Enzan

Study Overview

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Curriculum

Level 01Probability and Statistical Foundations14 topics
Mathematical Language, Functions, and ProofLessons available
  1. Statements, Sets, and Financial Notation
  2. Functions, Inverses, and Units
  3. Exponentials, Logarithms, and Summation
  4. Limits, Derivatives, Approximations, and Proof Habits
Probability Spaces, Events, and CountingLessons available
  1. Experiments and Event Spaces
  2. Counting and Combinatorial Probability
  3. Complement, Union, and Inclusion-Exclusion
  4. Independence and Dependence
  5. Conditional Probability Preview
Conditional Probability and Bayesian InferenceLessons available
  1. Conditional Probability as Restricted Information
  2. The Law of Total Probability
  3. Deriving Bayes' Rule
  4. Odds and Likelihood Ratios
  5. Bayesian Model Criticism
Discrete Random Variables and DistributionsLessons available
  1. Random Variables and Probability Mass
  2. Bernoulli and Binomial Models
  3. Geometric Waiting Times
  4. Poisson Counts and Model Limits
Continuous Distributions and TransformationsLessons available
  1. Densities and Cumulative Probability
  2. Uniform, Exponential, and Normal Families
  3. Expectations by Integration
  4. Transforms and Lognormality
  5. Tail Behavior and Distribution Criticism
Expectation, Linearity, and PayoffsLessons available
  1. Expectation as a Weighted Average
  2. Linearity Without Independence
  3. Expected Utility Versus Expected Money
  4. Model Error and Moment Existence
Variance, Covariance, and Diversification BasicsLessons available
  1. Dispersion Around the Mean
  2. Scaling Variance
  3. Covariance and Independence
  4. Variance of Combined Positions
Conditional Expectation and Decomposition LawsLessons available
  1. Conditional Mean as a Function of Information
  2. The Tower Property
  3. Conditioning and Forecast Errors
  4. Law of Total Variance
  5. Information and Conditional Independence
  6. Optional Mathematical View of Conditioning
Joint Distributions, Correlation, and Multivariate ProbabilityLessons available
  1. Joint, Marginal, and Conditional Laws
  2. Covariance and Correlation
  3. Multivariate Normal Modeling
  4. Dependence Beyond Linear Correlation
Convergence, LLN, and the Central Limit TheoremLessons available
  1. Types of Convergence
  2. Law of Large Numbers
  3. Central Limit Theorem
  4. When CLT Reasoning Breaks
  5. Asymptotics Versus Finite Samples
Sampling, Standard Errors, and Hypothesis TestingLessons available
  1. Sampling Distributions and Standard Error
  2. Why Sample Variance Uses n−1
  3. Student-t Tests and Intervals
  4. P-values, Errors, and Power
  5. Serial Dependence and Robust Standard Errors
  6. Confidence in Financial Decisions
Likelihood, Estimation, and Bayesian ParametersLessons available
  1. Likelihood Is Not Posterior Probability
  2. Deriving Maximum Likelihood Estimators
  3. Uncertainty and Fisher Information
  4. Bayesian Parameter Updating
  5. MAP, Regularization, and Sensitivity
Simple OLS, Correlation, and R-SquaredLessons available
  1. Regression as a Prediction Rule
  2. Deriving the OLS Normal Equations
  3. Residuals, Correlation, and R-Squared
  4. Statistical Inference on the Slope
  5. Confounding and Misspecified Functional Forms
  6. Optional Diagnostic Laboratory
Sharpe Inference, Multiple Testing, and Credible EvidenceLessons available
  1. Sharpe as Estimated Signal-to-Noise
  2. Sharpe and the Mean t-Statistic
  3. Dependence, Annualization, and Higher Moments
  4. Multiple Testing and Familywise Error
  5. False Discovery and Selective Reporting
  6. From Statistical Evidence to Economic Value
Level 02Linear Algebra, Calculus, and Optimization14 topics
Vectors, Matrices, and Linear SystemsLessons available
  1. Vectors as Financial Objects
  2. Matrix Algebra and Composition
  3. Solving Linear Systems
  4. Linear Maps and Economic Interpretation
Inner Products, Norms, and Orthogonal ProjectionLessons available
  1. Dot Products and Angles
  2. Orthogonality and Decomposition
  3. Least-Squares Projection
  4. Projection Matrices
Rank, Inverses, and Matrix ConditioningLessons available
  1. Rank and Linear Dependence
  2. Inverses and Exact Solvability
  3. Singular Values and Condition Numbers
  4. Numerical Methods and Pseudoinverse
