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Theory »

Overview

Video lectures

Theory

Introduction

About the ARPM Lab

Organization of the ARPM Lab

Learning the ARPM Lab by topic

Learning the ARPM Lab by channel

Audience and prerequisites

About quantitative finance: P and Q

Differences between P and Q

Commonalities between P and Q

Notation

Key notation tenets

Operators and special functions

Calculus

Probability and general distribution theory

Summary statistical features

Distributions

Stochastic processes

Time conventions and counting indexes

Factor models and learning

Views processing

Investor preferences/profile

I. The “Checklist”

[ 1 ] Risk drivers identification

[ 1.1 ] Equities

[ 1.2 ] Currencies

[ 1.2.1 ] Exchange rates

[ 1.2.2 ] Contracts

[ 1.3 ] Fixed-income

[ 1.3.1 ] Zero-coupon bond

[ 1.3.2 ] Rolling value

[ 1.3.3 ] Yield to maturity

[ 1.3.4 ] Alternative representations

[ 1.3.5 ] Parsimonious representations

[ 1.3.6 ] Spreads

[ 1.4 ] Derivatives

[ 1.4.1 ] Call option

[ 1.4.2 ] Rolling value

[ 1.4.3 ] Implied volatility

[ 1.4.4 ] Alternative representations

[ 1.4.5 ] Parsimonious representations

[ 1.4.6 ] Pure volatility products

[ 1.5 ] Credit

[ 1.5.1 ] Definitions

[ 1.5.2 ] Conditioning risk drivers

[ 1.5.3 ] Aggregate risk drivers

[ 1.6 ] Insurance

[ 1.7 ] Operations

[ 1.8 ] High frequency

[ 1.8.1 ] Market microstructure

[ 1.8.2 ] Activity time

[ 1.8.3 ] Time-changed variables

[ 1.9 ] Strategies

[ 1.10 ] Points of interest, pitfalls, practical tips

[ 1.10.1 ] Spurious heteroscedasticity

[ 1.10.2 ] Spot, par, yield, forward, zero, risk-free,...which rate and curve?

[ 2 ] Quest for invariance

[ 2.1 ] Efficiency: random walk

[ 2.1.1 ] Continuous-state random walk

[ 2.1.2 ] Discrete-state random walk

[ 2.1.3 ] Flexible combinations

[ 2.2 ] Mean-reversion (continuous state): ARMA

[ 2.2.1 ] AR(1) process

[ 2.2.2 ] AR(1) generalizations

[ 2.3 ] Mean-reversion (discrete state)

[ 2.3.1 ] Markov chains

[ 2.3.2 ] Markov chains for credit

[ 2.3.3 ] Structural models for credit

[ 2.3.4 ] Connection Markov chains/structural models

[ 2.4 ] Long memory: fractional integration

[ 2.5 ] Seasonality

[ 2.6 ] Volatility clustering

[ 2.6.1 ] GARCH

[ 2.6.2 ] Extensions of GARCH

[ 2.6.3 ] Stochastic volatility

[ 2.7 ] Multivariate quest

[ 2.7.1 ] Copula-marginal models

[ 2.7.2 ] Efficiency: multivariate random walk

[ 2.7.3 ] Mean reversion (continuous state): VAR(I)MA

[ 2.7.4 ] Mean reversion (discrete state)

