Quantitative Finance

Quantitative finance is a specialized field that applies advanced mathematical, statistical, and computational models to analyze financial markets and make investment and risk management decisions. Often referred to as financial engineering, this discipline is primarily concerned with the pricing of complex financial instruments like derivatives, the development of algorithmic trading strategies, and the precise measurement of financial risk. It serves as the analytical engine for institutions such as investment banks, hedge funds, and asset management firms, enabling them to navigate and capitalize on the complexities of the global financial system.

1. Foundations of Quantitative Finance
   1. Mathematical Prerequisites
      1. Calculus
         1. Limits and Continuity
            1. Definition of Limits
            2. Properties of Limits
            3. Continuity of Functions
            4. Intermediate Value Theorem
         2. Differentiation
            1. Rules of Differentiation
            2. Partial Derivatives
            3. Chain Rule
            4. Applications to Finance
         3. Integration
            1. Definite and Indefinite Integrals
            2. Integration by Parts
            3. Applications to Finance
         4. Taylor Series Expansions
            1. Taylor and Maclaurin Series
            2. Convergence of Series
            3. Applications in Approximations
         5. Multivariable Calculus
            1. Functions of Several Variables
            2. Gradient Vector
            3. Divergence and Curl
            4. Multiple Integrals
            5. Jacobian Matrices
            6. Hessian Matrices
         6. Differential Equations
            1. Ordinary Differential Equations
            2. Partial Differential Equations
            3. Analytical Solution Methods
            4. Numerical Solution Methods
            5. Applications in Option Pricing
      2. Linear Algebra
         1. Vectors and Matrices
            1. Vector Spaces
            2. Matrix Operations
            3. Linear Transformations
            4. Matrix Rank
            5. Matrix Inverse
         2. Eigenvalues and Eigenvectors
            1. Characteristic Equation
            2. Diagonalization
            3. Applications in Risk Analysis
            4. Applications in Portfolio Analysis
         3. Matrix Decompositions
            1. LU Decomposition
            2. Cholesky Decomposition
            3. Singular Value Decomposition
         4. Principal Component Analysis
            1. Covariance Matrix
            2. Eigenvalue Decomposition in PCA
            3. Dimensionality Reduction
            4. Applications in Risk Factors
      3. Probability Theory
         1. Probability Spaces and Measures
            1. Sample Spaces
            2. Events and Sigma-Algebras
            3. Probability Axioms
         2. Random Variables and Distributions
            1. Discrete Random Variables
            2. Continuous Random Variables
            3. Probability Mass Functions
            4. Probability Density Functions
            5. Cumulative Distribution Functions
            6. Normal Distribution
            7. Binomial Distribution
            8. Poisson Distribution
            9. Exponential Distribution
            10. Lognormal Distribution
         3. Expectation and Moments
            1. Expected Value
            2. Variance and Standard Deviation
            3. Skewness
            4. Kurtosis
            5. Covariance and Correlation
         4. Conditional Probability and Independence
            1. Conditional Probability
            2. Law of Total Probability
            3. Bayes' Theorem
            4. Independence of Events
         5. Limit Theorems
            1. Weak Law of Large Numbers
            2. Strong Law of Large Numbers
            3. Central Limit Theorem
            4. Applications in Finance
   2. Statistical and Econometric Methods
      1. Descriptive Statistics
         1. Measures of Central Tendency
            1. Mean
            2. Median
            3. Mode
         2. Measures of Dispersion
            1. Variance
            2. Standard Deviation
            3. Range
            4. Interquartile Range
         3. Distribution Shape
            1. Skewness
            2. Kurtosis
         4. Data Visualization
            1. Histograms
            2. Boxplots
            3. Scatterplots
      2. Inferential Statistics
         1. Sampling Methods
            1. Random Sampling
            2. Stratified Sampling
            3. Systematic Sampling
         2. Estimation Theory
            1. Point Estimation
            2. Interval Estimation
            3. Maximum Likelihood Estimation
         3. Hypothesis Testing
            1. Null and Alternative Hypotheses
            2. Type I and Type II Errors
            3. p-values and Significance Levels
            4. t-test
            5. Chi-square Test
            6. ANOVA
         4. Confidence Intervals
            1. Construction Methods
            2. Interpretation
            3. Applications in Finance
      3. Regression Analysis
         1. Linear Regression
            1. Model Specification
            2. Least Squares Estimation
            3. Goodness-of-Fit Measures
            4. Residual Analysis
         2. Multiple Regression
            1. Multicollinearity
            2. Model Selection Criteria
            3. Coefficient Interpretation
            4. Diagnostic Testing
            5. Assumption Validation
      4. Time Series Analysis
         1. Time Series Components
            1. Trend
            2. Seasonality
            3. Noise
         2. Stationarity
            1. Definition and Testing
            2. Unit Root Tests
            3. Differencing Methods
            4. Detrending Methods
         3. Autocorrelation Analysis
            1. Autocorrelation Function
            2. Partial Autocorrelation Function
         4. ARMA and ARIMA Models
            1. Autoregressive Models
            2. Moving Average Models
            3. ARMA Model Identification
            4. ARIMA Model Estimation
            5. Forecasting Methods
         5. Cointegration
            1. Definition and Concepts
            2. Cointegration Testing
            3. Error Correction Models
   3. Core Financial Concepts
      1. Financial Markets and Instruments
         1. Equity Markets
            1. Common Stocks
            2. Preferred Stocks
            3. Stock Exchanges
            4. Trading Mechanisms
         2. Fixed Income Markets
            1. Government Bonds
            2. Corporate Bonds
            3. Bond Pricing
            4. Yield Calculations
            5. Duration
            6. Convexity
         3. Foreign Exchange Markets
            1. Spot FX Markets
            2. Forward FX Markets
            3. Exchange Rate Quotations
         4. Commodity Markets
            1. Commodity Contracts
            2. Pricing Mechanisms
            3. Hedging Strategies
         5. Derivatives Markets
            1. Forward Contracts
            2. Futures Contracts
            3. Options Contracts
            4. Swap Contracts
            5. Hedging Applications
            6. Speculation Applications
      2. Fundamental Financial Principles
         1. Time Value of Money
            1. Present Value
            2. Future Value
            3. Discounting
            4. Compounding
            5. Annuities
            6. Perpetuities
         2. Risk and Return
            1. Risk Measurement
            2. Expected Return
            3. Risk-Return Tradeoff
            4. Volatility
         3. Arbitrage
            1. Definition and Concepts
            2. Arbitrage Opportunities
            3. Market Examples
         4. Market Efficiency
            1. Efficient Market Hypothesis
            2. Weak Form Efficiency
            3. Semi-Strong Form Efficiency
            4. Strong Form Efficiency