Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields in much the same way as the classical algebraic geometry arose from the study of polynomial equations with numerical coefficients. The first stage of the development of the theory is associated with its founder J. F. Ritt (1893 - 1951) and R. Cohn whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrew the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings. This book reflects the contemporary level of difference algebra; it contains a systematic study of partial difference algebraic structures and their applications, as well as the coverage of the classical theory of ordinary difference rings and field extensions. The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. It will be also of interest to researchers in computer algebra, theory of difference equations and equations of mathematical physics. The book is self-contained; it requires no prerequisites other than knowledge of basic algebraic concepts and mathematical maturity of an advanced undergraduate.

Mortgage Backed Securities (MBS) are among the most complex of all financial instruments. Analysis of MBS requires blending empirical analysis of borrower behavior with mathematical modeling of interest rates and home prices. Over the past 25 years, Davidson and Levin have been at the leading edge of MBS valuation and risk analysis. Mortgage Valuation Models: Embedded Options, Risk and Uncertainty is a detailed description of the sophisticated theories and advanced methods that the authors employ in real-world analysis of mortgage backed securities. Issues such as complexity, borrower options, uncertainty, and model risk play a central role in their approach to valuation of MBS. The book describes methods for modeling prepayments and defaults of borrowers. It explores closed form, backward induction and Monte Carlo valuation using the Option-Adjusted-Spread (OAS) approach, explains the origin of OAS and its relationship to model uncertainty. With reference to the classical CAPM and APT, the book advocates extending the concept of risk-neutrality to modeling home prices and borrower options, well beyond interest rates. The coverage spans the range of mortgage products from loans, TBA (to be announced) pass-through securities to subordinate tranches of subprime-mortgage securitizations and describes valuation methods for both agency and non-agency MBS including pricing new loans; Davidson and Levin put forth new approaches to prudent risk measurement, ranking, and decomposition that can help guide traders and risk managers. It reveals quantitative causes of the 2007-09 financial crisis and provides insights into the future of the US housing finance system and mortgage modeling. Despite the advances in mortgage modeling and valuation, this remains an ever-evolving field. Mortgage Valuation Models will serve as a foundation for the future development of models for mortgage-backed securities.

Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings.

The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. The book is self-contained; it requires no prerequisites other than the knowledge of basic algebraic concepts and a mathematical maturity of an advanced undergraduate.