marquardt calibration routine with neural network regression, replacing expensive MC simulations with cheap forward runs of a neural network trained to approximate the implied volatility map. Numerical experiments confirm the high accuracy and speed of our approach.
Author : S. R. S. Varadhan
Publisher : SIAM
ISBN 13 : 0898711894
Total Pages : 74 pages
Book Rating : 4.8/5 (987 download)
Download or read book Large Deviations and Applications written by S. R. S. Varadhan and published by SIAM. This book was released on 1984-01-31 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many situations exist in which solutions to problems are represented as function space integrals. Such representations can be used to study the qualitative properties of the solutions and to evaluate them numerically using Monte Carlo methods. The emphasis in this book is on the behavior of solutions in special situations when certain parameters get large or small.
Author : Makoto Takahashi
Publisher : Springer Nature
ISBN 13 : 981990935X
Total Pages : 120 pages
Book Rating : 4.8/5 (199 download)
Download or read book Stochastic Volatility and Realized Stochastic Volatility Models written by Makoto Takahashi and published by Springer Nature. This book was released on 2023-04-18 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This treatise delves into the latest advancements in stochastic volatility models, highlighting the utilization of Markov chain Monte Carlo simulations for estimating model parameters and forecasting the volatility and quantiles of financial asset returns. The modeling of financial time series volatility constitutes a crucial aspect of finance, as it plays a vital role in predicting return distributions and managing risks. Among the various econometric models available, the stochastic volatility model has been a popular choice, particularly in comparison to other models, such as GARCH models, as it has demonstrated superior performance in previous empirical studies in terms of fit, forecasting volatility, and evaluating tail risk measures such as Value-at-Risk and Expected Shortfall. The book also explores an extension of the basic stochastic volatility model, incorporating a skewed return error distribution and a realized volatility measurement equation. The concept of realized volatility, a newly established estimator of volatility using intraday returns data, is introduced, and a comprehensive description of the resulting realized stochastic volatility model is provided. The text contains a thorough explanation of several efficient sampling algorithms for latent log volatilities, as well as an illustration of parameter estimation and volatility prediction through empirical studies utilizing various asset return data, including the yen/US dollar exchange rate, the Dow Jones Industrial Average, and the Nikkei 225 stock index. This publication is highly recommended for readers with an interest in the latest developments in stochastic volatility models and realized stochastic volatility models, particularly in regards to financial risk management.
