Second Order Multiscale Stochastic Volatility Asymptotics

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Second Order Multiscale Stochastic Volatility Asymptotics

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Author : Jean-Pierre Fouque
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

Book Synopsis Second Order Multiscale Stochastic Volatility Asymptotics by : Jean-Pierre Fouque

Download or read book Second Order Multiscale Stochastic Volatility Asymptotics written by Jean-Pierre Fouque and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and S{ o}lna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -- one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that, to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.

Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives

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Author : Jean-Pierre Fouque
Publisher : Cambridge University Press
ISBN 13 : 113950245X
Total Pages : 456 pages
Book Rating : 4.1/5 (395 download)

Book Synopsis Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives by : Jean-Pierre Fouque

Download or read book Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives written by Jean-Pierre Fouque and published by Cambridge University Press. This book was released on 2011-09-29 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

American Options Under Stochastic Volatility

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Author : Ankush Agarwal
Publisher :
ISBN 13 :
Total Pages : 26 pages
Book Rating : 4.:/5 (13 download)

Book Synopsis American Options Under Stochastic Volatility by : Ankush Agarwal

Download or read book American Options Under Stochastic Volatility written by Ankush Agarwal and published by . This book was released on 2015 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate.

Algebraic Structures and Applications

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Author : Sergei Silvestrov
Publisher : Springer Nature
ISBN 13 : 3030418502
Total Pages : 976 pages
Book Rating : 4.0/5 (34 download)

Book Synopsis Algebraic Structures and Applications by : Sergei Silvestrov

Download or read book Algebraic Structures and Applications written by Sergei Silvestrov and published by Springer Nature. This book was released on 2020-06-18 with total page 976 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the latest advances in algebraic structures and applications, and focuses on mathematical concepts, methods, structures, problems, algorithms and computational methods important in the natural sciences, engineering and modern technologies. In particular, it features mathematical methods and models of non-commutative and non-associative algebras, hom-algebra structures, generalizations of differential calculus, quantum deformations of algebras, Lie algebras and their generalizations, semi-groups and groups, constructive algebra, matrix analysis and its interplay with topology, knot theory, dynamical systems, functional analysis, stochastic processes, perturbation analysis of Markov chains, and applications in network analysis, financial mathematics and engineering mathematics. The book addresses both theory and applications, which are illustrated with a wealth of ideas, proofs and examples to help readers understand the material and develop new mathematical methods and concepts of their own. The high-quality chapters share a wealth of new methods and results, review cutting-edge research and discuss open problems and directions for future research. Taken together, they offer a source of inspiration for a broad range of researchers and research students whose work involves algebraic structures and their applications, probability theory and mathematical statistics, applied mathematics, engineering mathematics and related areas.

Semiparametric Modeling of Implied Volatility

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Author : Matthias R. Fengler
Publisher : Springer Science & Business Media
ISBN 13 : 3540305912
Total Pages : 232 pages
Book Rating : 4.5/5 (43 download)

Book Synopsis Semiparametric Modeling of Implied Volatility by : Matthias R. Fengler

Download or read book Semiparametric Modeling of Implied Volatility written by Matthias R. Fengler and published by Springer Science & Business Media. This book was released on 2005-12-19 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers recent advances in the theory of implied volatility and refined semiparametric estimation strategies and dimension reduction methods for functional surfaces. The first part is devoted to smile-consistent pricing approaches. The second part covers estimation techniques that are natural candidates to meet the challenges in implied volatility surfaces. Empirical investigations, simulations, and pictures illustrate the concepts.

Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

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Author : Louis Paulot
Publisher :
ISBN 13 :
Total Pages : 27 pages
Book Rating : 4.:/5 (13 download)

Book Synopsis Asymptotic Implied Volatility at the Second Order with Application to the SABR Model by : Louis Paulot

Download or read book Asymptotic Implied Volatility at the Second Order with Application to the SABR Model written by Louis Paulot and published by . This book was released on 2016 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model.

Asymptotic Chaos Expansions in Finance

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Author : David Nicolay
Publisher : Springer
ISBN 13 : 9781447165057
Total Pages : 0 pages
Book Rating : 4.1/5 (65 download)

Book Synopsis Asymptotic Chaos Expansions in Finance by : David Nicolay

Download or read book Asymptotic Chaos Expansions in Finance written by David Nicolay and published by Springer. This book was released on 2014-12-05 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic instantaneous volatility models such as Heston, SABR or SV-LMM have mostly been developed to control the shape and joint dynamics of the implied volatility surface. In principle, they are well suited for pricing and hedging vanilla and exotic options, for relative value strategies or for risk management. In practice however, most SV models lack a closed form valuation for European options. This book presents the recently developed Asymptotic Chaos Expansions methodology (ACE) which addresses that issue. Indeed its generic algorithm provides, for any regular SV model, the pure asymptotes at any order for both the static and dynamic maps of the implied volatility surface. Furthermore, ACE is programmable and can complement other approximation methods. Hence it allows a systematic approach to designing, parameterising, calibrating and exploiting SV models, typically for Vega hedging or American Monte-Carlo. Asymptotic Chaos Expansions in Finance illustrates the ACE approach for single underlyings (such as a stock price or FX rate), baskets (indexes, spreads) and term structure models (especially SV-HJM and SV-LMM). It also establishes fundamental links between the Wiener chaos of the instantaneous volatility and the small-time asymptotic structure of the stochastic implied volatility framework. It is addressed primarily to financial mathematics researchers and graduate students, interested in stochastic volatility, asymptotics or market models. Moreover, as it contains many self-contained approximation results, it will be useful to practitioners modelling the shape of the smile and its evolution.

