Synopses & Reviews
Mathematical finance has grown into a huge area of research which requires a lot of care and a large number of sophisticated mathematical tools. The subject draws upon quite difficult results from the theory of stochastic processes, stochastic calculus and differential equations, among others, which can be daunting for the beginning researcher. This book simultaneously introduces the financial methodology and the relevant mathematical tools in a style that is mathematically rigorous and yet accessible to practitioners and mathematicians alike. It interlaces financial concepts such as arbitrage opportunities, admissible strategies, contingent claims, option pricing and default risk with the mathematical theory of Brownian motion, diffusion processes, and Lévy processes. The authors proceed by successive generalisations with increasing complexity assuming some basic knowledge of probability theory. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes. The extensive bibliography comprises a wealth of important references and the author index enables readers quickly to locate where the reference is cited within the book, making this volume an invaluable tool both for students and for those at the forefront of research and practice.
From the reviews: "The aim of this book is to explain the fundamental concepts of continuous-time finance ... . This text presents an up-to-date account of the powerful interplay between the two areas, which is accessible yet mathematically rigorous. ... This book is an accessible overview of the relevant sophisticated topics in the theory of processes, serves as an excellent guide through the literature and will doubtless become established as a standard work of reference for practitioners and researchers in the area of mathematical finance." (Aleksandar Mijatović, Mathematical Reviews, Issue 2011 h)
Fills the gap between modern finance books and mathematical books.
Stochastic processes of common use in mathematical finance are presented throughout this book, which consists of eleven chapters, interlacing on the one hand financial concepts and instruments, such as arbitrage opportunities, admissible strategies, contingent claims, option pricing, default risk, ruin, and on the other hand, Brownian motion, diffusion processes, LA(c)vy processes, together with the basic properties of these processes. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes.
Only basic knowledge of probability theory is assumed; the book is organized so that the mathematical facts pertaining to a given financial question are gathered close to the study of that question.
Table of Contents
Part I Continuous Path Processes.- 1. Continuous Path Random Processes: Mathematical Prerequisites.- 2. Basic Concepts and Examples in Finance.- 3. Hitting Times: A Mix of Mathematics and Finance.- 4. Complements on Brownian Motion.- 5. Complements on Continuous Path Processes.- 6. A Special Family of Diffusions: Bessel Processes.- Part II: Jump Processes.- 7. Default Risk: An Enlargement of Filtration Approach.- 8. Poisson Processes and Ruin Theory.- 9. General Processes: Mathematical Facts.- 10. Mixed Processes.- 11. Lévy Processes.- Appendices.- References.- Index