Martingales And Financial Mathematics In Discrete Time

This book is entirely devoted to discrete time and provides a detailed introduction to the construction of the rigorous mathematical tools required for the evaluation of options in financial markets. Both theoretical and practical aspects are explored through multiple examples and exercises, for which complete solutions are provided. Particular attention is paid to the Cox, Ross and Rubinstein model in discrete time. The book offers a combination of mathematical teaching and numerous exercises for wide appeal. It is a useful reference for students at the master’s or doctoral level who are specializing in applied mathematics or finance as well as teachers, researchers in the field of economics or actuarial science, or professionals working in the various financial sectors. Martingales and Financial Mathematics in Discrete Time is also for anyone who may be interested in a rigorous and accessible mathematical construction of the tools and concepts used in financial mathematics, or in the application of the martingale theory in finance

This book is an introduction to financial mathematics. The first part of the book studies a simple one-period model which serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of risk. In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Such models are typically incomplete: They involve intrinsic risks which cannot be hedged away completely. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk. In addition to many corrections and improvements, this second edition contains several new sections, including a systematic discussion of law-invariant risk measures and of the connections between American options, superhedging, and dynamic risk measures.

This self-contained book presents the theory underlying the valuation of derivative financial instruments, which is becoming a standard part of the professional toolbox in the financial industry. It provides great insight into the underlying economic ideas in a very readable form, putting the reader in an excellent position to proceed to the more general continuous-time theory.

A comprehensive and self-contained treatment of the theory and practice of option pricing. The role of martingale methods in financial modeling is exposed. The emphasis is on using arbitrage-free models already accepted by the market as well as on building the new ones. Standard calls and puts together with numerous examples of exotic options such as barriers and quantos, for example on stocks, indices, currencies and interest rates are analysed. The importance of choosing a convenient numeraire in price calculations is explained. Mathematical and financial language is used so as to bring mathematicians closer to practical problems of finance and presenting to the industry useful maths tools.

Since the publication of the first edition of this book, the area of mathematical finance has grown rapidly, with financial analysts using more sophisticated mathematical concepts, such as stochastic integration, to describe the behavior of markets and to derive computing methods. Maintaining the lucid style of its popular predecessor, Introduction

The book has been tested and refined through years of classroom teaching experience. With an abundance of examples, problems, and fully worked out solutions, the text introduces the financial theory and relevant mathematical methods in a mathematically rigorous yet engaging way. This textbook provides complete coverage of discrete-time financial models that form the cornerstones of financial derivative pricing theory. Unlike similar texts in the field, this one presents multiple problem-solving approaches, linking related comprehensive techniques for pricing different types of financial derivatives. Key features: In-depth coverage of discrete-time theory and methodology. Numerous, fully worked out examples and exercises in every chapter. Mathematically rigorous and consistent yet bridging various basic and more advanced concepts. Judicious balance of financial theory, mathematical, and computational methods. Guide to Material. This revision contains: Almost 200 pages worth of new material in all chapters. A new chapter on elementary probability theory. An expanded the set of solved problems and additional exercises. Answers to all exercises. This book is a comprehensive, self-contained, and unified treatment of the main theory and application of mathematical methods behind modern-day financial mathematics. Table of Contents List of Figures and Tables Preface I Introduction to Pricing and Management of Financial Securities 1 Mathematics of Compounding 2 Primer on Pricing Risky Securities 3 Portfolio Management 4 Primer on Derivative Securities II Discrete-Time Modelling 5 Single-Period Arrow–Debreu Models 6 Introduction to Discrete-Time Stochastic Calculus 7 Replication and Pricing in the Binomial Tree Model 8 General Multi-Asset Multi-Period Model Appendices A Elementary Probability Theory B Glossary of Symbols and Abbreviations C Answers and Hints to Exercises References Index Biographies Giuseppe Campolieti is Professor of Mathematics at Wilfrid Laurier University in Waterloo, Canada. He has been Natural Sciences and Engineering Research Council postdoctoral research fellow and university research fellow at the University of Toronto. In 1998, he joined the Masters in Mathematical Finance as an instructor and later as an adjunct professor in financial mathematics until 2002. Dr. Campolieti also founded a financial software and consulting company in 1998. He joined Laurier in 2002 as Associate Professor of Mathematics and as SHARCNET Chair in Financial Mathematics. Roman N. Makarov is Associate Professor and Chair of Mathematics at Wilfrid Laurier University. Prior to joining Laurier in 2003, he was an Assistant Professor of Mathematics at Siberian State University of Telecommunications and Informatics and a senior research fellow at the Laboratory of Monte Carlo Methods at the Institute of Computational Mathematics and Mathematical Geophysics in Novosibirsk, Russia.

The book is the first monograph on this highly important subject.

Risk Neutral Pricing and Financial Mathematics: A Primer provides a foundation to financial mathematics for those whose undergraduate quantitative preparation does not extend beyond calculus, statistics, and linear math. It covers a broad range of foundation topics related to financial modeling, including probability, discrete and continuous time and space valuation, stochastic processes, equivalent martingales, option pricing, and term structure models, along with related valuation and hedging techniques. The joint effort of two authors with a combined 70 years of academic and practitioner experience, Risk Neutral Pricing and Financial Mathematics takes a reader from learning the basics of beginning probability, with a refresher on differential calculus, all the way to Doob-Meyer, Ito, Girsanov, and SDEs. It can also serve as a useful resource for actuaries preparing for Exams FM and MFE (Society of Actuaries) and Exams 2 and 3F (Casualty Actuarial Society). Includes more subjects than other books, including probability, discrete and continuous time and space valuation, stochastic processes, equivalent martingales, option pricing, term structure models, valuation, and hedging techniques Emphasizes introductory financial engineering, financial modeling, and financial mathematics Suited for corporate training programs and professional association certification programs