Analytically Tractable Stochastic Stock Price Models

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Author : Archil Gulisashvili
Publisher : Springer
ISBN 13 : 9783642312137
Total Pages : 0 pages
Book Rating : 4.3/5 (121 download)

Book Synopsis Analytically Tractable Stochastic Stock Price Models by : Archil Gulisashvili

Download or read book Analytically Tractable Stochastic Stock Price Models written by Archil Gulisashvili and published by Springer. This book was released on 2012-09-05 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Asymptotic analysis of stochastic stock price models is the central topic of the present volume. Special examples of such models are stochastic volatility models, that have been developed as an answer to certain imperfections in a celebrated Black-Scholes model of option pricing. In a stock price model with stochastic volatility, the random behavior of the volatility is described by a stochastic process. For instance, in the Hull-White model the volatility process is a geometric Brownian motion, the Stein-Stein model uses an Ornstein-Uhlenbeck process as the stochastic volatility, and in the Heston model a Cox-Ingersoll-Ross process governs the behavior of the volatility. One of the author's main goals is to provide sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions, and implied volatilities in various stochastic volatility models. The author also establishes sharp asymptotic formulas for the implied volatility at extreme strikes in general stochastic stock price models. The present volume is addressed to researchers and graduate students working in the area of financial mathematics, analysis, or probability theory. The reader is expected to be familiar with elements of classical analysis, stochastic analysis and probability theory.

Parameter Estimation in Stochastic Volatility Models

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Author : Jaya P. N. Bishwal
Publisher : Springer Nature
ISBN 13 : 3031038614
Total Pages : 634 pages
Book Rating : 4.0/5 (31 download)

Book Synopsis Parameter Estimation in Stochastic Volatility Models by : Jaya P. N. Bishwal

Download or read book Parameter Estimation in Stochastic Volatility Models written by Jaya P. N. Bishwal and published by Springer Nature. This book was released on 2022-08-06 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.

Stochastic Volatility Asymptotics for Optimal Subsistence Consumption and Investment with Bankruptcy

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Author : Kexin Chen
Publisher :
ISBN 13 :
Total Pages : 23 pages
Book Rating : 4.:/5 (13 download)

Book Synopsis Stochastic Volatility Asymptotics for Optimal Subsistence Consumption and Investment with Bankruptcy by : Kexin Chen

Download or read book Stochastic Volatility Asymptotics for Optimal Subsistence Consumption and Investment with Bankruptcy written by Kexin Chen and published by . This book was released on 2018 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt: Subsistence consumption and investment problems with bankruptcy are classic constrained stochastic optimal control problem in financial economic, where the consumption rate should be greater than a positive number and the investor faces a bankruptcy payment. We derive novel asymptotic solution to this problem under the fast mean-reverting stochastic volatility model. We rigorously prove that the zeroth-order approximation for the optimal pair of consumption and investment strategies leads to the first-order accuracy of the objective function. In addition, this zeroth-order suboptimal consumption-investment pair is asymptotically optimal in a class of admissible trading strategy pairs.

Derivatives in Financial Markets with Stochastic Volatility

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Author : Jean-Pierre Fouque
Publisher : Cambridge University Press
ISBN 13 : 9780521791632
Total Pages : 222 pages
Book Rating : 4.7/5 (916 download)

Book Synopsis Derivatives in Financial Markets with Stochastic Volatility by : Jean-Pierre Fouque

Download or read book Derivatives in Financial Markets with Stochastic Volatility written by Jean-Pierre Fouque and published by Cambridge University Press. This book was released on 2000-07-03 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.

Data Analysis and Related Applications, Volume 1

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Author : Konstantinos N. Zafeiris
Publisher : John Wiley & Sons
ISBN 13 : 1394165501
Total Pages : 484 pages
Book Rating : 4.3/5 (941 download)

Book Synopsis Data Analysis and Related Applications, Volume 1 by : Konstantinos N. Zafeiris

Download or read book Data Analysis and Related Applications, Volume 1 written by Konstantinos N. Zafeiris and published by John Wiley & Sons. This book was released on 2022-08-17 with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt: The scientific field of data analysis is constantly expanding due to the rapid growth of the computer industry and the wide applicability of computational and algorithmic techniques, in conjunction with new advances in statistical, stochastic and analytic tools. There is a constant need for new, high-quality publications to cover the recent advances in all fields of science and engineering. This book is a collective work by a number of leading scientists, computer experts, analysts, engineers, mathematicians, probabilists and statisticians who have been working at the forefront of data analysis and related applications. The chapters of this collaborative work represent a cross-section of current concerns, developments and research interests in the above scientific areas. The collected material has been divided into appropriate sections to provide the reader with both theoretical and applied information on data analysis methods, models and techniques, along with related applications.

