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Theory »

Overview

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

Key notation tenets

Operators and special functions

Probability and general distribution theory

Summary statistical features

Stochastic processes

Time conventions and counting indexes

Factor models and learning

Views processing

Investor preferences/profile

I. The “Checklist”

Executive summary

[ 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

[ 1.5.1 ] 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 ] The final output

[ 1.11 ] Points of interest, pitfalls, practical tips

[ 1.11.1 ] Spurious heteroscedasticity

[ 1.11.2 ] 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(p) process

[ 2.2.3 ] MA(q) process

[ 2.2.4 ] ARMA(p,q) process

[ 2.2.5 ] ARIMA(p,d,q) process

[ 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 ] Vector autoregression of order 1: VAR(1)

[ 2.7.5 ] Mean reversion (discrete state)

[ 2.7.6 ] Volatility clustering

[ 2.7.7 ] Toward machine learning

[ 2.8 ] Cointegration

[ 2.8.1 ] Modeling

[ 2.8.2 ] Detection

[ 2.9 ] Relationships among processes

[ 2.9.1 ] Inversion of first-degree lag polynomial

[ 2.9.2 ] ARMA(p,q) processes as products of lag polynomials

[ 2.9.3 ] Relationships between ARMA, MA and AR

[ 2.9.4 ] AR(p) as VAR(1)

[ 2.9.5 ] Univariate processes as VAR(1)

[ 2.9.6 ] Relationships between VARMA, VMA and VAR

[ 2.9.7 ] VAR(p) as VAR(1)

[ 2.9.8 ] Multivariate processes as VAR(1)

[ 2.9.9 ] Stationary as MA(∞): Wold representation

[ 2.10 ] The final output

[ 2.10.1 ] Model for groups

[ 2.10.2 ] Fit and invariants extraction for groups

[ 2.10.3 ] Marginal test for invariance

[ 2.10.4 ] Green light toward joint estimation

[ 2.11 ] Points of interest, pitfalls, practical tips

[ 2.11.1 ] Model-free invariance extraction

[ 2.11.2 ] Generalized Wold’s theorem

[ 2.11.3 ] Information set

[ 2.11.4 ] Returns are not invariants

[ 2.11.5 ] 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

[ 3.3 ] 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

[ 3.6 ] 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 ] Dynamic copula

[ 3.9.3 ] Application: credit risk

[ 3.10 ] The final output

[ 3.10.1 ] Historical

[ 3.10.2 ] Analytical

[ 3.10.3 ] Copula-marginal

[ 3.11 ] Points of interest, pitfalls, practical tips

[ 3.11.1 ] Unconditional estimation

[ 3.11.2 ] Standardization

[ 3.11.3 ] Non-synchronous data

[ 3.11.4 ] High-frequency volatility/correlation

[ 3.11.5 ] Outlier detection

[ 3.11.6 ] Exponential moving moments and statistics

[ 3.11.7 ] Backward/forward exponential decay

[ 3.11.8 ] 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 ] Copula-marginal models

