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Option Pricing in Fractional Brownian Markets

Option Pricing in Fractional Brownian Markets
Author: Stefan Rostek
Publisher: Springer Science & Business Media
Total Pages: 146
Release: 2009-04-28
Genre: Business & Economics
ISBN: 3642003311

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Mandelbrot and van Ness (1968) suggested fractional Brownian motion as a parsimonious model for the dynamics of ?nancial price data, which allows for dependence between returns over time. Starting with Rogers(1997) there is an ongoing dispute on the proper usage of fractional Brownian motion in option pricing theory. Problems arise because fractional Brownian motion is not a semimartingale and therefore “no arbitrage pricing” cannot be applied. While this is consensus, the consequences are not as clear. The orthodox interpretation is simply that fractional Brownian motion is an inadequate candidate for a price process. However, as shown by Cheridito (2003) any theoretical arbitrage opportunities disappear by assuming that market p- ticipants cannot react instantaneously. This is the point of departure of Rostek’s dissertation. He contributes to this research in several respects: (i) He delivers a thorough introduction to fr- tional integration calculus and uses the binomial approximation of fractional Brownianmotion to give the reader a ?rst idea of this special market setting.


Implied Hurst Exponent and Fractional Implied Volatility

Implied Hurst Exponent and Fractional Implied Volatility
Author: Kinrey Li
Publisher:
Total Pages: 17
Release: 2014
Genre:
ISBN:

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Two methods to derive Hurst exponent from option prices are proposed in this paper. They are based on fractional Brownian market setting. The first method is to use fractional Black-Scholes model inversely to derive implied Hurst exponent. The second one depends on no specific option pricing model. It is a model-free approach which is applicable as long as asset price evolves with on jumps. The difficulty in deriving implied information from fractional Brownian market is due to the fact that both Hurst exponent and volatility are unobservable. So they can be derived as a whole from single-period option prices, but can hardly be separated from each other. In this paper, a method that integrates option prices of different maturities is suggested to solve this problem. We also make a comparison between volatility in classical Brownian market and that in fractional Brownian market, which reveals that variance term structures are fitted differently in two settings. Based on this result, we suggest two potential applications of implied Hurst exponent in this paper.


Stochastic Calculus for Fractional Brownian Motion and Applications

Stochastic Calculus for Fractional Brownian Motion and Applications
Author: Francesca Biagini
Publisher: Springer Science & Business Media
Total Pages: 331
Release: 2008-02-17
Genre: Mathematics
ISBN: 1846287979

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The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.


Quantitative Analysis in Financial Markets

Quantitative Analysis in Financial Markets
Author: Marco Avellaneda
Publisher: World Scientific
Total Pages: 372
Release: 1999
Genre: Mathematics
ISBN: 9789810246938

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Contains lectures presented at the Courant Institute's Mathematical Finance Seminar.


Market Risk and Financial Markets Modeling

Market Risk and Financial Markets Modeling
Author: Didier Sornette
Publisher: Springer Science & Business Media
Total Pages: 260
Release: 2012-02-03
Genre: Business & Economics
ISBN: 3642279317

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The current financial crisis has revealed serious flaws in models, measures and, potentially, theories, that failed to provide forward-looking expectations for upcoming losses originated from market risks. The Proceedings of the Perm Winter School 2011 propose insights on many key issues and advances in financial markets modeling and risk measurement aiming to bridge the gap. The key addressed topics include: hierarchical and ultrametric models of financial crashes, dynamic hedging, arbitrage free modeling the term structure of interest rates, agent based modeling of order flow, asset pricing in a fractional market, hedge funds performance and many more.


Basic Black-Scholes: Option Pricing and Trading

Basic Black-Scholes: Option Pricing and Trading
Author: Timothy Falcon Crack
Publisher: Timothy Crack
Total Pages: 264
Release: 2014-08-05
Genre: Business & Economics
ISBN: 9780994103857

