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Author: Imme van den Berg Publisher: World Scientific ISBN: 9789810243586 Category : Mathematics Languages : en Pages : 156
Book Description
There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level. The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude -- in the sense of nonstandard analysis -- are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.
Author: Imme van den Berg Publisher: World Scientific ISBN: 9789810243586 Category : Mathematics Languages : en Pages : 156
Book Description
There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level. The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude -- in the sense of nonstandard analysis -- are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.
Author: Imme Van Den Berg Publisher: World Scientific ISBN: 9814492779 Category : Business & Economics Languages : en Pages : 150
Book Description
There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level.The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude — in the sense of nonstandard analysis — are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.
Author: K.D. Stroyan Publisher: Elsevier ISBN: 0080960421 Category : Computers Languages : en Pages : 491
Book Description
This book gives a complete and elementary account of fundamental results on hyperfinite measures and their application to stochastic processes, including the *-finite Stieltjes sum approximation of martingale integrals. Many detailed examples, not found in the literature, are included. It begins with a brief chapter on tools from logic and infinitesimal (or non-standard) analysis so that the material is accessible to beginning graduate students.
Author: X. Sheldon Lin Publisher: John Wiley & Sons ISBN: 0471793205 Category : Mathematics Languages : en Pages : 224
Book Description
Incorporates the many tools needed for modeling and pricing infinance and insurance Introductory Stochastic Analysis for Finance and Insuranceintroduces readers to the topics needed to master and use basicstochastic analysis techniques for mathematical finance. The authorpresents the theories of stochastic processes and stochasticcalculus and provides the necessary tools for modeling and pricingin finance and insurance. Practical in focus, the book's emphasisis on application, intuition, and computation, rather thantheory. Consequently, the text is of interest to graduate students,researchers, and practitioners interested in these areas. While thetext is self-contained, an introductory course in probabilitytheory is beneficial to prospective readers. This book evolved from the author's experience as an instructor andhas been thoroughly classroom-tested. Following an introduction,the author sets forth the fundamental information and tools neededby researchers and practitioners working in the financial andinsurance industries: * Overview of Probability Theory * Discrete-Time stochastic processes * Continuous-time stochastic processes * Stochastic calculus: basic topics The final two chapters, Stochastic Calculus: Advanced Topics andApplications in Insurance, are devoted to more advanced topics.Readers learn the Feynman-Kac formula, the Girsanov's theorem, andcomplex barrier hitting times distributions. Finally, readersdiscover how stochastic analysis and principles are applied inpractice through two insurance examples: valuation of equity-linkedannuities under a stochastic interest rate environment andcalculation of reserves for universal life insurance. Throughout the text, figures and tables are used to help simplifycomplex theory and pro-cesses. An extensive bibliography opens upadditional avenues of research to specialized topics. Ideal for upper-level undergraduate and graduate students, thistext is recommended for one-semester courses in stochastic financeand calculus. It is also recommended as a study guide forprofessionals taking Causality Actuarial Society (CAS) and Societyof Actuaries (SOA) actuarial examinations.
Author: H. Jerome Keisler Publisher: American Mathematical Soc. ISBN: 0821822977 Category : Brownian motion processes Languages : en Pages : 197
Book Description
This monograph uses Robinson's infinitesimal (i.e., nonstandard) analysis to study stochastic integral equations with respect to a Brownian motion. By using a combination of standard and infinitesimal methods, we obtain new results about stochastic integral equations which can be stated in standard terms.
Author: Bruno Dinis Publisher: CRC Press ISBN: 1000012204 Category : Mathematics Languages : en Pages : 306
Book Description
Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers. Features Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system. Outlines complete algebraic rules for the neutrices and external numbers Conducts operational analysis of convergence and integration of functions known up to orders of magnitude Formalises a calculus of error propagation, covariant with algebraic operations Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox
Author: Cristoforo Sergio Bertuglia Publisher: OUP Oxford ISBN: 0191524441 Category : Mathematics Languages : en Pages : 402
Book Description
Covering a broad range of topics, this text provides a comprehensive survey of the modelling of chaotic dynamics and complexity in the natural and social sciences. Its attention to models in both the physical and social sciences and the detailed philosophical approach make this an unique text in the midst of many current books on chaos and complexity. Part 1 deals with the mathematical model as an instrument of investigation. The general meaning of modelling and, more specifically, questions concerning linear modelling are discussed. Part 2 deals with the theme of chaos and the origin of chaotic dynamics. Part 3 deals with the theme of complexity: a property of the systems and of their models which is intermediate between stability and chaos. Including an extensive index and bibliography along with numerous examples and simplified models, this is an ideal course text.
Author: Dmytro Gusak Publisher: Springer Science & Business Media ISBN: 0387878629 Category : Mathematics Languages : en Pages : 379
Book Description
Providing the necessary materials within a theoretical framework, this volume presents stochastic principles and processes, and related areas. Over 1000 exercises illustrate the concepts discussed, including modern approaches to sample paths and optimal stopping.
Author: Frederik S. Herzberg Publisher: Springer ISBN: 3642331491 Category : Mathematics Languages : en Pages : 125
Book Description
Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.
Author: Imme van den Berg
Author: Imme van den Berg
Author: Imme Van Den Berg
Author: I. van den Berg
Author: K.D. Stroyan
Author: X. Sheldon Lin
Author: H. Jerome Keisler
Author: Bruno Dinis
Author: Cristoforo Sergio Bertuglia
Author: Dmytro Gusak
Author: Frederik S. Herzberg