[ 4.4.4 ] Generalizations

[ 4.5 ] Historical

[ 4.5.1 ] Historical bootstrapping

[ 4.5.2 ] Hybrid Monte Carlo-historical

[ 4.6 ] Application: multivariate Markov chains

[ 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 ] The final output

[ 4.8.1 ] Analytical

[ 4.8.2 ] Scenarios

[ 4.9 ] Points of interest, pitfalls, practical tips

[ 4.9.1 ] Consecutive (non-)overlapping sequences

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

[ 4.9.3 ] GARCH generalized next-step

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

[ 4.9.5 ] Pitfalls for square-root rule

[ 4.9.6 ] Thick tails from thin tails

[ 4.9.7 ] Martingales

[ 4.9.8 ] Linear versus compounded returns

[ 4.9.9 ] FFT for moving averages

[ 4.9.10 ] Scenario projection enhancements by probability twisting

[ 4.9.11 ] 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

[ 5.1.7 ] Strategies

[ 5.2.1 ] Equities

[ 5.2.2 ] Currencies

[ 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 ] The final output

[ 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

[ 6.1.1 ] Portfolio

[ 6.1.2 ] Value

[ 6.1.3 ] Exposure

[ 6.1.4 ] Leverage

[ 6.2 ] Liquidity P&L/risk

[ 6.3 ] Static market/credit risk

[ 6.3.1 ] Standardized holdings and weights

[ 6.3.2 ] Scenario-probability distribution

[ 6.3.3 ] Elliptical distribution

[ 6.3.4 ] Quadratic-normal distribution

[ 6.4 ] Dynamic market/credit risk

[ 6.5 ] Stress-testing

[ 6.5.1 ] Theory

[ 6.5.2 ] Why have stress-tests

[ 6.5.3 ] Panic copula

[ 6.5.4 ] Extreme copula

[ 6.6 ] Enterprise risk management

[ 6.6.1 ] Portfolio: balance sheet

[ 6.6.2 ] Performance: income statement

[ 6.6.3 ] Operational risk

[ 6.6.4 ] Banking simplified regulatory framework

[ 6.6.5 ] Insurance simplified framework

[ 6.7 ] The final output

[ 6.7.1 ] Analytical approach

[ 6.7.2 ] Scenario-based approach

[ 6.8 ] Points of interest and pitfalls

[ 6.8.1 ] Solvency and collateral

[ 6.8.2 ] Credit value adjustment

[ 6.8.3 ] CreditRisk+ approximation

[ 7 ] Ex-ante evaluation

[ 7.1 ] Stochastic dominance

[ 7.2 ] Satisfaction/risk measures

[ 7.3 ] Mean-variance trade-off

[ 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 ] Expected utility and certainty-equivalent

[ 7.4.1 ] Common utility functions

[ 7.4.2 ] Computation

[ 7.5 ] Quantile (value at risk)

[ 7.5.1 ] Definition

[ 7.5.2 ] Computation

[ 7.6 ] Spectral satisfaction measures/Distortion expectations

[ 7.6.1 ] Definition

[ 7.6.2 ] Common spectral/distortion measures

[ 7.6.3 ] Computation

[ 7.7 ] Coherent satisfaction measures

[ 7.7.1 ] Definition

[ 7.7.2 ] Common coherent measures

[ 7.7.3 ] Computation

[ 7.8 ] Non-dimensional ratios

[ 7.8.1 ] Information ratio

[ 7.8.2 ] Downside ratios

[ 7.9 ] Additional measures

[ 7.9.2 ] Correlation

[ 7.9.3 ] Buhlmann expectation

[ 7.9.4 ] Esscher expectation

[ 7.10 ] Enterprise risk management

[ 7.11 ] The final output

[ 7.12 ] Pitfalls, points of interest and practical tips

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

[ 7.12.2 ] Utility versus quantile

[ 7.12.3 ] Utility versus spectrum functions

[ 7.12.4 ] Spectral/distortion versus coherent measures

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

[ 7.12.6 ] 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 ] The final output

[ 8a.7 ] Pitfalls and practical tips

[ 8a.7.1 ] Estimation versus attribution

[ 8a.7.2 ] 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 ] Homogenous measures and Euler decomposition

[ 8b.2.1 ] Standard deviation

[ 8b.2.2 ] Variance

[ 8b.2.3 ] Certainty-equivalent

[ 8b.2.4 ] Distortion/spectral measures

[ 8b.2.5 ] Economic capital

[ 8b.3 ] Esscher expectation

[ 8b.4 ] Buhlmann expectation

[ 8b.5 ] Minimum-torsion bets attribution of variance

[ 8b.5.1 ] Effective number of bets

[ 8b.5.2 ] Minimum-torsion bets

[ 8b.5.3 ] Traditional marginal contributions versus minimum-torsion contributions

[ 8b.6 ] The final output

[ 9 ] Construction

[ 9a ] Construction: portfolio optimization

[ 9a.1 ] A compromise: two-step mean-variance

[ 9a.2 ] Analytical solutions of the mean-variance problem

[ 9a.3 ] The final output

[ 9b ] Construction: cross-sectional strategies

[ 9b.1 ] Simplistic portfolio construction

[ 9b.1.1 ] Backtesting

[ 9b.2 ] Advanced portfolio construction

[ 9b.2.1 ] Signal-induced factor

[ 9b.2.2 ] Flexible factor

[ 9b.2.3 ] Backtesting

[ 9b.3 ] Relationship with FLAM and APT

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

[ 9b.3.2 ] Fundamental law of active management

[ 9b.3.3 ] APT assumption

[ 9b.4 ] Multiple portfolios

[ 9b.4.1 ] Characteristic portfolios

[ 9b.4.2 ] Signal-induced factors

[ 9b.4.3 ] Flexible factors

[ 9b.4.4 ] Relationship with FLAM and APT

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

[ 9b.5.1 ] Machine learning

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

[ 9c ] Construction: time series strategies

[ 9c.1 ] The market

[ 9c.1.1 ] Risky investment

[ 9c.1.2 ] Low-risk investment

[ 9c.1.3 ] Strategies

[ 9c.2 ] Expected utility maximization

[ 9c.2.1 ] The objective

[ 9c.2.2 ] Optimization

[ 9c.3 ] Option based portfolio insurance

[ 9c.3.1 ] Payoff design

[ 9c.3.2 ] Partial differential equation

[ 9c.3.3 ] Budget

[ 9c.3.4 ] Policy

[ 9c.3.5 ] A unified approach

[ 9c.4 ] Rolling horizon heuristics

[ 9c.4.1 ] Constant proportion portfolio insurance

[ 9c.4.2 ] Drawdown control

[ 9c.5 ] Signal induced strategy

[ 9c.6 ] Convexity analysis

[ 9d ] Construction: estimation and model risk

[ 9d.1 ] Allocation (=output) uncertainty: Robust allocation

[ 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 ] The final output

[ 10.5 ] Points of interest, pitfalls, practical tips

[ 10.5.1 ] Mean-variance optimization in complex models

[ 10.5.2 ] Price manipulation

[ 10.5.3 ] Testing

II. Factor models and learning

[ 11 ] Executive summary

[ 12 ] Linear factor models: foundations

[ 12.1 ] Overview

[ 12.1.1 ] Dominant-residual models

[ 12.1.2 ] Systematic-idiosyncratic models

[ 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 ] Symmetric regression

[ 12.2.7 ] Equivalent linear formulation for regression

[ 12.3 ] Principal component LFM’s

[ 12.3.1 ] Definition

[ 12.3.2 ] Solution: factor loadings and factor-construction matrix

[ 12.3.3 ] Prediction and fit

[ 12.3.4 ] Residuals features

[ 12.3.5 ] Natural scatter specification

[ 12.3.6 ] Identification issues

[ 12.4 ] Systematic-idiosyncratic LFM’s

[ 12.4.1 ] Definition

[ 12.4.2 ] Implications on parameters

[ 12.4.3 ] Implications on covariance

[ 12.4.4 ] Factor analysis: principal axis factorization

[ 12.4.5 ] Loadings unearthing and factor approximation

[ 12.4.6 ] Prediction and fit

[ 12.4.7 ] Identification issues

[ 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-idiosycratic assumption

[ 12.6 ] Points of interest, pitfalls, practical tips

[ 12.6.1 ] The r-squared

[ 12.6.2 ] Equivalent predictions

[ 12.6.3 ] Principal factors are not principal components

[ 13 ] Linear factor models: estimation

[ 13.1 ] Overview

[ 13.1.1 ] Time series models

[ 13.1.2 ] Data fit

[ 13.2 ] Static regression LFM’s

[ 13.3 ] Static principal component LFM’s

[ 13.4 ] Static systematic-idiosyncratic LFM’s

[ 13.5 ] Static cross-sectional LFM’s

[ 13.5.1 ] Factor construction

[ 13.5.2 ] Problems with pure cross-sectional estimation

[ 13.5.3 ] Hybrid cross-sectional estimation

[ 13.6 ] Truncation

[ 13.7 ] Wiener-Kolmogorov filtering

[ 13.7.1 ] Solution

[ 13.7.2 ] Estimation

[ 13.8 ] Dynamic principal component

[ 13.8.1 ] Solution

[ 13.8.2 ] Computational issue

[ 13.8.3 ] Estimation

[ 13.9 ] Linear state space model

[ 14 ] Linear factor models: enhancements

[ 14.1 ] Performance of regression versus principal component

[ 14.1.1 ] Regression revisited

[ 14.1.2 ] Principal component revisited

[ 14.1.3 ] The comparison

[ 14.2 ] Factor selection for regression

[ 14.2.1 ] Step-wise regression selection

[ 14.2.2 ] Lasso regression

[ 14.2.3 ] Ridge regression

[ 14.3 ] Sparse principal component

[ 14.4 ] Conditional principal component

[ 14.5 ] More general constraints

[ 15 ] Linear factor models: pitfalls

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

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

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

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

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

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

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

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

[ 15.9 ] LFM’s do not include factor analysis(14) [⋆⋆]

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

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

[ 16 ] Machine learning: foundations

[ 16.1 ] Key ideas from linear factor models

[ 16.1.1 ] Supervised linear predictors

[ 16.1.2 ] Unsupervised linear autoencoders

[ 16.1.3 ] Conditionally uncorrelated linear models

[ 16.2 ] Key concepts for machine learning

[ 16.2.1 ] Supervised, unsupervised, reinforcement learning

[ 16.2.2 ] Regression, classification, clustering, structures

[ 16.2.3 ] Point and probabilistic prediction

[ 16.2.4 ] Discriminant and generative probabilistic models

[ 16.3 ] Supervised point prediction: regression

[ 16.3.1 ] Mean regression

[ 16.3.2 ] Linear mean regression

[ 16.3.3 ] Analysis of variance

[ 16.3.4 ] Quantile regression

[ 16.3.5 ] Linear quantile regression

[ 16.4 ] Supervised point prediction: classification

[ 16.4.1 ] Mode classification

[ 16.4.2 ] Linear classification

[ 16.4.3 ] Receiver operating characteristic (ROC)