[ 2.7.5 ] Volatility clustering

[ 2.7.6 ] Toward machine learning

[ 2.8 ] Cointegration

[ 2.8.1 ] Modeling

[ 2.8.2 ] Detection

[ 2.9 ] Points of interest, pitfalls, practical tips

[ 2.9.1 ] Model-free invariance extraction

[ 2.9.2 ] Generalized Wold’s theorem

[ 2.9.3 ] Information set

[ 2.9.4 ] Returns are not invariants

[ 2.9.5 ] Sampling step size

[ 3 ] Estimation

[ 3.1 ] Setting the flexible probabilities

[ 3.1.1 ] Exponential decay and time conditioning

[ 3.1.2 ] Kernels and state conditioning

[ 3.1.3 ] Joint state and time conditioning

[ 3.1.4 ] Statistical power of flexible probabilities

[ 3.2 ] Historical

[ 3.2.1 ] From historical distribution to flexible probabilities

[ 3.2.2 ] Location-dispersion: HFP ellipsoid

[ 3.2.3 ] Kernel estimation with flexible probabilities

[ 3.2.4 ] Exponential moving moments and statistics

[ 3.3 ] Maximum likelihood

[ 3.3.1 ] From maximum likelihood to flexible probabilities

[ 3.3.2 ] Exponential family invariants

[ 3.3.3 ] Location-dispersion: normal MLFP ellipsoid

[ 3.3.4 ] Location-dispersion: t MLFP ellipsoid

[ 3.4 ] Bayesian

[ 3.4.1 ] Exponential family invariants

[ 3.4.2 ] Normal invariants

[ 3.5 ] Shrinkage

[ 3.5.1 ] Mean shrinkage: James-Stein

[ 3.5.2 ] Covariance shrinkage: Ledoit-Wolf

[ 3.5.3 ] Correlation shrinkage: random matrix theory

[ 3.5.4 ] Covariance shrinkage: sparse eigenvector rotations

[ 3.5.5 ] Covariance shrinkage: glasso

[ 3.5.6 ] Covariance shrinkage: factor analysis

[ 3.6 ] Generalized method of moments

[ 3.6.1 ] Method of moments

[ 3.6.2 ] Generalized method of moments - exact specification

[ 3.6.3 ] Generalized method of moments - over specification

[ 3.7 ] Robustness

[ 3.7.1 ] Local robustness

[ 3.7.2 ] Global robustness

[ 3.8 ] Missing data

[ 3.8.1 ] Randomly missing data

[ 3.8.2 ] Times series of different length

[ 3.8.3 ] Missing series

[ 3.9 ] (Dynamic) copula-marginal

[ 3.9.1 ] Static copula

[ 3.9.2 ] Application: credit risk

[ 3.9.3 ] Dynamic copula

[ 3.10 ] Points of interest, pitfalls, practical tips

[ 3.10.1 ] Unconditional estimation

[ 3.10.2 ] Standardization

[ 3.10.3 ] Non-synchronous data

[ 3.10.4 ] High-frequency volatility/correlation

[ 3.10.5 ] Outlier detection

[ 3.10.6 ] Backward/forward exponential decay

[ 3.10.7 ] Combining estimation techniques

[ 4 ] Projection

[ 4.1 ] One-step historical projection

[ 4.2 ] Univariate analytical projection

[ 4.3 ] Multivariate analytical projection

[ 4.4 ] Monte Carlo

[ 4.4.1 ] Markov models

[ 4.4.2 ] State-space models

[ 4.4.3 ] Generalizations

[ 4.5 ] Historical

[ 4.5.1 ] Historical bootstrapping

[ 4.5.2 ] Hybrid Monte Carlo-historical

[ 4.6 ] Application to credit risk

[ 4.6.1 ] Copula-marginal approach

[ 4.7 ] Square-root rule and generalizations

[ 4.7.1 ] Thin-tailed random walk

[ 4.7.2 ] Thick-tailed random walk

[ 4.7.3 ] Multivariate random walk

[ 4.7.4 ] General processes

[ 4.8 ] Points of interest, pitfalls, practical tips

[ 4.8.1 ] Consecutive (non-)overlapping sequences

[ 4.8.2 ] Semi-analytical random walk projection via fast Fourier transform

[ 4.8.3 ] GARCH generalized next-step

[ 4.8.4 ] Projection step at a later stage of the Checklist

[ 4.8.5 ] Pitfalls for square-root rule

[ 4.8.6 ] Thick tails from thin tails

[ 4.8.7 ] Martingales

[ 4.8.8 ] Linear versus compounded returns

[ 4.8.9 ] FFT for moving averages

[ 4.8.10 ] Scenario projection enhancements by probability twisting

[ 4.8.11 ] Jumps versus diffusion

[ 5 ] Pricing at the horizon

[ 5.1 ] Exact repricing

[ 5.1.1 ] Equities

[ 5.1.2 ] Currencies

[ 5.1.3 ] Fixed-income

[ 5.1.4 ] Derivatives

[ 5.1.5 ] Credit

[ 5.1.6 ] High frequency

[ 5.2.3 ] Fixed-income

[ 5.2.4 ] Derivatives

[ 5.2.5 ] Other asset classes

[ 5.3 ] Taylor approximations

[ 5.3.1 ] Equities

[ 5.3.2 ] Fixed-income

[ 5.3.3 ] Derivatives

[ 5.3.4 ] Other asset classes

[ 5.4 ] Hybrid Taylor/repricing approximation

[ 5.4.1 ] Global quadratic approximation

[ 5.4.2 ] Interpolated residual

[ 5.5 ] Joint P&L distribution

[ 5.5.1 ] Analytical approach

[ 5.5.2 ] Scenario-based approach

[ 5.6 ] Testing the pricing function

[ 5.7 ] Pitfalls and practical tips

[ 5.7.1 ] Pricing at the horizon and arbitrage

[ 5.7.2 ] Path dependence

[ 5.7.3 ] “Pricing at the horizon”versus “asset pricing/valuation theory”

[ 5.7.4 ] Black-Merton-Scholes is exactly correct!

