Overview

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 economicsand 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.

Full Product Details

Author:   Belal Ehsan Baaquie
Publisher:   Springer Verlag, Singapore
Imprint:   Springer Verlag, Singapore
Edition:   1st ed. 2020
Weight:   0.694kg
ISBN:  

9789811566134

ISBN 10:   9811566135
Pages:   432
Publication Date:   12 August 2021
Audience:   College/higher education ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

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 to integrals 6.3 Definite and indefinite integrals 6.4 Applications in economics 6.5 Multiple Integrals 6.5.1 Change of variables 6.6 Gaussian integration 6.6.1 N-dimensional Gaussian integration 6.7 Problems   7 Differentiation 7.1 Introduction 7.2 Inverse of Integration 7.3 Rules of differentiation 7.4 Integration by parts 7.5 Taylor expansion 7.6 Minimum and maximum 7.6.1 Maximizing profit 7.7 Integration; change of variable 7.8 Partial derivatives 7.8.1 Chain rule; Jacobian 7.8.2 Polar coordinates; Gaussian integration 7.9 Hessian matrix: critical points 7.10 Constrained optimization: Lagrange multiplier 7.10.1 Interpretation of λc 7.11 Line integral; Exact and inexact differentials 7.12 Problems   8 Functional analysis 8.1 Dirac bracket and vector notation 8.2 Continuous basis states 8.3 Dirac delta function 8.4 Basis states for function space 8.5 Operators on function space 8.6 Gaussian kernel 8.7 Fourier Transform 8.8 Taylor expansion 8.9 Gaussian functional integration 8.10 Problems   9 Ordinary Differential Equations 9.1 Introduction 9.2 Separable differential equations 9.3 Linear differential equations 9.4 Bernoulli differential equation 9.5 Homegeneous differential equation 9.6 Second order linear differential equations 9.6.1 Single eigenvalue 9.7 Ricatti differential equation 9.8 Inhomogeneous second order differential equations 9.8.1 Green’s function 9.9 System of linear differential equations 9.10 Strum-Louisville theorem; special functions 9.11 Problems   PART IV : PROBABILITY THEORY 10 Random variables 10.1 Introduction: Risk 10.1.1 Example 10.2 Key ideas of probability 10.3 Discrete random variables 10.3.1 Bernoulli random variable 10.3.2 Binomial random variable 10.3.3 Poisson random variable 10.4 Continuous random variables 10.4.1 Uniform random variable 10.4.2 Exponential random variable 10.4.3 Normal (Gaussian) random variable 10.5 Problems   11 Probability distribution functions 11.0.1 Cumulative density 11.1 Axioms of probability theory 11.2 Joint probability density 11.3 Independent random variables 11.3.1 Law of large numbers 11.4 Correlated random variables 11.5 Marginal probability density 11.6 Conditional expectation value 11.6.1 Discrete random variable 11.6.2 Continuous random variables 11.7 Problems 12 Stochastic processes & Option pricing 12.1 Gaussian white noise 12.1.1 Integrals of White Noise 12.2 Ito Calculus 12.3 Lognormal Stock Price 12.4 Black-Scholes Equation; Hedged Portfolio 12.4.1 Assumptions in the Derivation of Black-Scholes 12.5 Risk-Neutral Martingale Solution of the Black-Scholes Equation 12.6 Black-Scholes-Schrodinger equation 12.7 Linear Langevin Equation 12.7.1 Random Paths 12.8 Problems   13 Appendix 13.1 Introduction 13.2 Integers 13.3 Real numbers 13.4 Cantor’s Diagonal Argument 13.5 Higher Order Infinities 13.6 Mathematical Logic

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Author Information

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.

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