[ 16.5 ] Supervised probabilistic prediction

[ 16.5.1 ] Regression

[ 16.5.2 ] Binary classification

[ 16.5.3 ] Multiple classification

[ 16.5.4 ] Generalized linear models

[ 16.6 ] Unsupervised learning: autoencoders

[ 16.6.1 ] Principal component linear factor models

[ 16.6.2 ] Cross-sectional linear factor models

[ 16.6.3 ] k-means clustering

[ 16.6.4 ] Independent component analysis

[ 16.7 ] Probabilistic graphical models

[ 16.7.1 ] Probabilistic factor analysis

[ 16.7.2 ] Mixture models

[ 16.7.3 ] Naive Bayes models

[ 16.7.4 ] Graphs

[ 16.7.5 ] Markov random fields

[ 16.7.6 ] Bayes networks

[ 17 ] Machine learning: estimation

[ 17.1 ] Probabilistic linear state-space models

[ 17.1.1 ] Model

[ 17.1.2 ] Estimation

[ 17.2 ] Probabilistic state space models

[ 17.2.1 ] Hidden Markov models

[ 18 ] Machine learning: enhancements

[ 18.1 ] Feature engineering

[ 18.1.1 ] Interaction

[ 18.1.2 ] Encoding

[ 18.1.3 ] Trees

[ 18.1.4 ] Segmentation

[ 18.1.5 ] Feature basis

[ 18.1.6 ] Exponential family

[ 18.2 ] Deep learning

[ 18.2.1 ] Point prediction

[ 18.2.2 ] Probabilistic prediction

[ 18.3 ] Gradient boosting

[ 18.4 ] Regularization

[ 18.4.1 ] Step-wise features selection

[ 18.4.2 ] Ridge, lasso, elastic nets

[ 18.4.3 ] Glasso

[ 18.4.4 ] Bayesian prior

[ 18.5 ] Ensemble learning

[ 18.5.1 ] Bagging

[ 18.5.2 ] Flexible probabilities as random-variables

[ 18.5.3 ] Flexible probabilities through conditioning

[ 18.5.4 ] Ensemble weighting

[ 19 ] From probabilistic regression to Gaussian processes

[ 19.1 ] Time series models

[ 19.2 ] Maximum likelihood

[ 19.2.1 ] The model

[ 19.2.2 ] Normal assumption

[ 19.2.3 ] Student t assumption

[ 19.3 ] Bayesian

[ 19.3.1 ] Normal conditional likelihood

[ 19.3.2 ] Normal-inverse-Wishart prior distribution

[ 19.3.3 ] Normal-inverse-Wishart posterior distribution

[ 19.3.4 ] Student t predictive distribution

[ 19.3.5 ] Classical equivalent

[ 19.3.6 ] Uncertainty

[ 19.4 ] Mixed approach

[ 20 ] Application: principal component analysis of the yield curve

[ 20.1 ] Cross-sectional structure of the yield curve covariance

[ 20.2 ] Finite set of times to maturity

[ 20.3 ] The continuum limit

[ 21 ] Application: credit default classification

[ 21.1 ] Background

[ 21.2 ] Fit and assessment

[ 21.3 ] Logistic regression

[ 21.4 ] Interactions

[ 21.5 ] Encoding

[ 21.6 ] Regularization

[ 21.8 ] Gradient boosting

[ 21.9 ] Cross-validation

[ 22 ] Application: clustering