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

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Author : Chloe Lacombe
Publisher :
ISBN 13 :
Total Pages : 28 pages
Book Rating : 4.:/5 (13 download)

Book Synopsis Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models by : Chloe Lacombe

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.

Handbook of Volatility Models and Their Applications

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Author : Luc Bauwens
Publisher : John Wiley & Sons
ISBN 13 : 1118272056
Total Pages : 566 pages
Book Rating : 4.1/5 (182 download)

Book Synopsis Handbook of Volatility Models and Their Applications by : Luc Bauwens

Download or read book Handbook of Volatility Models and Their Applications written by Luc Bauwens and published by John Wiley & Sons. This book was released on 2012-03-22 with total page 566 pages. Available in PDF, EPUB and Kindle. Book excerpt: A complete guide to the theory and practice of volatility models in financial engineering Volatility has become a hot topic in this era of instant communications, spawning a great deal of research in empirical finance and time series econometrics. Providing an overview of the most recent advances, Handbook of Volatility Models and Their Applications explores key concepts and topics essential for modeling the volatility of financial time series, both univariate and multivariate, parametric and non-parametric, high-frequency and low-frequency. Featuring contributions from international experts in the field, the book features numerous examples and applications from real-world projects and cutting-edge research, showing step by step how to use various methods accurately and efficiently when assessing volatility rates. Following a comprehensive introduction to the topic, readers are provided with three distinct sections that unify the statistical and practical aspects of volatility: Autoregressive Conditional Heteroskedasticity and Stochastic Volatility presents ARCH and stochastic volatility models, with a focus on recent research topics including mean, volatility, and skewness spillovers in equity markets Other Models and Methods presents alternative approaches, such as multiplicative error models, nonparametric and semi-parametric models, and copula-based models of (co)volatilities Realized Volatility explores issues of the measurement of volatility by realized variances and covariances, guiding readers on how to successfully model and forecast these measures Handbook of Volatility Models and Their Applications is an essential reference for academics and practitioners in finance, business, and econometrics who work with volatility models in their everyday work. The book also serves as a supplement for courses on risk management and volatility at the upper-undergraduate and graduate levels.

Mathematics of Finance

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Author : George Yin
Publisher : American Mathematical Soc.
ISBN 13 : 0821834126
Total Pages : 414 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Mathematics of Finance by : George Yin

Download or read book Mathematics of Finance written by George Yin and published by American Mathematical Soc.. This book was released on 2004 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains papers based on talks given at the first AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance held at Snowbird. This book includes such topics as modeling, estimation, optimization, control, and risk assessment and management. It is suitable for students interested in mathematical finance.

Hybrid Switching Diffusions

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Author : G. George Yin
Publisher : Springer Science & Business Media
ISBN 13 : 1441911057
Total Pages : 398 pages
Book Rating : 4.4/5 (419 download)

Book Synopsis Hybrid Switching Diffusions by : G. George Yin

Download or read book Hybrid Switching Diffusions written by G. George Yin and published by Springer Science & Business Media. This book was released on 2009-10-03 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book encompasses the study of hybrid switching di usion processes and their applications. The word \hybrid" signi es the coexistence of c- tinuous dynamics and discrete events, which is one of the distinct features of the processes under consideration. Much of the book is concerned with the interactions of the continuous dynamics and the discrete events. Our motivations for studying such processes originate from emerging and - isting applications in wireless communications, signal processing, queueing networks, production planning, biological systems, ecosystems, nancial engineering, and modeling, analysis, and control and optimization of lar- scale systems, under the in uence of random environments. Displaying mixture distributions, switching di usions may be described by the associated operators or by systems of stochastic di erential eq- tions together with the probability transition laws of the switching actions. We either have Markov-modulated switching di usions or processes with continuous state-dependent switching. The latter turns out to be much more challenging to deal with. Viewing the hybrid di usions as a number of di usions joined together by the switching process, they may be se- ingly not much di erent from their di usion counterpart. Nevertheless, the underlying problems become more di cult to handle, especially when the switching processes depend on continuous states. The di culty is due to the interaction of the discrete and continuous processes and the tangled and hybrid information pattern.

Implied Volatility Asymptotics Under Affine Stochastic Volatility Models

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Author : Antoine Jacquier
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (93 download)

Book Synopsis Implied Volatility Asymptotics Under Affine Stochastic Volatility Models by : Antoine Jacquier

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: