Fundamental Mathematical Topics for QF

 

Mathematical Topics

  1. Calculus & Series
    • Derivative Rules.
    • Taylor Series: Very important in quantitative finance—especially in the context of Itô–stochastic calculus.
    • Maclaurin Series & Euler’s Formula/Identity
  2. Combinatorics & Complex Numbers
    • Combinatorics: Techniques for counting (permutations, combinations, etc.).
    • Complex Numbers.
  3. Differential Equations
    • Ordinary and Partial Differential Equations: diferential equations, P.D.E., “Euler” classic solution methods for ODEs.
    • Fourier Series: often used in solving differential equations.
  4. Financial Models
    • Black–Scholes Model: Essential in option pricing and financial engineering.
  5. Linear Algebra in Finance
    • Matrix Multiplication: Represents mapping from one vector space to another.
    • Eigenvalues and Eigenvectors:
      • Expressed as: Av = λv
      • With the characteristic equation: (A – λI) = 0
      • Includes eigenvector normalization.
    • Additional Techniques:
      • Pivoting in a matrix
      • Tridiagonal (Banded) Matrices: Used for solving PDEs with implicit methods.
      • Gaussian Elimination.
      • Lagrange Methods: (Often applied in optimization and interpolation.)
      • Matrix Inversion: Not frequently done in finance due to computational complexity.

References & Suggested Reading

  • Primer for Financial Engineering by Dan Stefanica
  • Thomas’ Calculus
  • Numerical Analysis by Burden & Faires
  • Differential Equations and Boundary Value Problems by Cullen & Zill 

 

Topic Extension

Calculus

  • Functions and Limits
  • Differentiation and Integration
  • Complex Numbers
  • Functions of Several Variables

Differential Equations

  • First Order Equations
  • Second and Higher Order Equations

Linear Algebra

  • Matrices and Vectors
  • Systems of Linear Equations
  • Eigenvalues and Eigenvectors

Probability

  • Probability Distribution Function
  • Cumulative Distribution
  • Expectation Algebra
  • Key Discrete and Continuous Distributions (including the Normal Distribution)
  • Central Limit Theorem

Statistics

  • General Summary Statistics
  • Maximum Likelihood Estimator
  • Regression and Correlation

 


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