[ 5.7.5 ] Greeks for intra-day updates

[ 5.7.6 ] Greeks at the horizon

[ 5.7.7 ] Bond carry versus accrued interest

[ 5.7.8 ] Option carry versus theta

[ 6 ] Aggregation

[ 6.1 ] Stock variables

[ 6.2 ] Credit value adjustment

[ 6.3 ] Liquidity value adjustment

[ 6.4 ] Static market/credit risk

[ 6.4.1 ] Standardized holdings and weights

[ 6.4.2 ] Scenario-probability distribution

[ 6.4.3 ] Elliptical distribution

[ 6.4.4 ] Quadratic-normal distribution

[ 6.5 ] Dynamic market/credit risk

[ 6.6 ] Stress-testing

[ 6.6.1 ] Theory

[ 6.6.2 ] Why have stress-tests

[ 6.6.3 ] Panic copula

[ 6.6.4 ] Extreme copula

[ 6.7 ] Enterprise risk management

[ 6.7.1 ] Portfolio: balance sheet

[ 6.7.2 ] Performance: income statement

[ 6.7.3 ] Operational risk

[ 6.7.4 ] Banking simplified regulatory framework

[ 6.7.5 ] Insurance simplified framework

[ 6.8 ] Points of interest and pitfalls

[ 6.8.1 ] Solvency and collateral

[ 6.8.2 ] CreditRisk+ approximation

[ 7 ] Ex-ante evaluation

[ 7.1 ] Stochastic dominance

[ 7.2 ] Satisfaction/risk measures

[ 7.3 ] Mean-variance trade-off

[ 7.3.1 ] Mean

[ 7.3.2 ] Variance

[ 7.3.3 ] Standard deviation

[ 7.3.4 ] Mean-variance trade-off

[ 7.3.5 ] A strange success story

[ 7.4 ] The fundamental risk quadrangle

[ 7.4.1 ] Relevant cases

[ 7.4.2 ] Generalizations

[ 7.5 ] Expected utility and certainty-equivalent

[ 7.5.1 ] Common examples

[ 7.5.2 ] Computation

[ 7.6 ] Value at Risk and quantile

[ 7.6.1 ] Definition

[ 7.6.2 ] Computation

[ 7.7 ] Expected shortfall and sub-quantile

[ 7.7.1 ] Definition

[ 7.7.2 ] Computation

[ 7.8 ] Spectral/distortion satisfaction measures

[ 7.8.1 ] Definition

[ 7.8.2 ] Common examples

[ 7.8.3 ] Computation

[ 7.9 ] Coherent satisfaction measures

[ 7.9.1 ] Definition

[ 7.9.2 ] Common examples

[ 7.9.3 ] Computation

[ 7.10 ] Induced expectations

[ 7.10.1 ] Definition

[ 7.10.2 ] Common examples

[ 7.10.3 ] Computation

[ 7.11 ] Non-dimensional ratios

[ 7.11.1 ] Information ratio

[ 7.11.2 ] Downside ratios

[ 7.11.3 ] Correlation

[ 7.12 ] Enterprise risk management

[ 7.13 ] Pitfalls, points of interest and practical tips

[ 7.13.1 ] The Arrow-Pratt approximation of the certainty-equivalent

[ 7.13.2 ] Utility versus quantile

[ 7.13.3 ] Utility versus spectrum functions

[ 7.13.4 ] The Buhlmann and Esscher expectations are not distortion expectations

[ 7.13.5 ] Satisfaction measures under normality

[ 8 ] Ex-ante attribution

[ 8a ] Ex-ante attribution: performance

[ 8a.1 ] Bottom-up exposures

[ 8a.1.1 ] Pricing factors

[ 8a.1.2 ] Style factors/smart beta

[ 8a.2 ] Top-down exposures: factors on demand

[ 8a.2.1 ] Analytical computation

[ 8a.2.2 ] Cardinality constraints

[ 8a.3 ] Relationship between bottom-up and top-down exposures

[ 8a.3.1 ] Subportfolios

[ 8a.3.2 ] Style factors/smart beta

[ 8a.4 ] Joint distribution

[ 8a.4.1 ] Elliptical distribution

[ 8a.4.2 ] Scenario-probability distribution

[ 8a.5 ] Application: hedging

[ 8a.6 ] Pitfalls and practical tips

[ 8a.6.1 ] Estimation versus attribution

[ 8a.6.2 ] The ex-ante attribution is not a regression on past data

[ 8b ] Ex-ante attribution: risk

[ 8b.1 ] General criteria

[ 8b.1.1 ] Isolated/“first in”proportional attribution

[ 8b.1.2 ] “Last in”proportional attribution

[ 8b.1.3 ] Sequential attribution

[ 8b.1.4 ] Shapley attribution

[ 8b.2 ] Euler decomposition

[ 8b.2.1 ] Standard deviation and variance

[ 8b.2.2 ] Certainty-equivalent

[ 8b.2.3 ] Quantile

[ 8b.2.4 ] Sub-quantile

[ 8b.2.5 ] Spectral satisfaction measures

[ 8b.2.6 ] Coherent measures

[ 8b.2.7 ] Economic capital

[ 8b.3 ] Linear attribution for induced expectations

[ 8b.3.1 ] Actuarial pricing

[ 8b.4 ] Minimum-torsion bets attribution of variance

[ 8b.4.1 ] Minimum-torsion bets

[ 8b.4.2 ] Effective number of bets

[ 9 ] Construction

[ 9a ] Construction: portfolio optimization

[ 9a.1 ] Mean-variance principles

[ 9a.2 ] Sensitivity analysis

[ 9a.3 ] Analytical solutions of the mean-variance problem

[ 9a.4 ] Portfolio replication

[ 9a.5 ] Benchmark allocation

[ 9a.6 ] Pitfalls and practical tips

[ 9b ] Construction: estimation and model risk

[ 9b.1 ] Estimation risk measurement

[ 9b.2 ] Sample-based allocation

[ 9b.3 ] Prior allocation

[ 9b.4 ] Estimation (=input) uncertainty: Bayesian allocation

[ 9b.5 ] Allocation (=output) uncertainty: Robust allocation

[ 9c ] Construction: cross-sectional strategies

[ 9c.1 ] Simplistic portfolio construction

[ 9c.1.1 ] Backtesting

[ 9c.2 ] Advanced portfolio construction

[ 9c.2.1 ] Signal-induced factor

[ 9c.2.2 ] Flexible factor

[ 9c.2.3 ] Backtesting

[ 9c.3 ] Relationship with FLAM and APT

[ 9c.3.1 ] Signal-induced moments of the P&L

[ 9c.3.2 ] Fundamental law of active management

[ 9c.3.3 ] APT assumption

[ 9c.4 ] Multiple portfolios

[ 9c.4.1 ] Characteristic portfolios

[ 9c.4.2 ] Signal-induced factors

[ 9c.4.3 ] Flexible factors

[ 9c.4.4 ] Relationship with FLAM and APT

[ 9c.5 ] Points of interest, pitfalls, practical tips

[ 9c.5.1 ] Machine learning

[ 9c.5.2 ] Generalized fundamental law of active management

[ 9d ] Construction: time series strategies

[ 9d.1 ] The market

[ 9d.1.1 ] Risky investment

[ 9d.1.2 ] Low-risk investment

[ 9d.1.3 ] Strategies

[ 9d.2 ] Expected utility maximization

[ 9d.2.1 ] The objective

[ 9d.2.2 ] Optimization

[ 9d.3 ] Option based portfolio insurance

[ 9d.3.1 ] Payoff design

[ 9d.3.2 ] Partial differential equation

[ 9d.3.3 ] Budget

[ 9d.3.4 ] Policy

[ 9d.3.5 ] A unified approach

[ 9d.4 ] Rolling horizon heuristics

[ 9d.4.1 ] Constant proportion portfolio insurance

[ 9d.4.2 ] Drawdown control

[ 9d.5 ] Signal induced strategy

[ 9d.6 ] Convexity analysis

[ 10 ] Execution

[ 10.1 ] Market impact modeling

[ 10.1.1 ] Liquidity curve

[ 10.1.2 ] Exogenous impact

[ 10.1.3 ] Endogenous impact

[ 10.2 ] Order scheduling

[ 10.2.1 ] Trading P&L decomposition

[ 10.2.2 ] Model P&L

[ 10.2.3 ] Moments of model P&L

[ 10.2.4 ] Model P&L optimization

[ 10.2.5 ] Quasi-optimal P&L distribution

[ 10.3 ] Order placement

[ 10.3.1 ] Step 1: order scheduling

[ 10.3.2 ] Step 2: order placement

[ 10.4 ] Points of interest, pitfalls, practical tips

[ 10.4.1 ] Mean-variance optimization in complex models

[ 10.4.2 ] Price manipulation

[ 10.4.3 ] Testing

II. Factor models and learning

[ 11 ] Executive summary

[ 12 ] Linear factor models

[ 12.1 ] Overview

[ 12.1.1 ] The r-squared

[ 12.1.2 ] Dominant-residual models

[ 12.1.3 ] Systematic-idiosyncratic models

[ 12.1.4 ] Estimation

[ 12.2 ] Regression LFM’s

[ 12.2.1 ] Definition

[ 12.2.2 ] Solution: factor loadings

[ 12.2.3 ] Prediction and fit

[ 12.2.4 ] Residuals features

[ 12.2.5 ] Natural scatter specification

[ 12.2.6 ] The ∥ projection operator

[ 12.2.7 ] Estimation

[ 12.3 ] Principal-component LFM’s

[ 12.3.1 ] Definition

[ 12.3.2 ] Solution: factor loadings and constructed factors

[ 12.3.3 ] Identification issues

[ 12.3.4 ] Prediction and fit

[ 12.3.5 ] Residuals features

[ 12.3.6 ] Natural scatter specification

[ 12.3.7 ] Estimation

[ 12.4 ] Factor-analysis LFM’s

[ 12.4.1 ] Definition

[ 12.4.2 ] Solution: factor loadings and idiosyncratic variances

[ 12.4.3 ] Special case: isotropic variances

[ 12.4.4 ] Identification issues

[ 12.4.5 ] Factor approximation

[ 12.4.6 ] Prediction and fit

[ 12.4.7 ] Residuals features

[ 12.4.8 ] Natural scatter specification

[ 12.4.9 ] Estimation

[ 12.5 ] Cross-sectional LFM’s

[ 12.5.1 ] Definition

[ 12.5.2 ] Solution: factor-construction matrix

[ 12.5.3 ] Prediction and fit

[ 12.5.4 ] Residuals features

[ 12.5.5 ] Natural scatter specification

[ 12.5.6 ] Systematic-idiosyncratic assumption

[ 12.5.7 ] Estimation

[ 12.6 ] Points of interest, pitfalls, practical tips

[ 12.6.1 ] LFM’s are not a regression on past data(1) [⋆⋆]

[ 12.6.2 ] LFM’s are not about returns(2-3) [⋆⋆]

[ 12.6.3 ] LFM’s are not about stocks(4) [⋆⋆]

[ 12.6.4 ] LFM’s “factors”are not “factors returns”(5) [⋆]

[ 12.6.5 ] LFM’s are not systematic-idiosyncratic(6-7) [⋆⋆⋆]

[ 12.6.6 ] LFM’s are not horizon-independent(8) [⋆⋆]

[ 12.6.7 ] LFM’s are not a dimension reduction technique(9) [⋆]

[ 12.6.8 ] LFM’s are not APT and CAPM(10-11-12) [⋆⋆⋆]

[ 12.6.9 ] Factor analysis LFM’s are not idiosyncratic(14) [⋆⋆]

[ 12.6.10 ] LFM’s do not extract premia-generating factors(16)[⋆⋆]

[ 12.6.11 ] LFM’s are not always necessary(13-15-17-18) [⋆⋆⋆]

