Mathematical Methods and Quantum Mathematics for Economics and Finance, 1st ed. 2020

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

2.5 Interest rates; bonds

2.6 Problems

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

5.6.1 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

6.1 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

7.4 Integration by parts

7.5 Taylor expansion

7.6 Minimum and maximum

7.7 Integration; change of variable

7.8 Partial derivatives

7.8.1 Chain rule; Jacobian

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

9.3 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

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

11.2 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

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

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.

Provides a concise and up-to-date introduction for all graduate students studying finance/quantitative finance

Includes a wealth of examples and exercises to promote student comprehension

Discusses the Black-Scholes equation to help students understand the mathematical underpinning of financial instruments