[ 22.1 ] k-means clustering

[ 22.2 ] Shrinkage

III. Valuation

[ 23 ] Executive summary

[ 24 ] Background definitions

[ 24.1 ] Valuation foundations

[ 24.1.1 ] Instruments

[ 24.1.2 ] Value

[ 24.1.3 ] Cash-flows

[ 24.1.4 ] Re-invested cash-flows

[ 24.1.5 ] Cash-flow adjusted value

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

[ 24.1.7 ] Payoff

[ 24.2 ] Points of interest and pitfalls

[ 24.2.1 ] Value versus price

[ 24.2.2 ] Multi-currency conversions

[ 24.2.3 ] Actual versus simple P&L

[ 25 ] Linear pricing theory

[ 25a ] Linear pricing theory: core

[ 25a.1 ] Fundamental axioms

[ 25a.1.1 ] Law of one price

[ 25a.1.2 ] Linearity

[ 25a.1.3 ] No arbitrage

[ 25a.2 ] Stochastic discount factor

[ 25a.2.1 ] Identification

[ 25a.2.2 ] Misidentification

[ 25a.3 ] Fundamental theorem of asset pricing

[ 25a.4 ] Risk-neutral pricing

[ 25a.4.1 ] General case

[ 25a.4.2 ] No rebalancing limit: forward measure

[ 25a.4.3 ] Continuous rebalancing limit

[ 25a.5 ] Capital asset pricing model framework

[ 25a.5.1 ] Maximum Sharpe ratio portfolio

[ 25a.5.2 ] Security market line

[ 25a.5.3 ] Alternative derivation: linear factor model for stochastic discount factor

[ 25a.6 ] Covariance principle

[ 25a.7 ] Point of interest and pitfalls

[ 25a.7.1 ] Relationships among fundamental axioms

[ 25b ] Linear pricing theory: further assumptions

[ 25b.1 ] Completeness

[ 25b.1.1 ] Definition

[ 25b.1.2 ] Pricing

[ 25b.1.3 ] Arrow-Debreu securities

[ 25b.1.4 ] Stochastic discount factor

[ 25b.2 ] Equilibrium: pure capital asset pricing model

[ 25b.3 ] Equilibrium: Buhlmann pricing equation

[ 25b.4 ] Arbitrage pricing theory

[ 25b.4.1 ] Standard derivation: linear factor model for instruments

[ 25b.4.2 ] Alternative derivation: linear factor model for stochastic discount factor

[ 25b.5 ] Intertemporal consistency

[ 25b.5.1 ] Continuous time variables

[ 25b.5.2 ] Martingales

[ 25b.5.3 ] Heuristic for stochastic discount factor time consistency

[ 25b.5.4 ] Heuristic for numeraire martingale

[ 26 ] Non-linear pricing theory

[ 26.1 ] Fundamental axioms

[ 26.1.1 ] Law of one price

[ 26.1.2 ] Non-linearity

[ 26.1.3 ] Arbitrage

[ 26.2 ] Valuation as evaluation

[ 26.2.1 ] Variance and other shift principles

[ 26.2.2 ] Certainty-equivalent principle

[ 26.2.3 ] Distortion principles

[ 26.2.4 ] Esscher principle

[ 26.3 ] Intertemporal consistency

[ 26.3.1 ] Continuous time variables

[ 26.3.2 ] Non-linear "martingales"?