[ 12.6.12 ] Affine versus linear formulation

[ 12.6.13 ] Linear regression: a success story

[ 12.6.14 ] Principal factors are not principal components

[ 12.6.15 ] Performance of regression versus principal component

[ 12.6.16 ] Conditional principal component

[ 12.6.17 ] More general constraints

[ 13 ] Machine learning foundations

[ 13.1 ] Models

[ 13.1.1 ] Supervised modeling

[ 13.1.2 ] Unsupervised modeling

[ 13.1.3 ] Reinforcement modeling

[ 13.2 ] Prediction

[ 13.2.1 ] Point prediction

[ 13.2.2 ] Probabilistic prediction

[ 13.2.3 ] Discriminant vs generative models

[ 13.2.4 ] Point/probabilistic connections

[ 13.3 ] Inference and learning

[ 13.3.1 ] Inference

[ 13.3.2 ] Learning

[ 14 ] Point machine learning models

[ 14.1 ] Least squares regression

[ 14.1.1 ] Error

[ 14.1.2 ] Theoretical optimum

[ 14.1.3 ] Optimization in practice

[ 14.1.4 ] Linear basis

[ 14.1.5 ] Trees

[ 14.1.6 ] Neural networks

[ 14.1.7 ] Gradient boosting

[ 14.1.8 ] Kernel trick

[ 14.1.9 ] ANOVA

[ 14.1.10 ] Geometrical interpretation

[ 14.2 ] Non-least squares regression

[ 14.2.1 ] Error

[ 14.2.2 ] Theoretical optimum

[ 14.2.3 ] Optimization in practice

[ 14.2.4 ] Linear basis

[ 14.2.5 ] Advanced methods

[ 14.2.6 ] Generalized point regression

[ 14.3 ] Binary classification

[ 14.3.1 ] Joint distribution

[ 14.3.2 ] Error

[ 14.3.3 ] Theoretical optimum

[ 14.3.4 ] Receiver operating characteristic (ROC)

[ 14.3.5 ] Optimization in practice

[ 14.3.6 ] Perceptron

[ 14.3.7 ] Support vector machines

[ 14.3.8 ] Fisher discriminant analysis

[ 14.4 ] Multinomial classification

[ 14.4.1 ] Joint distribution

[ 14.4.2 ] Error

[ 14.4.3 ] Theoretical optimum

[ 14.4.4 ] Discriminant classification

[ 14.4.5 ] Optimization in practice

[ 14.4.6 ] Leveraging binary classifiers

[ 14.5 ] Least squares autoencoders

[ 14.5.1 ] Principal component analysis

[ 14.5.2 ] Minimum torsion variables

[ 14.5.3 ] k-means clustering

[ 14.5.4 ] Kernel trick

[ 14.5.5 ] Independent component analysis

[ 14.6 ] Points of interest, pitfalls, practical tips

[ 14.6.1 ] Binary classification: alternative errors

[ 15 ] Probabilistic machine learning models

[ 15.1 ] Discriminant regression

[ 15.1.1 ] Error

[ 15.1.2 ] Theoretical optimum

[ 15.1.3 ] Optimization in practice

[ 15.1.4 ] Linear regression

[ 15.1.5 ] Non-linear regression

[ 15.1.6 ] Generalized linear models

[ 15.1.7 ] Further generalizations

[ 15.2 ] Discriminant classification

[ 15.2.1 ] Error

[ 15.2.2 ] Theoretical optimum

[ 15.2.3 ] Alternative approaches

[ 15.2.4 ] Simplex maps

[ 15.2.5 ] Optimization in practice

[ 15.2.6 ] Logistic regression

[ 15.2.7 ] Probit regression

[ 15.2.8 ] Neural networks

[ 15.2.9 ] Trees

[ 15.2.10 ] Gradient boosting

[ 15.3 ] Graphical models

[ 15.3.1 ] Probabilistic factor analysis

[ 15.3.2 ] Mixture models

[ 15.3.3 ] Naive Bayes models

[ 15.3.4 ] Graphs

[ 15.3.5 ] Markov random fields

[ 15.3.6 ] Bayes networks

[ 15.4 ] Points of interest

[ 15.4.1 ] Alternative errors

[ 15.4.2 ] Exponential tilting

[ 15.4.3 ] Linear regression for classification

[ 16 ] Generalized probabilistic inference

[ 16.1 ] Minimum relative entropy

[ 16.1.1 ] Base distribution and view variables

[ 16.1.2 ] Point views

[ 16.1.3 ] Distributional views

[ 16.1.4 ] Partial views

[ 16.1.5 ] Partial views on generalized expectations

[ 16.1.6 ] Sanity check

[ 16.1.7 ] Confidence

[ 16.1.8 ] Relationship with Bayesian updating

[ 16.2 ] Analytical implementation

[ 16.2.1 ] Base distribution

[ 16.2.2 ] Views

[ 16.2.3 ] Updated distribution

[ 16.2.4 ] Confidence

[ 16.2.5 ] Relevant special cases

[ 16.3 ] Flexible probabilities implementation

[ 16.3.1 ] Base distribution

[ 16.3.2 ] Views

[ 16.3.3 ] Updated distribution

[ 16.3.4 ] Confidence

[ 16.4 ] Factor-based implementations

[ 16.5 ] Copula opinion pooling

[ 16.5.1 ] Base distribution

[ 16.5.2 ] Views

[ 16.5.3 ] Updated distribution

[ 16.5.4 ] Confidence

[ 16.5.5 ] The algorithm

[ 16.6 ] Generalized shrinkage

[ 16.6.1 ] Intuition

[ 16.6.2 ] Classical shrinkage

[ 16.6.3 ] Bayesian updating

[ 16.6.4 ] Minimum relative entropy

[ 16.6.5 ] Shrinkage

[ 16.6.6 ] Regularization

[ 17 ] Dynamic and spatial models

[ 17.1 ] Overview

[ 17.2 ] Least squares dynamic models

[ 17.3 ] Levinson filtering

[ 17.4 ] Wiener-Kolmogorov filtering

[ 17.4.1 ] Solution

[ 17.4.2 ] Causal filtering

[ 17.5 ] Kriging

[ 17.6 ] Dynamic principal component analysis

[ 17.6.1 ] Solution

[ 17.6.2 ] Computational issue

[ 17.7 ] Functional principal component analysis

[ 17.8 ] Probabilistic linear state-space models

[ 17.8.1 ] Model

[ 17.8.2 ] Estimation

[ 17.9 ] Probabilistic state space models

[ 17.9.1 ] Hidden Markov models

[ 17.9.2 ] Hidden Markov VAR(1)

III. Valuation

[ 18 ] Executive summary

[ 19 ] Background definitions

[ 19.1 ] Valuation foundations

[ 19.1.1 ] Instruments

[ 19.1.2 ] Value

[ 19.1.3 ] Cash-flows

[ 19.1.4 ] Re-invested cash-flows

[ 19.1.5 ] Cash-flow adjusted value

[ 19.1.6 ] Profit-and-loss (P&L)

