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Markov Representation of the Heath-Jarrow-Morton Model

Markov Representation of the Heath-Jarrow-Morton Model
Author: Oren Cheyette
Publisher:
Total Pages: 13
Release: 1999
Genre:
ISBN:

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This paper provides a derivation of an arbitrage free approximation to any HJM model as a continuous time Markov model with a finite number of state variables. Arbitrage freedom is maintained exactly at the cost of approximating any particular term structure of volatility. Using a large enough set of state variables, any ``reasonable'' term structure of volatility can be approximated to any desired level of accuracy in a systematic fashion. The state variables fall into two sets. One set specifies the values of a finite number of forward rates of different maturities. The other set determines how to ``interpolate'' to other maturities; their values depend on the integrated covariances of the variables in the first set. A restriction of the model to volatility term structures dependent only on time and relative maturity is derived: a general volatility structure with this property is approximated as a sum of exponentials in the relative maturity, weighted by arbitrary functions of the time and the current term structure. A three factor, 19 state version of this model is fitted to 12 years of weekly US Treasury yield curve data. The results indicate that it is possible to achieve an accurate representation of the empirical term structure factors with only two or three exponential terms per factor. NOTE: This paper is available in TeX (with macros) or postscript form.


Consistency Problems for Heath-Jarrow-Morton Interest Rate Models

Consistency Problems for Heath-Jarrow-Morton Interest Rate Models
Author: Damir Filipovic
Publisher: Springer
Total Pages: 141
Release: 2004-11-02
Genre: Mathematics
ISBN: 354044548X

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Bond markets differ in one fundamental aspect from standard stock markets. While the latter are built up to a finite number of trade assets, the underlying basis of a bond market is the entire term structure of interest rates: an infinite-dimensional variable which is not directly observable. On the empirical side, this necessitates curve-fitting methods for the daily estimation of the term structure. Pricing models, on the other hand, are usually built upon stochastic factors representing the term structure in a finite-dimensional state space. Written for readers with knowledge in mathematical finance (in particular interest rate theory) and elementary stochastic analysis, this research monograph has threefold aims: to bring together estimation methods and factor models for interest rates, to provide appropriate consistency conditions and to explore some important examples.


The LIBOR Market Model in Practice

The LIBOR Market Model in Practice
Author: Dariusz Gatarek
Publisher: John Wiley & Sons
Total Pages: 290
Release: 2007-01-30
Genre: Business & Economics
ISBN: 0470060417

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The LIBOR Market Model (LMM) is the first model of interest rates dynamics consistent with the market practice of pricing interest rate derivatives and therefore it is widely used by financial institution for valuation of interest rate derivatives. This book provides a full practitioner's approach to the LIBOR Market Model. It adopts the specific language of a quantitative analyst to the largest possible level and is one of first books on the subject written entirely by quants. The book is divided into three parts - theory, calibration and simulation. New and important issues are covered, such as various drift approximations, various parametric and nonparametric calibrations, and the uncertain volatility approach to smile modelling; a version of the HJM model based on market observables and the duality between BGM and HJM models. Co-authored by Dariusz Gatarek, the 'G' in the BGM model who is internationally known for his work on LIBOR market models, this book offers an essential perspective on the global benchmark for short-term interest rates.


Paris-Princeton Lectures on Mathematical Finance 2003

Paris-Princeton Lectures on Mathematical Finance 2003
Author: Tomasz R. Bielecki
Publisher: Springer
Total Pages: 259
Release: 2004-08-30
Genre: Mathematics
ISBN: 3540444688

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The Paris-Princeton Lectures in Financial Mathematics, of which this is the second volume, will, on an annual basis, publish cutting-edge research in self-contained, expository articles from outstanding - established or upcoming! - specialists. The aim is to produce a series of articles that can serve as an introductory reference for research in the field. It arises as a result of frequent exchanges between the finance and financial mathematics groups in Paris and Princeton. This volume presents the following articles: "Hedging of Defaultable Claims" by T. Bielecki, M. Jeanblanc, and M. Rutkowski; "On the Geometry of Interest Rate Models" by T. Björk; "Heterogeneous Beliefs, Speculation and Trading in Financial Markets" by J.A. Scheinkman, and W. Xiong.


Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time
Author: Tomas Björk
Publisher: OUP Oxford
Total Pages: 600
Release: 2009-08-06
Genre: Business & Economics
ISBN: 0191610291

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The third edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton's fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. In this substantially extended new edition Bjork has added separate and complete chapters on the martingale approach to optimal investment problems, optimal stopping theory with applications to American options, and positive interest models and their connection to potential theory and stochastic discount factors. More advanced areas of study are clearly marked to help students and teachers use the book as it suits their needs.


Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time
Author: Tomas Bjork
Publisher: Oxford University Press, USA
Total Pages: 584
Release: 2020-01-16
Genre: Arbitrage
ISBN: 0198851618

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The fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Bjork has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. This edition includes an entirely new section presenting dynamic equilibrium theory, covering unit net supply endowments models and the Cox-Ingersoll-Ross equilibrium factor model. Providing two full treatments of arbitrage theory-the classical delta hedging approach and the modern martingale approach-this book is written so that these approaches can be studied independently of each other, thus providing the less mathematically-oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action. This textbook is a natural choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in the market.


Enterprise Risk Analytics for Capital Markets

Enterprise Risk Analytics for Capital Markets
Author: Raghurami Reddy Etukuru
Publisher: iUniverse
Total Pages: 433
Release: 2014-10-09
Genre: Business & Economics
ISBN: 1491744928

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While quantitative models can help predict the trends in Capital Markets, forecasts don't always hold up and can quickly cause things to spiral out of control and can lead to global risk. In order to reduce systemic risk, the G20 committed to a fundamental reform of the financial system, to correct the fault lines, and to rebuild the financial system as a safer, more resilient source of finance that better serves the real economy. This requires Financial Institutions to develop sound Risk Management practices. In straightforward language, you'll learn about key components of risk management, including risk knowledge, risk quantification, risk data management, risk data aggregation, risk architectures, risk analytics and reporting, risk regulation. You'll also get definitions explaining how different financial products work, mathematical formulas with explanations, and insights on different asset classes, different approaches to hedging, and much more. This book Enterprise Risk Analytics for Capital Markets will help whether you are just beginning a career in risk management or advancing your career with in risk management.


Mathematical Finance

Mathematical Finance
Author: Christian Fries
Publisher: John Wiley & Sons
Total Pages: 512
Release: 2007-10-19
Genre: Mathematics
ISBN: 9780470179772

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A balanced introduction to the theoretical foundations and real-world applications of mathematical finance The ever-growing use of derivative products makes it essential for financial industry practitioners to have a solid understanding of derivative pricing. To cope with the growing complexity, narrowing margins, and shortening life-cycle of the individual derivative product, an efficient, yet modular, implementation of the pricing algorithms is necessary. Mathematical Finance is the first book to harmonize the theory, modeling, and implementation of today's most prevalent pricing models under one convenient cover. Building a bridge from academia to practice, this self-contained text applies theoretical concepts to real-world examples and introduces state-of-the-art, object-oriented programming techniques that equip the reader with the conceptual and illustrative tools needed to understand and develop successful derivative pricing models. Utilizing almost twenty years of academic and industry experience, the author discusses the mathematical concepts that are the foundation of commonly used derivative pricing models, and insightful Motivation and Interpretation sections for each concept are presented to further illustrate the relationship between theory and practice. In-depth coverage of the common characteristics found amongst successful pricing models are provided in addition to key techniques and tips for the construction of these models. The opportunity to interactively explore the book's principal ideas and methodologies is made possible via a related Web site that features interactive Java experiments and exercises. While a high standard of mathematical precision is retained, Mathematical Finance emphasizes practical motivations, interpretations, and results and is an excellent textbook for students in mathematical finance, computational finance, and derivative pricing courses at the upper undergraduate or beginning graduate level. It also serves as a valuable reference for professionals in the banking, insurance, and asset management industries.