[ 26.4 ] Point of interest and pitfalls

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

[ 27 ] Valuation implementation

[ 27.1 ] Portfolio value

[ 27.1.1 ] Long positions

[ 27.1.2 ] Short positions

[ 27.1.3 ] Generic positions

[ 27.1.4 ] Funds

[ 27.1.5 ] Sum-of-parts

[ 27.1.6 ] Valuation recipe

[ 27.2 ] Equities

[ 27.2.1 ] Discounted cash-flows

[ 27.2.2 ] Multiples

[ 27.3 ] Options

[ 27.3.1 ] Bachelier

[ 27.3.2 ] Black-Scholes

[ 27.3.3 ] Heston

[ 27.3.4 ] Valuation recipe

[ 27.4 ] Fixed-income

[ 27.4.1 ] Vasicek

[ 27.4.2 ] Other models

[ 27.4.3 ] Valuation recipe

[ 27.5 ] Insurance

[ 27.5.1 ] Life insurance

[ 27.5.2 ] Non-life insurance

[ 27.6 ] Real assets

IV. Performance analysis

[ 28 ] Executive summary

[ 29 ] Performance definitions

[ 29.1 ] Holding P&L of a portfolio

[ 29.2 ] Trading P&L

[ 29.2.1 ] Single transaction

[ 29.2.2 ] Multiple transactions in one position

[ 29.2.3 ] Portfolio rebalancing

[ 29.3 ] Implementation shortfall

[ 29.4 ] Returns

[ 29.4.1 ] Basic definitions

[ 29.4.2 ] Standard linear returns and weights

[ 29.4.3 ] Generalized linear returns

[ 29.4.4 ] Generalized weights and aggregation

[ 29.4.5 ] Investments with capital injection

[ 29.4.6 ] Log-returns

[ 29.5 ] Excess performance

[ 29.5.1 ] Benchmark

[ 29.5.2 ] Excess return

[ 29.6 ] Path analysis

[ 29.7 ] Pitfalls and practical tips

[ 29.7.1 ] Linear versus compounded returns

[ 30 ] Performance attribution

V. Quant toolbox

[ 32 ] Distributions

[ 32.1 ] Representations of a distribution

[ 32.1.1 ] Univariate distributions

[ 32.1.2 ] Multivariate distributions

[ 32.2 ] Marginalization

[ 32.3 ] Conditioning

[ 32.3.1 ] Conditional variables

[ 32.3.2 ] Conditional features

[ 32.3.3 ] Deterministic versus stochastic conditioning

[ 32.4 ] Elliptical distributions

[ 32.4.1 ] Fundamental concepts

[ 32.4.2 ] Stochastic representations

[ 32.4.3 ] Moments and dependence

[ 32.4.4 ] Affine equivariance

[ 32.4.5 ] Notable elliptical distributions

[ 32.4.6 ] Generation of elliptical scenarios

[ 32.4.7 ] Scenario generation with dimension reduction

[ 32.5 ] Scenario-probability distributions

[ 32.5.1 ] Types of scenario-probability distributions

[ 32.5.2 ] Probability density function

[ 32.5.3 ] Transformations and generalized expectations

[ 32.5.4 ] Cumulative distribution function

[ 32.5.5 ] Continuous cumulative distribution function

[ 32.5.6 ] Quantile

[ 32.5.7 ] Continuous quantile

[ 32.5.8 ] Moments and other statistical features

[ 32.5.9 ] Probabilities parametrization

[ 32.6 ] Mixture distributions

[ 32.7 ] Exponential family distributions

[ 32.7.1 ] Normal distribution

[ 32.7.2 ] Scenario-probability distribution

[ 32.8 ] Other special classes of distributions

[ 32.8.1 ] Stable distributions

[ 32.8.2 ] Infinitely divisible distributions

[ 32.9 ] Notable specific distributions

[ 32.9.1 ] Uniform distribution

[ 32.9.2 ] Quadratic-normal distribution

[ 32.9.3 ] Wishart distribution

[ 33 ] Geometry of distributions

[ 33.1 ] Distributions geometry

[ 33.1.1 ] Fisher metric: length and volume

[ 33.1.2 ] Flatness and geodesics

[ 33.1.3 ] Duality: potentials and Legendre transformations

[ 33.1.4 ] Distance and divergence

[ 33.2 ] Exponential distributions geometry

[ 33.3 ] Scenario-probability distribution geometry

[ 34.1 ] Univariate results

[ 34.2 ] Definition and properties

[ 34.2.1 ] Grades

[ 34.2.2 ] Copula

[ 34.2.3 ] Sklar’s theorem

[ 34.2.4 ] Copula invariance

[ 34.3 ] Special classes of copulas

[ 34.3.1 ] Elliptical copulas

[ 34.3.2 ] Archimedean copulas

[ 34.4 ] Implementation

[ 34.4.1 ] Copula-marginal separation

[ 34.4.2 ] Copula-marginal combination

[ 35 ] Correlation and generalizations

[ 35.1 ] Measures of dependence

[ 35.1.1 ] Schweizer-Wolff measure

[ 35.1.2 ] Mutual information

[ 35.2 ] Measures of concordance

[ 35.2.1 ] Kendall’s tau

[ 35.2.2 ] Spearman’s rho

[ 35.3 ] Correlation

[ 35.4 ] Related definitions

[ 35.5 ] Points of interest, pitfalls, practical tips

[ 35.5.1 ] Schweizer and Wolff measure via simulations

[ 36 ] Stochastic dominance

[ 36.1 ] Strong dominance (order zero dominance)

[ 36.2 ] Weak dominance (first order stochastic dominance)