[ 19.1.7 ] Payoff

[ 19.2 ] Points of interest and pitfalls

[ 19.2.1 ] Value versus price

[ 19.2.2 ] Multi-currency conversions

[ 19.2.3 ] Actual versus simple P&L

[ 20 ] Linear pricing theory

[ 20a ] Linear pricing theory: core

[ 20a.1 ] Fundamental axioms

[ 20a.1.1 ] Law of one price

[ 20a.1.2 ] Linearity

[ 20a.1.3 ] Absence of arbitrage

[ 20a.1.4 ] Relationships among fundamental axioms

[ 20a.2 ] Fundamental theorem of asset pricing

[ 20a.2.1 ] Linear pricing equation

[ 20a.2.2 ] Numeraire

[ 20a.2.3 ] Identification issues

[ 20a.3 ] Risk-neutral pricing

[ 20a.3.1 ] Discrete-time rebalancing

[ 20a.3.2 ] No rebalancing: forward measure

[ 20a.3.3 ] Continuous rebalancing limit

[ 20a.4 ] Capital asset pricing framework

[ 20a.4.1 ] Maximum Sharpe ratio portfolio

[ 20a.4.2 ] Security market line

[ 20a.5 ] Covariance principle

[ 20b ] Linear pricing theory: further assumptions

[ 20b.1 ] Completeness

[ 20b.1.1 ] General statement

[ 20b.1.2 ] Arrow-Debreu securities

[ 20b.1.3 ] European options

[ 20b.2 ] Equilibrium: capital asset pricing model

[ 20b.3 ] Arbitrage pricing theory

[ 20b.3.1 ] Standard derivation: linear factor model for instruments

[ 20b.3.2 ] Alternative derivation: linear factor model for stochastic discount factor

[ 20b.4 ] Intertemporal consistency

[ 20b.4.1 ] The framework

[ 20b.4.2 ] Intertemporal linear pricing equation

[ 20b.4.3 ] Intertemporal fundamental theorem of asset pricing

[ 21 ] Non-linear pricing theory

[ 21.1 ] Fundamental axioms

[ 21.1.1 ] Law of one price

[ 21.1.2 ] Non-linearity

[ 21.1.3 ] Arbitrage

[ 21.2 ] Valuation as evaluation

[ 21.2.1 ] Variance and other shift principles

[ 21.2.2 ] Certainty-equivalent principle

[ 21.2.3 ] Distortion principles

[ 21.2.4 ] Esscher principle

[ 21.3 ] Intertemporal consistency

[ 21.3.1 ] Continuous time variables

[ 21.3.2 ] Non-linear “martingales”?

