Martingales and Financial Mathematics in Discrete Time

Benoîte de Saporta, Mounir Zili

• Publisher: Wiley-ISTE
• Year: 2021
• ISBN:
  9781119885023
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This book is entirely devoted to discrete time and provides a detailed introduction to the construction of the rigorous mathematical tools required for the evaluation of options in financial markets. Both theoretical and practical aspects are explored through multiple examples and exercises, for which complete solutions are provided. Particular attention is paid to the Cox, Ross and Rubinstein model in discrete time. The book offers a combination of mathematical teaching and numerous exercises for wide appeal. It is a useful reference for students at the master's or doctoral level who are specializing in applied mathematics or finance as well as teachers, researchers in the field of economics or actuarial science, or professionals working in the various financial sectors. Martingales and Financial Mathematics in Discrete Time is also for anyone who may be interested in a rigorous and accessible mathematical construction of the tools and concepts used in financial mathematics, or in the application of the martingale theory in finance

1
Elementary Probabilities and an Introduction to Stochastic Processes

This chapter reviews the basic concepts related to probability and random variables which will be useful for the rest of this text. For a more detailed explanation as well as demonstrations, the readers may refer to [BAR 07, DAC 82, FOA 03, OUV 08, OUV 09] in French and [BIL 12, CHU 01, DUR 10, KAL 02, SHI 00] in English. The readers who are already familiar with these concepts may proceed straight to section 1.3, which introduces the concept of stochastic processes.

This chapter begins with a brief summary of the concepts of a σ-algebra in section 1.1. These concepts will help in understanding the construction of the properties of conditional expectation in Chapter 2. We then study the chief definitions and properties of random variables and their distribution in section 1.2. There is an emphasis on discrete random variables as this entire book essentially studies discrete cases. Section 1.3 defines a stochastic process, which is the main subject studied in this book...