  5. Interpretation Under Ill-Conditioning
Matrix Least Squares and Multivariate RegressionLessons available
  1. Building a Design Matrix
  2. Normal Equations in Matrix Form
  3. Projection, Fitted Values, and R-Squared
  4. Sampling Variance and Robustness
  5. Regularization and Numerically Stable Fits
Covariance Matrices and Positive SemidefinitenessLessons available
  1. Random Vectors and Mean Vectors
  2. Covariance Matrix Entries
  3. Positive Semidefinite Property
  4. Sample Covariance and Rank
Portfolio Variance, Correlation, and Risk DecompositionLessons available
  1. Linear Portfolio Payoff
  2. Deriving Quadratic Portfolio Variance
  3. Two-Asset Diversification Curves
  4. Marginal and Component Risk
  5. Limits of Diversification
Eigenvalues, Eigenvectors, and Principal ComponentsLessons available
  1. Eigenvectors as Invariant Directions
  2. Spectral Decomposition for Covariance
  3. Principal Components from Variance Maximization
  4. Explained Variance and Scaling
  5. Stability and Interpretability
Covariance Estimation and ShrinkageLessons available
  1. Why Sample Covariances Are Noisy
  2. High-Dimensional Singular Risk
  3. Shrinkage Toward a Structured Target
  4. Choosing an Intensity Without Leakage
  5. Downstream Portfolio Sensitivity
Derivatives, Gradients, Jacobians, and HessiansLessons available
  1. Rates of Change and Chain Rule
  2. Partial Derivatives and Gradients
  3. Jacobians and Multivariate Chain Rule
  4. Second Derivatives and the Hessian
  5. Elasticity and Error Propagation
Taylor Expansion and Local ApproximationLessons available
  1. First-Order Taylor Expansion
  2. Second-Order Curvature Correction
  3. Multivariable Taylor Formula
  4. Remainder and Validity Regions
Convexity, Concavity, and Jensen’s InequalityLessons available
  1. Convex Sets and Functions
  2. Second-Order Convexity Tests
  3. Jensen’s Inequality
  4. Convexity and Economic Decisions
  5. Global Versus Local Optimality
Gradient-Based and Numerical OptimizationLessons available
  1. Stationary Points and Necessary Conditions
  2. Deriving Gradient Descent
  3. Learning Rates and Convergence
  4. Conditioning and Algorithm Choice
  5. Stopping, Numerical Error, and Reproducibility
Constrained Optimization, Lagrange Multipliers, and KKTLessons available
  1. From Unconstrained to Feasible Choices
  2. Equality-Constrained Multipliers
  3. Shadow Prices and Sensitivity
  4. Inequalities and Complementary Slackness
  5. Constraint Qualifications and Sufficiency
  6. Optional Duality and Quadratic Programming
Mean–Variance Portfolio Optimization and Efficient FrontiersLessons available
  1. Mean–Variance Objective
  2. Unconstrained Closed-Form Weights
  3. Fully Invested Minimum-Variance Portfolio
  4. Efficient Frontier and Tangency
  5. Realistic Constraints and Transaction Costs
  6. Estimation Risk and Robust Decisions
Level 03Time Series and Stochastic Mathematics13 topics
Returns, Compounding, and Financial Time IndexingLessons available
  1. Simple and Gross Returns
  2. Log Returns and Their Approximation
  3. Multi-Period Aggregation
  4. Time Stamps, Rebalancing, and Bias
Autocorrelation, Dependence, and ForecastabilityLessons available
  1. Lag Notation and Time-Series Dependence
  2. Autocovariance and Autocorrelation
  3. White Noise, Predictability, and Sampling Error
  4. Dependence and Long-Run Variance
Stationary Processes, ARMA Models, and ForecastingLessons available
  1. Strict Versus Weak Stationarity
  2. Autoregressive Dynamics
  3. Moving Averages and ARMA Behavior
  4. Multi-Step Forecasts and Forecast Error
  5. Diagnostics and Structural Instability
Unit Roots, Differencing, and ADF TestingLessons available
  1. Roots, Persistence, and Nonstationarity
  2. Differencing and Integrated Orders
  3. Dickey–Fuller Test Logic
  4. Augmentation, Trends, and Lag Selection
  5. Interpreting Evidence and Structural Breaks
Cointegration, Error Correction, and Mean-Reversion Half-LifeLessons available
  1. Spurious Regression in Trending Prices
  2. Cointegration as a Stable Linear Combination
  3. Engle–Granger and Error-Correction Logic
  4. AR(1) Half-Life and Oscillation
  5. Trading Limitations and Stability Tests
Conditional Heteroskedasticity and GARCH(1,1)Lessons available
  1. Conditional Versus Unconditional Volatility
