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.

Hans Föllmer is Professor for Mathematics at the Humboldt University in Berlin, Germany.

Alexander Schied is Professor at the Institute for Mathematics of the Technical University Berlin, Germany.

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community.

Please submit any book proposals to Niels Jacob.

Titles in planning include

Flavia Smarazzo and Alberto Tesei, Measure Theory: Radon Measures, Young Measures, and Applications to Parabolic Problems (2019)
Elena Cordero and Luigi Rodino, Time-Frequency Analysis of Operators (2019)
Mark M. Meerschaert, Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational Models for Fractional Calculus, second edition (2020)
Mariusz Lemaczyk, Ergodic Theory: Spectral Theory, Joinings, and Their Applications (2020)
Marco Abate, Holomorphic Dynamics on Hyperbolic Complex Manifolds (2021)
Miroslava Anti, Joeri Van der Veken, and Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds of Almost Complex Spaces and Almost Product Spaces (2021)
Kai Liu, Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference Equations (2021)
Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi Miyanishi, Affine Space Fibrations (2022)