[ 21.4 ] Point of interest and pitfalls

[ 21.4.1 ] Linear (mis)uses of non-linear pricing

[ 22 ] Valuation implementation

[ 22.1 ] Portfolio value

[ 22.1.1 ] Long positions

[ 22.1.2 ] Short positions

[ 22.1.3 ] Generic positions

[ 22.1.4 ] Portfolios, funds

[ 22.1.5 ] Sum-of-parts

[ 22.1.6 ] Valuation recipe

[ 22.2 ] Equities

[ 22.2.1 ] Discounted cash-flows

[ 22.2.2 ] Multiples

[ 22.3 ] Options

[ 22.3.1 ] Bachelier

[ 22.3.2 ] Black-Scholes

[ 22.3.3 ] Heston

[ 22.3.4 ] Valuation recipe

[ 22.4 ] Fixed-income

[ 22.4.1 ] Vasicek

[ 22.4.2 ] Other models

[ 22.4.3 ] Valuation recipe

[ 22.5 ] Insurance

[ 22.5.1 ] Life insurance

[ 22.5.2 ] Non-life insurance

[ 22.6 ] Real assets

IV. Performance analysis

[ 23 ] Executive summary

[ 24 ] Performance definitions

[ 24.1 ] Holding P&L of a portfolio

[ 24.1.1 ] Long positions

[ 24.1.2 ] Short positions

[ 24.1.3 ] Generic positions

[ 24.1.4 ] Portfolios, funds

[ 24.2 ] Trading P&L

[ 24.2.1 ] Single transaction

[ 24.2.2 ] Multiple transactions in one position

[ 24.2.3 ] Portfolio rebalancing

[ 24.3 ] Implementation shortfall

[ 24.4 ] Returns

[ 24.4.1 ] Basic definitions

[ 24.4.2 ] Standard linear returns and weights

[ 24.4.3 ] Generalized linear returns

[ 24.4.4 ] Generalized weights and aggregation

[ 24.4.5 ] Investments with capital injection

[ 24.4.6 ] Log-returns

[ 24.5 ] Excess performance

[ 24.5.1 ] Benchmark

[ 24.5.2 ] Excess return

[ 24.6 ] Path analysis

[ 24.7 ] Pitfalls and practical tips

[ 24.7.1 ] Linear versus compounded returns

[ 25 ] Performance attribution

V. Quant toolbox

[ 26 ] Summary

[ 27 ] Linear algebra primer

[ 27.1 ] Vector spaces

[ 27.1.1 ] Vector operations

[ 27.1.2 ] Basis and coordinates

[ 27.1.3 ] Vector subspaces

[ 27.2 ] Linear transformations

[ 27.2.1 ] Matrix representation

[ 27.2.2 ] Composition

[ 27.2.3 ] Invertibility

[ 27.2.4 ] Positivity

[ 27.3 ] Inner product spaces

[ 27.3.1 ] Length, distance and angle

[ 27.3.2 ] Orthogonal projection

[ 27.3.3 ] Best prediction

[ 27.3.4 ] Rotations

[ 27.4 ] Metric and normed spaces

[ 27.4.1 ] Norm

[ 27.4.2 ] Distance

[ 27.4.3 ] Divergence

[ 27.5 ] Spectral decomposition

[ 27.5.1 ] Eigenvalues and eigenvectors

[ 27.5.2 ] Spectral theorem

[ 27.5.3 ] Matrix notation

[ 27.5.4 ] Generalizations

[ 27.6 ] Matrix transpose-square-root

[ 27.6.1 ] Gramian

[ 27.6.2 ] Transpose-square-roots

[ 27.6.3 ] Orthonormalization

[ 27.6.4 ] Spectrum/principal components

[ 27.6.5 ] Cholesky/Gram-Schmidt

[ 27.6.6 ] Riccati/minimum torsion

[ 27.7 ] Matrix operations

[ 27.7.1 ] The vector space of matrices

[ 27.7.2 ] Key operations

[ 27.7.3 ] Pseudo-inverse

[ 27.7.4 ] Useful identities

[ 27.8 ] Matrix polynomials

[ 27.8.1 ] Matrix polynomials factorization

[ 27.8.2 ] Matrix polynomial inversion

[ 28 ] Calculus primer

[ 28.1 ] Differentiation

[ 28.1.1 ] Univariate functions

[ 28.1.2 ] Multivariate functions

[ 28.1.3 ] Matrix-variate functions

[ 28.2 ] Numerical derivatives

[ 28.2.1 ] Univariate approximations

[ 28.2.2 ] Multivariate approximations

[ 28.3 ] Taylor expansion

[ 28.3.1 ] Univariate functions

[ 28.3.2 ] Multivariate functions

[ 28.4 ] Integration

[ 28.4.1 ] Univariate integration

[ 28.4.2 ] Fundamental theorem of calculus

[ 28.4.3 ] Multivariate integration

[ 28.5 ] Monotone functions

[ 28.5.1 ] Univariate monotonicity

[ 28.5.2 ] Entrywise monotonicity

[ 28.5.3 ] Monotone maps

[ 28.6 ] Convexity

[ 28.6.1 ] Univariate convexity

[ 28.6.2 ] Multivariate convexity

[ 29 ] Functional analysis primer

[ 29.1 ] Measure theory

[ 29.1.1 ] Domains

[ 29.1.2 ] Measures

[ 29.1.3 ] Lebesgue’s decomposition

[ 29.2 ] Functional algebra

[ 29.2.1 ] Function spaces

[ 29.2.2 ] Linear operators

[ 29.2.3 ] Kernel representation

[ 29.3 ] L2 spaces

[ 29.3.1 ] Inner product

[ 29.3.2 ] Dirac delta

[ 29.3.3 ] Convolution

[ 29.3.4 ] Unitary operators

[ 29.4 ] Fourier transform

[ 29.4.1 ] The convolution theorem

[ 29.4.2 ] Fourier integral transform

[ 29.4.3 ] Discrete time Fourier transform

[ 29.4.4 ] Fourier series

[ 29.4.5 ] Discrete Fourier transform

[ 29.5 ] Spectral theorem

[ 29.5.1 ] Mercer kernels

[ 29.5.2 ] Eigenfunctions and eigenvalues

[ 29.5.3 ] Motivation

[ 29.5.4 ] Matrix-valued kernels

[ 29.6 ] Bochner’s theorem

[ 29.6.1 ] Spectral representation

[ 29.6.2 ] Matrix-valued kernels

[ 29.7 ] Mercer’s theorem

[ 29.7.1 ] Spectral representation

[ 29.7.2 ] Reproducing kernel Hilbert spaces

[ 29.7.3 ] Matrix-valued kernels

[ 29.8 ] Functional calculus

[ 29.8.1 ] Gateaux derivative

[ 29.8.2 ] Frechet derivative

[ 29.8.3 ] Second order derivative

[ 30 ] Optimization primer

[ 30.1 ] Continuous programming

[ 30.1.1 ] General purpose tools

[ 30.1.2 ] Convex programming

[ 30.1.3 ] Conic programming

[ 30.1.4 ] Semidefinite programming

[ 30.1.5 ] Second-order cone programming

[ 30.1.6 ] Quadratic programming

[ 30.1.7 ] Ridge, lasso, elastic nets

[ 30.1.8 ] Linear programming

[ 30.2 ] Integer ¯n-choose-k selection

[ 30.2.1 ] Notation

[ 30.2.2 ] Exact solution

[ 30.2.3 ] General heuristics principles

[ 30.2.4 ] Naive selection

[ 30.2.5 ] Forward step-wise selection

[ 30.2.6 ] Step-wise generalizations

[ 30.2.7 ] Ridge, lasso, elastic net heuristics

[ 30.3 ] Points of interest

[ 30.3.1 ] Equivalent optimization problems

[ 31 ] Distributions

[ 31.1 ] Representations of a distribution

[ 31.1.1 ] Random variables

[ 31.1.2 ] Univariate distributions

[ 31.1.3 ] Multivariate distributions

[ 31.2 ] Marginalization

[ 31.3 ] Conditioning

[ 31.3.1 ] Conditional variables

[ 31.3.2 ] Bayes theorem

[ 31.3.3 ] Conditional features

[ 31.3.4 ] Deterministic versus stochastic conditioning

[ 31.4 ] Normal distribution

[ 31.4.1 ] Properties of the normal distribution

[ 31.4.2 ] Extensions of the normal distribution

[ 31.5 ] Notable univariate distributions

[ 31.5.1 ] Quadratic-normal distribution

[ 31.5.2 ] Gamma distribution

[ 31.6 ] Notable multivariate distributions

[ 31.6.1 ] Student t

[ 31.6.2 ] Cauchy

[ 31.6.3 ] Uniform distribution

[ 31.6.4 ] Uniform inside the ellipsoid

[ 31.6.5 ] Uniform on the ellipsoid

[ 31.6.6 ] Lognormal distribution

[ 31.6.7 ] Wishart distribution

[ 31.7 ] Elliptical distributions

[ 31.7.1 ] Fundamental concepts

