Mathematical Methods and Quantum Mathematics for Economics and Finance

1. Auflage 2020
ISBN: 978-981-1566-13-4
Verlag: Springer Nature Singapore

Buch, Englisch, 432 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 686 g

Given the rapid pace of development in economics and finance, a concise and up-to-date introduction to mathematical methods has become a prerequisite for all graduate students, even those not specializing in quantitative finance. This book offers an introductory text on mathematical methods for graduate students of economics and finance–and leading to the more advanced subject of quantum mathematics. The content is divided into five major sections: mathematical methods are covered in the first four sections, and can be taught in one semester. The book begins by focusing on the core subjects of linear algebra and calculus, before moving on to the more advanced topics of probability theory and stochastic calculus. Detailed derivations of the Black-Scholes and Merton equations are provided – in order to clarify the mathematical underpinnings of stochastic calculus. Each chapter of the first four sections includes a problem set, chiefly drawn from economics and finance. In turn, section five addresses quantum mathematics. The mathematical topics covered in the first four sections are sufficient for the study of quantum mathematics; Black-Scholes option theory and Merton’s theory of corporate debt are among topics analyzed using quantum mathematics.

Zielgruppe

Graduate

Autoren/Hrsg.

Baaquie, Belal Ehsan

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Interdisziplinär Finanz- und Versicherungsmathematik
• Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Wirtschaftstheorie, Wirtschaftsphilosophie
• Wirtschaftswissenschaften Betriebswirtschaft Wirtschaftsmathematik und -statistik

Inhaltsverzeichnis

PART I: INTRODUCTION

1 Introduction

1.1 Introduction

1.2 Elementary Algebra

1.2.1 Quadratic polynomial

1.3 Finite Series

1.4 Infinite Series

1.4.1 Cauchy convergence

1.5 Problems



2 Functions

2.1 Introduction

2.2 Exponential function

2.3 Demand and supply function

2.4 Option theory payoff

2.5 Interest rates; bonds

2.6 Problems



PART II: LINEAR ALGEBRA

3 Simultaneous linear equations

3.1 Introduction

3.2 Two commodities

3.3 Vectors
3.4 Basis vectors
3.4.1 Scalar product
3.5 Linear transformations; matrices

3.6 EN: N-dimensional linear vector space

3.7 Linear transformations of EN

3.8 Problems



4 Matrices

4.1 Introduction

4.2 Matrix multiplication

4.3 Properties of N × N matrices

4.4 System of linear equations

4.5 Determinant: 2 × 2 case

4.6 Inverse of a 2 × 2 matrix

4.7 Outer product; transpose

4.7.1 Transpose

4.8 Eigenvalues and eigenvectors

4.8.1 Spectral decomposition

4.9 Problems



5 Square matrices

5.1 Determinant: 3 × 3 case

5.2 Properties of determinants

5.3 N × N determinant

5.3.1 Inverse of a N × N matrix

5.4 Leontief input-output model

5.4.1 Hawkins-Simon condition

5.5 Symmetric matrices

5.6 Symmetric matrix: diagonalization

5.6.1 Functions of a symmetric matrix

5.7 Hermitian matrices

5.8 Diagonalizable matrices

5.8.1 Non-symmetric matrix

5.9 Change of Basis states

5.9.1 Symmetric matrix: change of basis

5.9.2 Hermitian matrix: change of basis

5.10 Problems



PART III: CALCULUS

6 Integration

6.1 Introduction

6.2 Sums leading

Über Autor(innen)

Prof. Belal Ehsan Baaquie holds a B.S. in Physics from Caltech and a Ph.D. in Theoretical Physics from Cornell University, USA. His main research interest is in the study and application of mathematical methods from quantum field theory. He has applied the mathematical formalism of field theory to finance and been a major contributor to the emerging field of quantum finance. His current focus is on developing the formalism of quantum finance and applying it to option pricing, corporate coupon bonds, and the theory of interest rates, as well as the study of equity, foreign exchange, and commodities. He is also applying methodologies from statistical mechanics and quantum field theory to the study of microeconomics and macroeconomics.