[ 36.3 ] Second order stochastic dominance

[ 36.4 ] Order q stochastic dominance

[ 36.5 ] Points of interest, pitfalls, practical tips

[ 37 ] Location and dispersion

[ 37.1 ] Expectation and variance

[ 37.1.1 ] Key definitions

[ 37.1.2 ] Visualization: uncertainty band

[ 37.1.3 ] Affine equivariance

[ 37.1.4 ] Variational principles

[ 37.2 ] Expectation and covariance

[ 37.2.1 ] Key definitions

[ 37.2.2 ] Affine equivariance

[ 37.2.3 ] Linear algebra: spectral decomposition

[ 37.2.4 ] Visualization: ellipsoid

[ 37.2.5 ] Statistics: principal component analysis

[ 37.2.6 ] Calculus: most important summary statistics

[ 37.2.7 ] Probability: Chebyshev’s inequality

[ 37.2.8 ] Variational principles

[ 37.3 ] L2 geometry

[ 37.3.1 ] Expectation inner product

[ 37.3.2 ] Covariance (improper) inner product

[ 37.3.3 ] Covariance versus expectation inner product

[ 37.3.4 ] Visualization: Euclidean vectors

[ 37.4 ] Generalized location-dispersion: affine equivariance

[ 37.4.1 ] Univariate case

[ 37.4.2 ] Multivariate case

[ 37.5 ] Generalized location-dispersion: variational principles

[ 37.5.1 ] Key definitions

[ 37.5.2 ] Moment-based location-dispersion

[ 37.5.3 ] Lp-based location-dispersion

[ 37.5.4 ] Partial moment-based location-dispersion

[ 37.5.5 ] Quantile-based location-dispersion

[ 37.5.6 ] Multivariate generalizations

[ 37.6 ] Points of interest, pitfalls, practical tips

[ 37.6.1 ] Alternative visualizations in low dimension

[ 37.6.2 ] Lp geometry

[ 38 ] Spectral analysis

[ 38.1 ] The Fourier transform

[ 38.1.1 ] Linear algebra perspective

[ 38.1.2 ] Functional analysis perspective

[ 38.2 ] Spectrum

[ 38.3 ] Spectral representation

[ 38.3.1 ] Spectral representation restated

[ 38.3.2 ] Finite sample principal component analysis

[ 38.3.3 ] Infinite Toeplitz matrices

[ 38.3.4 ] Infinite sample principal component analysis

[ 38.4 ] Filtering

[ 38.5 ] Estimation

[ 39 ] Decision theory with model uncertainty

[ 39.1 ] Foundations of decision theory

[ 39.1.1 ] Fundamental concepts

[ 39.1.2 ] Frequentist approach

[ 39.1.3 ] Bayesian approach

[ 39.1.4 ] Model risk, estimation risk

[ 40 ] Estimation techniques

[ 40.1 ] Maximum likelihood

[ 40.1.1 ] General theory

[ 40.1.2 ] Hidden variables

[ 40.1.3 ] Relevant cases

[ 40.1.4 ] Numerical methods

[ 40.2 ] Bayesian statistics

[ 40.2.1 ] General theory

[ 40.2.2 ] Bayesian estimation

[ 40.2.3 ] Bayesian prediction

[ 40.2.4 ] Analytical results

[ 40.2.5 ] Numerical methods

[ 41 ] Views processing

[ 41.1 ] Minimum relative entropy

[ 41.1.1 ] Base distribution and view variables

[ 41.1.2 ] Point views

[ 41.1.3 ] Distributional views

[ 41.1.4 ] Partial views

[ 41.1.5 ] Partial views on generalized expectations

[ 41.1.6 ] Sanity check

[ 41.1.7 ] Confidence

[ 41.1.8 ] Relationship with Bayesian updating

[ 41.2 ] Analytical implementation

[ 41.2.1 ] Base distribution

[ 41.2.2 ] Views

[ 41.2.3 ] Updated distribution

[ 41.2.4 ] Confidence

[ 41.2.5 ] Relevant special cases

[ 41.3 ] Flexible probabilities implementation

[ 41.3.1 ] Base distribution

[ 41.3.2 ] Views

[ 41.3.3 ] Updated distribution

[ 41.3.4 ] Confidence

[ 41.4 ] Factor-based implementations

[ 41.5 ] Copula opinion pooling

[ 41.5.1 ] Base distribution

[ 41.5.2 ] Views

[ 41.5.3 ] Updated distribution

[ 41.5.4 ] Confidence

[ 41.5.5 ] The algorithm

[ 41.6 ] Generalized shrinkage

[ 41.6.1 ] Intuition

[ 41.6.2 ] Classical shrinkage

[ 41.6.3 ] Bayesian updating

[ 41.6.4 ] Minimum relative entropy

[ 41.6.5 ] Shrinkage

[ 41.6.6 ] Regularization

[ 42 ] Black-Litterman

[ 42.1 ] Equilibrium prior distribution

[ 42.1.1 ] Prior distribution of expected returns

[ 42.1.2 ] Prior predictive distribution

[ 42.2 ] Active views

[ 42.2.1 ] Setting the parameters