[ 31.7.2 ] Moments and dependence

[ 31.7.3 ] Stochastic representations

[ 31.7.4 ] Affine equivariance

[ 31.7.5 ] Generation of elliptical scenarios

[ 31.7.6 ] Scenario generation with dimension reduction

[ 31.8 ] Scenario-probability distributions

[ 31.8.1 ] Types of scenario-probability distributions

[ 31.8.2 ] Probability mass and density function

[ 31.8.3 ] Transformations and generalized expectations

[ 31.8.4 ] Cumulative distribution function

[ 31.8.5 ] Continuous cumulative distribution function

[ 31.8.6 ] Quantile

[ 31.8.7 ] Continuous quantile

[ 31.8.8 ] Moments and other statistical features

[ 31.9 ] Categorical distribution

[ 31.9.1 ] Discriminant variables

[ 31.9.2 ] Probabilities parametrization

[ 31.10 ] Exponential family distributions

[ 31.10.1 ] Normal distribution

[ 31.10.2 ] Scenario-probability distribution

[ 31.11 ] Mixture distributions

[ 31.11.1 ] Binary case

[ 31.11.2 ] Multi-class case

[ 31.12 ] Other special classes of distributions

[ 31.12.1 ] Stable distributions

[ 31.12.2 ] Infinitely divisible distributions

[ 31.13 ] Distributions cheat sheet

[ 31.13.1 ] Univariate distributions

[ 31.13.2 ] Multivariate distributions

[ 31.13.3 ] Matrix-valued distributions

[ 31.14 ] Points of interest, pitfalls, practical tips

[ 31.14.1 ] Probability spaces

[ 31.14.2 ] Finite spaces

[ 31.14.3 ] Discretized random variables

[ 32 ] Geometry of distributions

[ 32.1 ] Distributions geometry

[ 32.1.1 ] Fisher metric: length and volume

[ 32.1.2 ] Flatness and geodesics

[ 32.1.3 ] Duality: potentials and Legendre transformations

[ 32.1.4 ] Distance and divergence

[ 32.2 ] Exponential distributions geometry

[ 32.3 ] Scenario-probability distribution geometry

[ 33 ] Copulas

[ 33.1 ] Univariate results

[ 33.2 ] Definition and properties

[ 33.2.1 ] Grades

[ 33.2.2 ] Copula

[ 33.2.3 ] Sklar’s theorem

[ 33.2.4 ] Copula invariance

[ 33.3 ] Special classes of copulas

[ 33.3.1 ] Elliptical copulas

[ 33.3.2 ] Archimedean copulas

[ 33.4 ] Implementation

[ 33.4.1 ] Copula-marginal separation

[ 33.4.2 ] Copula-marginal combination

[ 34 ] Correlation and generalizations

[ 34.1 ] Measures of dependence

[ 34.1.1 ] Schweizer-Wolff measure

[ 34.1.2 ] Mutual information

[ 34.2 ] Measures of concordance

[ 34.2.1 ] Kendall’s tau

[ 34.2.2 ] Spearman’s rho

[ 34.3 ] Correlation

[ 34.3.1 ] Related definitions

[ 34.4 ] Points of interest, pitfalls, practical tips

[ 34.4.1 ] Schweizer and Wolff measure via simulations

[ 35 ] Location and dispersion

[ 35.1 ] Expectation and variance

[ 35.1.1 ] Key definitions

[ 35.1.2 ] Visualization: uncertainty band

[ 35.1.3 ] Affine equivariance

[ 35.1.4 ] Variational principles

[ 35.2 ] Expectation and covariance

[ 35.2.1 ] Key definitions

[ 35.2.2 ] Affine equivariance

[ 35.2.3 ] Linear algebra: spectral decomposition

[ 35.2.4 ] Visualization: ellipsoid

[ 35.2.5 ] Statistics: principal component analysis

[ 35.2.6 ] Calculus: most important summary statistics

[ 35.2.7 ] Probability: Chebyshev’s inequality

[ 35.2.8 ] Variational principles

[ 35.3 ] L2 geometry

[ 35.3.1 ] Expectation inner product

[ 35.3.2 ] Covariance (improper) inner product

[ 35.3.3 ] Covariance versus expectation inner product

[ 35.3.4 ] Least squares prediction

[ 35.3.5 ] Orthonormal sets

[ 35.3.6 ] Visualization: Euclidean vectors

[ 35.4 ] Generalized affine equivariance

[ 35.4.1 ] Univariate case

[ 35.4.2 ] Multivariate case

[ 35.5 ] Generalized variational principles

[ 35.5.1 ] Key definitions

[ 35.5.2 ] Relevant cases

[ 35.5.3 ] Multivariate extensions

[ 35.6 ] Points of interest, pitfalls, practical tips

[ 35.6.1 ] Alternative visualizations in low dimension

[ 35.6.2 ] Lp geometry

[ 36 ] Bias reduction

[ 36.1 ] Functional fit

[ 36.1.1 ] Machine learning as functional optimization

[ 36.1.2 ] Parametric functional optimization

[ 36.2 ] Linear basis

[ 36.2.1 ] Interactions/polynomials

[ 36.2.2 ] Orthogonal series

[ 36.2.3 ] Error minimization

[ 36.3 ] Trees

[ 36.3.1 ] CART

[ 36.3.2 ] Splines

[ 36.3.3 ] Voronoi diagrams

[ 36.3.4 ] Error minimization

[ 36.4 ] Neural networks

[ 36.4.1 ] Neurons

[ 36.4.2 ] Neural networks

[ 36.4.3 ] Projection pursuit

[ 36.4.4 ] Error minimization

[ 36.5 ] Gradient boosting

[ 36.6 ] Kernel trick

[ 37 ] Statistical decision theory

[ 37.1 ] Stochastic dominance fundamentals

[ 37.1.1 ] Strong dominance

[ 37.1.2 ] Weak dominance

[ 37.1.3 ] Second and higher order dominance

[ 37.2 ] No-data decision problem

[ 37.2.1 ] Stochastic dominance failure

[ 37.2.2 ] Bayesian evaluation

[ 37.2.3 ] Robust evaluation

[ 37.2.4 ] General functionals evaluation

[ 37.3 ] Data-driven decision problem

[ 37.3.1 ] Frequentist approach

[ 37.3.2 ] Bayesian approach

[ 37.3.3 ] Admissibility

[ 37.3.4 ] Model and estimation risk

[ 37.3.5 ] Unified approach

[ 37.4 ] Sequential decision theory and dynamic programming

[ 38 ] Probabilistic estimation and inference techniques

[ 38.1 ] Maximum likelihood

[ 38.1.1 ] Observable processes

[ 38.1.2 ] Latent variables

[ 38.1.3 ] EM algorithm

[ 38.1.4 ] IID latent processes

[ 38.1.5 ] Markov state-space processes

[ 38.1.6 ] Networks

[ 38.2 ] Bayesian statistics

[ 38.2.1 ] Estimation

[ 38.2.2 ] Prediction

[ 38.2.3 ] Analytical solutions

[ 38.3 ] Inference via Monte Carlo

[ 38.3.1 ] Metropolis-Hastings

[ 38.4 ] Inference and learning via variational techniques

[ 38.4.1 ] IM projection

[ 38.4.2 ] Inference

[ 38.4.3 ] Learning

[ 38.4.4 ] Analytical solution: exponential family

[ 38.4.5 ] Variational solution

[ 38.4.6 ] EM algorithm in population

[ 38.4.7 ] Dimension reduction

[ 39 ] Estimation and assessment

[ 39.1 ] Probabilistic prediction assessment for invariants

[ 39.1.1 ] Estimators as decisions

[ 39.1.2 ] Frequentist approach

[ 39.1.3 ] Bayesian approach

[ 39.1.4 ] Analytical results

[ 39.1.5 ] Monte Carlo simulations

[ 39.1.6 ] Cross-validation

[ 39.2 ] Bias versus variance

[ 39.3 ] Point prediction assessment

[ 39.3.1 ] Estimators as decisions

[ 39.3.2 ] Frequentist approach

[ 39.3.3 ] Bayesian approach

[ 39.3.4 ] Historical with flexible probabilities estimators

[ 39.3.5 ] Cross-validation

[ 39.4 ] Probabilistic prediction assessment