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THE AUTHOR: Dr. Crack studied PhD-level option pricing at MIT and Harvard Business School, taught undergraduate and MBA option pricing at Indiana University (winning many teaching awards), was an independent consultant to the New York Stock Exchange, worked as an asset management practitioner in London, and has traded options for over 15 years. This unique mixture of learning, teaching, consulting, practice, and trading is reflected in every page. SUMMARY OVERVIEW: This revised third edition of Basic Black-Scholes gives extremely clear explanations of Black-Scholes option pricing theory, and discusses direct applications of the theory to option trading. The presentation does not go far beyond basic Black-Scholes for three reasons: First, a novice need not go far beyond Black-Scholes to make money in the options markets; Second, all high-level option pricing theory is simply an extension of Black-Scholes; and Third, there already exist many books that look far beyond Black-Scholes without first laying the firm foundation given here. The trading advice does not go far beyond elementary call and put positions because more complex trades are simply combinations of these. WHAT MAKES THIS BOOK SPECIAL OR UNIQUE?: -It contains the basic intuition you need to trade options for the first time, or interview for an options job. -Honest advice about trading: there is no simple way to beat the markets, but if you have skill this advice can help make you money, and if you have no skill but still choose to trade, this advice can reduce your losses. -Full immersion treatment of transactions costs (T-costs). -Lessons from trading stated in simple terms. -Stylized facts about the markets (e.g., how to profit from reversals, when are T-costs highest/lowest during the trading day, implications of the market for corporate control, etc.). -How to apply (European-style) Black-Scholes pricing to the trading of (American-style) options. -Leverage through margin trading compared to leverage through options. -Black-Scholes option pricing code for the HP17B, HP19B, and HP12C. -Two downloadable spreadsheets. The first allows the user to forecast T-costs for option positions using simple models. The second allows the user to explore option sensitivities including the Greeks. -Practitioner Bloomberg Terminal screenshots to aid learning. -Simple discussion of continuously-compounded returns. -Introduction to "paratrading" (trading stocks side-by-side with options to generate additional profit). -Unique "regrets" treatment of early exercise decisions and trade-offs for American-style calls and puts. -Unique discussion of put-call parity and option pricing. -How to calculate Black-Scholes in your head in 10 seconds (also in Heard on The Street: Quantitative Questions from Wall Street Job Interviews). -Special attention to arithmetic Brownian motion with general pricing formulae and comparisons to Bachelier (1900) and Black-Scholes. -Careful attention to the impact of dividends in analytical American option pricing. -Dimensional analysis and the adequation formula (relating FX call and FX put prices through transformed Black-Scholes formulae). -Intuitive review of risk-neutral pricing/probabilities and how and why these are related to physical pricing/probabilities. -Careful distinction between the early Merton (non-risk-neutral) hedging-type argument and later Cox-Ross/Harrison-Kreps risk-neutral pricing -Simple discussion of Monte-Carlo methods in science and option pricing. -Simple interpretations of the Black-Scholes formula and PDE and implications for trading. -Careful discussion of conditional probabilities as they relate to Black-Scholes. -Intuitive treatment of high-level topics e.g., bond-numeraire interpretation of Black-Scholes (where N(d2) is P*(ITM)) versus the stock-numeraire interpretation (where N(d1) is P**(ITM)).


Recent Developments in Mathematical Finance

Recent Developments in Mathematical Finance
Author: Jiongmin Yong
Publisher: World Scientific
Total Pages: 286
Release: 2002
Genre: Business & Economics
ISBN: 9812799575

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The book deals with topics such as the pricing of various contingent claims within different frameworks, risk-sensitive problems, optimal investment, defaultable term structure, etc. It also reflects on some recent developments in certain important aspects of mathematical finance. Contents: Intensity-Based Valuation of Basket Credit Derivatives (T R Bielecki & M Rutkowski); Comonotonicity of Backward Stochastic Differential Equations (Z Chen & X Wang); Some Lookback Option Pricing Problems (X Guo); Optimal Investment and Consumption with Fixed and Proportional Transaction Costs (H Liu); Filtration Consistent Nonlinear Expectations (F Coquet et al.); A Theory of Volatility (A Savine); Discrete Time Markets with Transaction Costs (L Stettner); Options on Dividend Paying Stocks (R Beneder & T Vorst); Risk: From Insurance to Finance (H Yang); Arbitrage Pricing Systems in a Market Driven by an It Process (S Luo et al.); and other papers. Readership: Graduate students and researchers in mathematical finance and economics.


Recent Developments In Mathematical Finance - Proceedings Of The International Conference On Mathematical Finance

Recent Developments In Mathematical Finance - Proceedings Of The International Conference On Mathematical Finance
Author: Jiongmin Yong
Publisher: World Scientific
Total Pages: 286
Release: 2001-12-28
Genre: Mathematics
ISBN: 9814489697

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The book deals with topics such as the pricing of various contingent claims within different frameworks, risk-sensitive problems, optimal investment, defaultable term structure, etc. It also reflects on some recent developments in certain important aspects of mathematical finance.


Discrete-Time Approximations and Limit Theorems

Discrete-Time Approximations and Limit Theorems
Author: Yuliya Mishura
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 222
Release: 2021-10-25
Genre: Mathematics
ISBN: 3110652994

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The De Gruyter Series in Probability and Stochastics is devoted to the publication of high-level monographs and specialized graduate texts in any branch of modern probability theory and stochastics, along with their numerous applications in other parts of mathematics, physics and informatics, in economics and finance, and in the life sciences. The aim of the series is to present recent research results in the form of authoritative and comprehensive works that will serve the probability and stochastics community as basis for further research. Editorial Board Itai Benjamini, Weizmann Institute of Science, Israel Jean Bertoin, Universität Zürich, Switzerland Michel Ledoux, Université de Toulouse, France René L. Schilling, Technische Universität Dresden, Germany


Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities

Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities
Author: Anatoli? Vital?evich Svishchuk
Publisher: World Scientific
Total Pages: 326
Release: 2013
Genre: Business & Economics
ISBN: 9814440132

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Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities is devoted to the modeling and pricing of various kinds of swaps, such as those for variance, volatility, covariance, correlation, for financial and energy markets with different stochastic volatilities, which include CIR process, regime-switching, delayed, mean-reverting, multi-factor, fractional, Levy-based, semi-Markov and COGARCH(1,1). One of the main methods used in this book is change of time method. The book outlines how the change of time method works for different kinds of models and problems arising in financial and energy markets and the associated problems in modeling and pricing of a variety of swaps. The book also contains a study of a new model, the delayed Heston model, which improves the volatility surface fitting as compared with the classical Heston model. The author calculates variance and volatility swaps for this model and provides hedging techniques. The book considers content on the pricing of variance and volatility swaps and option pricing formula for mean-reverting models in energy markets. Some topics such as forward and futures in energy markets priced by multi-factor Levy models and generalization of Black-76 formula with Markov-modulated volatility are part of the book as well, and it includes many numerical examples such as S&P60 Canada Index, S&P500 Index and AECO Natural Gas Index.