Introduction to the Mathematics of Finance (Graduate Studies in Mathematics, Vol. 72) (Graduate Studies in Mathematics, 72)

The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors' approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications. In particular, Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities. The study of global properties of geometric objects leads to the far-reaching development of topology, including topology and geometry of fiber bundles. Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics. Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory. Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology. The modern subject of mathematical finance has undergone considerable development, both in theory and practice, since the seminal work of Black and Scholes appeared a third of a century ago. This book is intended as an introduction to some elements of the theory that will enable students and researchers to go on to read more advanced texts and research papers. The book begins with the development of the basic ideas of hedging and pricing of European and American derivatives in the discrete (i.e., discrete time and discrete state) setting of binomial tree models. Then a general discrete finite market model is introduced, and the fundamental theorems of asset pricing are proved in this setting. Tools from probability such as conditional expectation, filtration, (super)martingale, equivalent martingale measure, and martingale representation are all used first in this simple discrete framework. This provides a bridge to the continuous (time and state) setting, which requires the additional concepts of Brownian motion and stochastic calculus. The simplest model in the continuous setting is the famous Black-Scholes model, for which pricing and hedging of European and American derivatives are developed. The book concludes with a description of the fundamental theorems for a continuous market model that generalizes the simple Black-Scholes model in several directions. The Modern Subject Of Mathematical Finance Has Undergone Considerable Development, Both In Theory And Practice, Since The Seminal Work Of Black And Scholes Appeared A Third Of A Century Ago. This Book Is Intended As An Introduction To Some Elements Of The Theory That Will Enable Students And Researchers To Go On To Read More Advanced Texts And Research Papers.--book Jacket. Chapter 1. Financial Markets And Derivatives Chapter 2. Binomial Model Chapter 3. Finite Market Model Chapter 4. Black-scholes Model Chapter 5. Multi-dimensional Black-scholes Model Appendix A. Conditional Expectation And $l^p$-spaces Appendix B. Discrete Time Stochastic Processes Appendix C. Continuous Time Stochastic Processes Appendix D. Brownian Motion And Stochastic Integration R.j. Williams. Includes Bibliographical References (p. 145-147) And Index.