  2. ARCH Mechanism and Squared Shocks
  3. GARCH(1,1) Recursion
  4. Parameter Constraints and Long-Run Variance
  5. Multi-Step Volatility Forecasting
  6. Diagnostics, Heavy Tails, and Extensions
Random Walks and Discrete-to-Continuous ScalingLessons available
  1. Random Walk Recursions
  2. Mean and Variance Across Horizons
  3. Diffusive Rescaling and CLT
  4. Market Efficiency Versus Statistical Randomness
Markov Chains, Hitting Times, and Gambler’s RuinLessons available
  1. Discrete Markov Property
  2. Transition Matrices and Multi-Step Dynamics
  3. Gambler's Ruin via Difference Equations
  4. Hitting Times and Limits of Gambling Models
Brownian Motion and Gaussian Continuous-Time PathsLessons available
  1. Defining Brownian Motion
  2. Mean, Variance, and Covariance
  3. Scaling and Path Roughness
  4. First Passage and Barriers
  5. Discrete Approximation to Brownian Motion
Filtrations, Martingales, and Optional StoppingLessons available
  1. Information Sets and Adaptedness
  2. Martingales, Submartingales, and Drift
  3. Discounting and Financial Martingale Intuition
  4. Stopping Times and Stopped Processes
  5. Optional Stopping and Gambler’s Ruin Revisited
Quadratic Variation and Stochastic IntegrationLessons available
  1. Why Ordinary Variation Fails
  2. Deriving Quadratic Variation
  3. Itô Integral from Adapted Step Processes
  4. Itô Isometry and Mean-Zero Properties
  5. Formal Differential Multiplication Rules
Itô’s Lemma: Derivation, Extensions, and ValidationLessons available
  1. Revisiting Taylor at Brownian Scale
  2. Deriving the Scalar Itô Formula
  3. Itô on the Square of Brownian Motion
  4. Itô on the Logarithm
  5. Product and Quotient Rules
  6. Two-Dimensional Itô and Correlation
  7. Diffusion Generator and Expected Dynamics
  8. Optional Rigorous Proof Roadmap
Stochastic Differential Equations and Geometric Brownian MotionLessons available
  1. Reading an SDE
  2. Deriving Geometric Brownian Motion
  3. Mean, Median, and Lognormal Returns
  4. Numerical SDE Discretization
  5. Model Failure and Alternative Dynamics
Level 04Derivatives, Portfolio Theory, and Risk13 topics
Derivative Contracts, Payoffs, and No-ArbitrageLessons available
  1. Contingent Claims and Contract Terms
  2. Forwards and Time Value
  3. Law of One Price and Replication
  4. Statewise Bounds and Practical Frictions
Put–Call Parity and Synthetic PositionsLessons available
  1. Payoff Identity at Expiry
  2. European Put–Call Parity
  3. Dividend Adjustments and Synthetic Forwards
  4. American Exercise and Market Frictions
Discrete Hedging, Binomial Trees, and Risk-Neutral PricesLessons available
  1. One-Step State-Contingent Hedging
  2. No-Arbitrage Probability Bounds
  3. Risk-Neutral Transition Probability
  4. Multi-Step Backward Induction
  5. Discrete Limits and Hedging Constraints
Physical Versus Risk-Neutral Measures and Stochastic DiscountingLessons available
  1. Two Roles for Probability
  2. Stochastic Discounting and Numeraires
  3. Market Price of Risk
  4. Girsanov Bridge and its Limits
  5. Completeness, Uniqueness, and Model Risk
Black–Scholes–Merton Pricing: Full DerivationLessons available
  1. Model Assumptions and Contract Definition
  2. Itô Dynamics of an Option Value
  3. Delta-Hedged Replicating Portfolio
  4. Deriving the Black–Scholes PDE
  5. Risk-Neutral Terminal Distribution
  6. Evaluating the European Call Integral
  7. European Put and Boundary Checks
  8. Model Breakdowns and Hedge Discretization
Option Greeks and Sensitivity GeometryLessons available
  1. Delta and Replication Meaning
  2. Gamma and Nonlinear Exposure
  3. Vega and Implied-Volatility Risk
  4. Theta and Rho
  5. Greek Interactions and Risk Surfaces
Implied Volatility, Smile, and Surface ConsistencyLessons available
  1. From Option Price to Implied Sigma
  2. Vega, Uniqueness, and Numerical Stability
  3. Smile and Skew Geometry
  4. Term Structure and Static Arbitrage
  5. Model Calibration and Realized-Vol Distinction
Delta Hedging, Gamma P&L, and Realized Versus Implied VarianceLessons available
  1. Delta-Neutral Portfolio Setup
  2. Second-Order Option P&L
  3. Relating Theta to Implied Variance
  4. Gamma P&L and Realized Quadratic Variation
  5. Discrete Hedging, Fees, and Surface Moves
Capital Asset Pricing Model and Equilibrium BetaLessons available
