Long Range Stochastic Volatility with Two Scales in Option Pricing

Long Range Stochastic Volatility with Two Scales in Option Pricing

Author: Li Kong

Publisher:

Published: 2012

Total Pages: 79

ISBN-13: 9781124685823

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We exploit a general framework, a martingale approach method, to estimate the derivative price for different stochastic volatility models. This method is a very useful tool for handling non-markovian volatility models. With this method, we get the order of the approximation error by evaluating the orders of three error correction terms. We also summarize some challenges in using the martingale approach method to evaluate the derivative prices. We propose two stochastic volatility models. Our goal is to get the analytical solution for the derivative prices implied by the models. Another goal is to obtain an explicit model for the implied volatility and in particular how it depends on time to maturity. The first model we propose involves the increments of a standard Brownian Motion for a short time increment. The second model involves fractional Brownian Motion(fBm) and two scales. By using fBm in our model, we naturally incorporate a long-range dependence feature of the volatility process. In addition, the implied volatility corresponding to our second model capture a feature of the volatility as observed in the paper Maturity cycles in implied volatility by Fouque, which analyzed the S & P 500 option price data and observed that for long dated options the implied volatility is approximately affine in the reciprocal of time to maturity, while for short dated options the implied volatility is approximately affine in the reciprocal of square root of time to maturity. The leading term in the implied volatility also matches the case when we have time-dependent volatility in the Black-Scholes equation.


Option Pricing with Long Memory Stochastic Volatility Models

Option Pricing with Long Memory Stochastic Volatility Models

Author: Zhigang Tong

Publisher: LAP Lambert Academic Publishing

Published: 2013

Total Pages: 184

ISBN-13: 9783659346279

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It is now known that long memory stochastic volatility models can capture the well-documented evidence of volatility persistence. However, due to the complex structures of the long memory processes, the analytical formulas for option prices are not available yet. In this book, we propose two fractional continuous time stochastic volatility models which are built on the popular short memory stochastic volatility models. Using the tools from stochastic calculus, fractional calculus and Fourier transform, we derive the (approximate) analytical solutions for option prices. We also numerically study the effects of long memory on option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter. We also find that long memory models can accommodate the short term options and the decay of volatility skew better than the corresponding short memory models. These findings would appeal to the researchers and practitioners in the areas of quantitative finance.


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

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

Author: Jean-Pierre Fouque

Publisher: Cambridge University Press

Published: 2011-09-29

Total Pages: 456

ISBN-13: 113950245X

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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.


Complex Systems in Finance and Econometrics

Complex Systems in Finance and Econometrics

Author: Robert A. Meyers

Publisher: Springer Science & Business Media

Published: 2010-11-03

Total Pages: 919

ISBN-13: 1441977007

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Finance, Econometrics and System Dynamics presents an overview of the concepts and tools for analyzing complex systems in a wide range of fields. The text integrates complexity with deterministic equations and concepts from real world examples, and appeals to a broad audience.


Handbook of Quantitative Finance and Risk Management

Handbook of Quantitative Finance and Risk Management

Author: Cheng-Few Lee

Publisher: Springer Science & Business Media

Published: 2010-06-14

Total Pages: 1700

ISBN-13: 0387771174

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Quantitative finance is a combination of economics, accounting, statistics, econometrics, mathematics, stochastic process, and computer science and technology. Increasingly, the tools of financial analysis are being applied to assess, monitor, and mitigate risk, especially in the context of globalization, market volatility, and economic crisis. This two-volume handbook, comprised of over 100 chapters, is the most comprehensive resource in the field to date, integrating the most current theory, methodology, policy, and practical applications. Showcasing contributions from an international array of experts, the Handbook of Quantitative Finance and Risk Management is unparalleled in the breadth and depth of its coverage. Volume 1 presents an overview of quantitative finance and risk management research, covering the essential theories, policies, and empirical methodologies used in the field. Chapters provide in-depth discussion of portfolio theory and investment analysis. Volume 2 covers options and option pricing theory and risk management. Volume 3 presents a wide variety of models and analytical tools. Throughout, the handbook offers illustrative case examples, worked equations, and extensive references; additional features include chapter abstracts, keywords, and author and subject indices. From "arbitrage" to "yield spreads," the Handbook of Quantitative Finance and Risk Management will serve as an essential resource for academics, educators, students, policymakers, and practitioners.


Recent Developments in Data Science and Business Analytics

Recent Developments in Data Science and Business Analytics

Author: Madjid Tavana

Publisher: Springer

Published: 2018-03-27

Total Pages: 494

ISBN-13: 3319727451

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This edited volume is brought out from the contributions of the research papers presented in the International Conference on Data Science and Business Analytics (ICDSBA- 2017), which was held during September 23-25 2017 in ChangSha, China. As we all know, the field of data science and business analytics is emerging at the intersection of the fields of mathematics, statistics, operations research, information systems, computer science and engineering. Data science and business analytics is an interdisciplinary field about processes and systems to extract knowledge or insights from data. Data science and business analytics employ techniques and theories drawn from many fields including signal processing, probability models, machine learning, statistical learning, data mining, database, data engineering, pattern recognition, visualization, descriptive analytics, predictive analytics, prescriptive analytics, uncertainty modeling, big data, data warehousing, data compression, computer programming, business intelligence, computational intelligence, and high performance computing among others. The volume contains 55 contributions from diverse areas of Data Science and Business Analytics, which has been categorized into five sections, namely: i) Marketing and Supply Chain Analytics; ii) Logistics and Operations Analytics; iii) Financial Analytics. iv) Predictive Modeling and Data Analytics; v) Communications and Information Systems Analytics. The readers shall not only receive the theoretical knowledge about this upcoming area but also cutting edge applications of this domains.


Mathematics of Finance

Mathematics of Finance

Author: George Yin

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 414

ISBN-13: 0821834126

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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.


The Heston Model and its Extensions in Matlab and C#

The Heston Model and its Extensions in Matlab and C#

Author: Fabrice D. Rouah

Publisher: John Wiley & Sons

Published: 2013-08-01

Total Pages: 437

ISBN-13: 1118695178

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Tap into the power of the most popular stochastic volatility model for pricing equity derivatives Since its introduction in 1993, the Heston model has become a popular model for pricing equity derivatives, and the most popular stochastic volatility model in financial engineering. This vital resource provides a thorough derivation of the original model, and includes the most important extensions and refinements that have allowed the model to produce option prices that are more accurate and volatility surfaces that better reflect market conditions. The book's material is drawn from research papers and many of the models covered and the computer codes are unavailable from other sources. The book is light on theory and instead highlights the implementation of the models. All of the models found here have been coded in Matlab and C#. This reliable resource offers an understanding of how the original model was derived from Ricatti equations, and shows how to implement implied and local volatility, Fourier methods applied to the model, numerical integration schemes, parameter estimation, simulation schemes, American options, the Heston model with time-dependent parameters, finite difference methods for the Heston PDE, the Greeks, and the double Heston model. A groundbreaking book dedicated to the exploration of the Heston model—a popular model for pricing equity derivatives Includes a companion website, which explores the Heston model and its extensions all coded in Matlab and C# Written by Fabrice Douglas Rouah a quantitative analyst who specializes in financial modeling for derivatives for pricing and risk management Engaging and informative, this is the first book to deal exclusively with the Heston Model and includes code in Matlab and C# for pricing under the model, as well as code for parameter estimation, simulation, finite difference methods, American options, and more.