Description
Book SynopsisGiven 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.
Table of ContentsPART 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
