Mathematical Methods for Financial Markets

Mathematical finance has grown into a huge area of research which requires a large number of sophisticated mathematical tools. 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. Stochastic processes of common use in Mathematical finance are presented throughout the book, which consists of eleven chapters, interlacing on 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, Levy 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.