[ 39.4.1 ] Estimators as decisions

[ 39.4.2 ] Frequentist approach

[ 39.4.3 ] Bayesian approach

[ 39.4.4 ] Maximum likelihood with flexible probabilities

[ 39.4.5 ] Cross-validation

[ 40 ] Estimation and regularization

[ 40.1 ] Background

[ 40.1.1 ] Panels

[ 40.1.2 ] Assumptions

[ 40.1.3 ] Estimation

[ 40.1.4 ] Testing

[ 40.1.5 ] Bias and variance

[ 40.2 ] Regularization

[ 40.2.1 ] Stepwise features selection

[ 40.2.2 ] Ridge, lasso, elastic nets

[ 40.2.3 ] Glasso

[ 40.2.4 ] Categorical factors selection

[ 40.2.5 ] Bayesian prior

[ 40.3 ] Sparse principal component

[ 40.4 ] Ensemble learning

[ 40.4.1 ] Bagging

[ 40.4.2 ] Flexible probabilities as random-variables

[ 40.4.3 ] Flexible probabilities through conditioning

[ 40.4.4 ] Ensemble weighting

[ 41 ] Hypothesis testing

[ 41.1 ] Hypothesis testing for invariants

[ 41.1.1 ] Statistics

[ 41.1.2 ] P-value

[ 41.1.3 ] Univariate testing: the z-statistic

[ 41.1.4 ] Multivariate testing: the Hotelling statistic

[ 42 ] Stochastic processes primer

[ 42.1 ] Main definitions

[ 42.1.1 ] Weak white noise

[ 42.1.2 ] White noise

[ 42.1.3 ] Stationary process

[ 42.1.4 ] Integrated processes

[ 42.1.5 ] Ergodic process

[ 42.1.6 ] Cointegrated process

[ 42.1.7 ] Martingale process

[ 42.1.8 ] State process and Markov process

[ 42.1.9 ] Random fields

[ 42.1.10 ] Orthogonal increments process

[ 42.2 ] Relationships among processes

[ 42.2.1 ] White noise (with finite variance) ⇒ weak white noise

[ 42.2.2 ] White noise ⇒ stationary

[ 42.2.3 ] White noise ⇒ ergodic

[ 42.2.4 ] Ergodic ⇒ stationary

[ 42.2.5 ] Stationary ⇒ covariance stationary

[ 42.2.6 ] Weak white noise ⇒ integrated

[ 42.2.7 ] Weak white noise ⇒ covariance stationary

[ 42.2.8 ] Integrated ⇒ covariance stationary

[ 42.3 ] Pitfalls

[ 42.4 ] Discrete time processes

[ 42.4.1 ] Filtrations

[ 42.4.2 ] Iterated expectations

[ 42.4.3 ] Adapted processes

[ 42.4.4 ] Martingales

[ 42.4.5 ] Approximations of processes

[ 43 ] Invariance tests

[ 43.1 ] Simple tests

[ 43.2 ] Refinements and pitfalls

[ 43.2.1 ] Circle-like covariance (not data)

[ 43.2.2 ] Stronger tests based on copulas

[ 44 ] Continuous time processes

[ 44.1 ] Efficiency: Lévy processes

[ 44.1.1 ] Numerical implementation

[ 44.1.2 ] Infinite divisibility

[ 44.1.3 ] Continuous state: Brownian diffusion

[ 44.1.4 ] Discrete state: Poisson jumps

[ 44.1.5 ] Lévy-Khintchine representation

[ 44.1.6 ] Subordination

[ 44.2 ] Mean-reversion (continuous state)

[ 44.2.1 ] Ornstein-Uhlenbeck

[ 44.2.2 ] Square-root process and other generalizations

[ 44.3 ] Mean-reversion (discrete state)

[ 44.3.1 ] Time-homogeneous Markov chain

[ 44.3.2 ] Time-inhomogeneous Markov chains

[ 44.3.3 ] Stationarity and unconditional distributions

[ 44.4 ] Long memory: fractional Brownian motion

[ 44.4.1 ] Fractional Brownian motion

[ 44.5 ] Volatility clustering

[ 44.5.1 ] Stochastic volatility

[ 44.5.2 ] Time change

[ 44.5.3 ] Connection between time-changed Brownian motion and stochastic volatility

[ 44.6 ] Multivariate mean reversion

[ 44.6.1 ] Definitions

[ 44.6.2 ] Conditional distribution of MVOU

[ 44.6.3 ] Stationarity and unconditional distribution of MVOU

[ 44.6.4 ] Geometrical interpretation∗

[ 44.6.5 ] Cointegrated Ornstein-Uhlenbeck

[ 44.6.6 ] Relationship between (V)AR and (MV)OU

[ 45 ] Covariance stationary processes

[ 45.1 ] L2 processes

[ 45.1.1 ] Non-stationary moments

[ 45.1.2 ] Stationary moments

[ 45.1.3 ] Geometry

[ 45.2 ] Order-one autoregression

[ 45.2.1 ] White noise

[ 45.2.2 ] Random walk

[ 45.2.3 ] Autoregression

[ 45.2.4 ] Stationarity and moments

[ 45.2.5 ] Prediction

[ 45.2.6 ] Multivariate extensions

[ 45.2.7 ] Cointegrated VAR(1)

[ 45.3 ] VARMA processes

[ 45.3.1 ] ARMA(p,q)

[ 45.3.2 ] Multivariate extensions

[ 45.4 ] Linear state space models

[ 45.4.1 ] Definition

[ 45.4.2 ] Stationarity and moments

[ 45.5 ] Filtering

[ 45.5.1 ] Formal definition

[ 45.5.2 ] Impulse response function

[ 45.5.3 ] Frequency response function

[ 45.5.4 ] Inversion

[ 45.5.5 ] Causality

[ 45.5.6 ] Time domain filters

[ 45.5.7 ] Frequency domain filters

[ 45.6 ] Spectral representation

[ 45.6.1 ] Intuition

[ 45.6.2 ] Spectral theorem

[ 45.6.3 ] Heuristics verification

[ 45.6.4 ] The Cramer decomposition

[ 45.6.5 ] Heuristics verification

[ 45.6.6 ] Spectral representation of notable processes

[ 45.7 ] Wold representation

[ 45.7.1 ] Intuition

[ 45.7.2 ] Formal statement

[ 45.7.3 ] Weak white noise via Gram-Schmidt

[ 45.7.4 ] Linearly regular component

[ 45.7.5 ] Linearly deterministic component

[ 45.8 ] Prediction

[ 45.9 ] Points of interest

[ 45.9.1 ] VAR(1) as universal approximation

[ 46 ] Signals

[ 46.1 ] Carry signals

[ 46.1.1 ] Fixed-income

[ 46.1.2 ] Foreign exchange

[ 46.2 ] Value signals

[ 46.2.1 ] Book

[ 46.2.2 ] Pricing

[ 46.3 ] Technical signals

[ 46.3.1 ] Momentum

[ 46.3.2 ] Filters

[ 46.3.3 ] Cointegration

[ 46.4 ] Microstructure signals

[ 46.4.1 ] Trade autocorrelation

[ 46.4.2 ] Order imbalance

[ 46.4.3 ] Price prediction

[ 46.4.4 ] Volume clustering

[ 46.5 ] Fundamental and other signals

[ 46.6 ] Signal processing

[ 46.6.1 ] Smoothing

[ 46.6.2 ] Scoring

[ 46.6.3 ] Ranking

[ 47 ] Black-Litterman

[ 47.1 ] Equilibrium distribution

[ 47.1.1 ] Performance model

[ 47.1.2 ] Prior distribution of expected returns

[ 47.1.3 ] Prior predictive performance distribution

[ 47.2 ] Active views

[ 47.2.1 ] Active views model

[ 47.2.2 ] Active views statement

[ 47.2.3 ] Posterior distribution of the expected returns

[ 47.3 ] Black-Litterman posterior

[ 47.4 ] Limit cases and generalizations

[ 47.4.1 ] High confidence in prior

[ 47.4.2 ] Low confidence in views

[ 47.4.3 ] High confidence in views

[ 47.4.4 ] Generalizations

[ 47.4.5 ] From linear returns to risk drivers

[ 47.4.6 ] From stock-like to generic asset classes

[ 47.4.7 ] From normal to non-normal markets

[ 47.4.8 ] From linear equality views to partial flexible views

[ 48 ] Useful algorithms

[ 48.1 ] Moment-matching scenarios

[ 48.1.1 ] Twisting scenarios

[ 48.1.2 ] Twisting probabilities

[ 48.2 ] Normalized empirical histogram

[ 48.3 ] Minimum-torsion optimization algorithm

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