  1. Diversifiable Versus Systematic Risk
  2. Deriving Covariance Beta
  3. CAPM Expected-Return Equation
  4. Alpha and Empirical Tests
  5. Assumption Critique and Alternatives
Fama–French Factors and Portfolio AttributionLessons available
  1. Factor Portfolios and Excess Returns
  2. Estimating Three-Factor Betas
  3. Factor Return Attribution
  4. Uncertainty and Factor Selection
  5. Extensions and Economic Interpretation
Performance Measurement, Sharpe, and Downside AlternativesLessons available
  1. Sharpe and Units
  2. Annualization Under Dependence
  3. Information and Downside Ratios
  4. Limits of Ratio Optimization
Kelly Criterion, Log-Optimal Growth, and Drawdown ExposureLessons available
  1. Log Growth as a Decision Objective
  2. Binary Kelly Fraction
  3. Continuous-Time Kelly Allocation
  4. Parameter Uncertainty and Fractional Kelly
  5. Risk of Ruin and Path Dependence
Value at Risk, Expected Shortfall, Drawdown, and StressLessons available
  1. Defining Loss and Value at Risk
  2. Gaussian VaR and Parameter Risk
  3. Expected Shortfall and Coherence
  4. Maximum Drawdown and Sequence Risk
  5. Stress Testing and Model Risk
Level 05Market Microstructure and Advanced Quantitative Research12 topics
Market Structure, Bid–Ask Spread, and MidpriceLessons available
  1. Orders, Quotes, and Matching Rules
  2. Best Bid, Ask, and Midprice
  3. Spread as an Execution Cost
  4. Market Frictions and Price Conventions
Order Books, Imbalance, and Adverse SelectionLessons available
  1. Depth, Levels, and Queue Position
  2. Deriving Volume Imbalance
  3. Microprice and Short-Horizon Fair Value
  4. Queue Dynamics, Cancellation, and Fill Uncertainty
  5. Adverse Selection and Toxic Flow
Square-Root Impact and Empirical Price ResponseLessons available
  1. Mechanical Versus Informational Impact
  2. The Empirical Square-Root Relation
  3. Impact Estimation and Endogeneity
  4. Impact in Execution Decisions
VWAP, TWAP, and Implementation ShortfallLessons available
  1. Volume-Weighted Average Price
  2. Time-Weighted Execution Schedules
  3. Implementation Shortfall Decomposition
  4. Benchmark Gaming and Execution Tradeoffs
Optimal Execution and Trading-Cost ControlLessons available
  1. The Execution Control Problem
  2. Temporary Impact and Cost
  3. Inventory Risk During Trading
  4. Balancing Risk and Impact
  5. Constraints, Forecasts, and Model Failure
Avellaneda–Stoikov Market-Making MathematicsLessons available
  1. Dealer Objective and Inventory Constraints
  2. Brownian Reference Price and Inventory Risk
  3. Deriving Reservation Price
  4. Order-Fill Intensities and Quote Distance
  5. Optimal Spread Under Stylized Assumptions
  6. Model Criticism and Calibration
Backtesting Design and Bias ControlsLessons available
  1. Research Question and Falsifiable Rule
  2. Look-Ahead and Point-in-Time Data
  3. Survivorship and Selection Bias
  4. Fees, Slippage, and Capacity
  5. Audit Trail and Reproducibility
Research Design, Hypothesis Discipline, and Robust EvidenceLessons available
  1. Formulating a Tradable Hypothesis
  2. Baselines, Benchmarks, and Negative Controls
  3. Discovery, Confirmation, and Selection
  4. Economic Materiality and Robustness
  5. Evidence Dossier and Go/No-Go Decision
Walk-Forward, Purged Validation, and Temporal GeneralizationLessons available
  1. Why Random Shuffles Leak Financial Information
  2. Rolling and Expanding Window Design
  3. Purging Overlapping Outcomes
  4. Embargo and Feature Availability
  5. Nested Tuning and Final Holdout
Deflated Sharpe Ratio, Trial Counts, and Selection-Aware InferenceLessons available
  1. Probabilistic Sharpe and Sampling Shape
  2. Winner's Curse Across Many Trials
  3. Expected Maximum Benchmark
  4. Deflated Sharpe and Effective Tests
  5. When DSR Is Not Enough
Statistical Learning and Classification MetricsLessons available
  1. Targets, Scores, and Confusion Matrices
  2. Precision, Recall, and F1
  3. Thresholds, ROC/PR, and Base Rates
  4. Probability Calibration and Proper Scoring
  5. Economic Decision Rules and Robustness
Hidden Markov Models and Market RegimesLessons available
  1. Hidden States and Markov Transitions
  2. Emission Distributions and Likelihood
  3. Forward Filtering Recursion
  4. Smoothing, Viterbi, and Look-Ahead
  5. Model Selection, Label Switching, and Drift