[ 42.2.2 ] Sanity check

[ 42.3 ] Posterior distribution

[ 42.3.1 ] Posterior distribution of expected returns

[ 42.3.2 ] Posterior predictive distribution

[ 42.4 ] Limit cases

[ 42.4.1 ] High confidence in prior

[ 42.4.2 ] Low confidence in views

[ 42.4.3 ] Full confidence in views

[ 42.5 ] Generalizations

[ 42.5.1 ] From linear returns to risk drivers

[ 42.5.2 ] From stock-like to generic asset classes

[ 42.5.3 ] From normal to non-normal markets

[ 42.5.4 ] From linear equality views to partial flexible views

[ 43 ] Estimation and assessment

[ 43.1 ] Background

[ 43.1.1 ] Panels

[ 43.1.2 ] Assumptions

[ 43.1.3 ] Estimation

[ 43.1.4 ] Testing

[ 43.2 ] Probabilistic prediction assessment for invariants

[ 43.2.1 ] Estimators as decisions

[ 43.2.2 ] Frequentist approach

[ 43.2.3 ] Bayesian approach

[ 43.2.4 ] Analytical results

[ 43.2.5 ] Monte Carlo simulations

[ 43.2.6 ] Cross-validation

[ 43.3 ] Bias versus variance

[ 43.4 ] Point prediction assessment

[ 43.4.1 ] Estimators as decisions

[ 43.4.2 ] Frequentist approach

[ 43.4.3 ] Bayesian approach

[ 43.4.4 ] Historical with flexible probabilities estimators

[ 43.4.5 ] Cross-validation

[ 43.5 ] Probabilistic prediction assessment

[ 43.5.1 ] Estimators as decisions

[ 43.5.2 ] Frequentist approach

[ 43.5.3 ] Bayesian approach

[ 43.5.4 ] Maximum likelihood with flexible probabilities

[ 43.5.5 ] Cross-validation

[ 44 ] Hypothesis testing

[ 44.1 ] Hypothesis testing for invariants

[ 44.1.1 ] Statistics

[ 44.1.2 ] P-value

[ 44.1.3 ] Univariate testing: the z-statistic

[ 44.1.4 ] Multivariate testing: the Hotelling statistic

[ 45 ] Stochastic processes cheat sheet

[ 45.1 ] Main definitions

[ 45.1.1 ] Weak white noise

[ 45.1.2 ] White noise

[ 45.1.3 ] Stationary process

[ 45.1.4 ] Integrated processes

[ 45.1.5 ] Ergodic process

[ 45.1.6 ] Cointegrated process

[ 45.1.7 ] Martingale process

[ 45.1.8 ] State process and Markov process

[ 45.1.9 ] Random fields

[ 45.1.10 ] Orthogonal increments process

[ 45.2 ] Relationships among processes

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

[ 45.2.2 ] White noise ⇒ stationary

[ 45.2.3 ] White noise ⇒ ergodic

[ 45.2.4 ] Ergodic ⇒ stationary

[ 45.2.5 ] Stationary ⇒ covariance-stationary

[ 45.2.6 ] Weak white noise ⇒ integrated

[ 45.2.7 ] Weak white noise ⇒ covariance-stationary

[ 45.2.8 ] Covariance-stationary ⇒ integrated

[ 45.3 ] Pitfalls

[ 46 ] Invariance tests

[ 46.1 ] Simple tests

[ 46.2 ] Refinements and pitfalls

[ 46.2.1 ] Circle-like covariance (not data)

[ 46.2.2 ] Stronger tests based on copulas

[ 47 ] Continuous time processes

[ 47.1 ] Efficiency: Lévy processes

[ 47.1.1 ] Infinite divisibility

[ 47.1.2 ] Continuous state: Brownian diffusion

[ 47.1.3 ] Discrete state: Poisson jumps

[ 47.1.4 ] Lévy-Khintchine representation

[ 47.1.5 ] Subordination

[ 47.2 ] Mean-reversion (continuous state)

[ 47.2.1 ] Ornstein-Uhlenbeck process

[ 47.2.2 ] Square-root process and other generalizations

[ 47.3 ] Mean-reversion (discrete state)

[ 47.3.1 ] Time-homogeneous generator

[ 47.3.2 ] Time-inhomogeneous generator

[ 47.3.3 ] Projection of a Markov chain

[ 47.4 ] Long memory: fractional Brownian motion

[ 47.4.1 ] Fractional Brownian motion

[ 47.5 ] Volatility clustering

[ 47.5.1 ] Stochastic volatility

[ 47.5.2 ] Time change

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

[ 48 ] First order autoregression

[ 48.1.1 ] Conditional distribution of AR(1)

[ 48.1.2 ] Stationarity and unconditional distribution of AR(1)

[ 48.2 ] VAR(1)

[ 48.2.1 ] Conditional distribution of VAR(1)

[ 48.2.2 ] Stationarity and unconditional distribution of VAR(1)

[ 48.2.3 ] Cointegrated VAR(1)

[ 48.3 ] Ornstein-Uhlenbeck

[ 48.3.1 ] Conditional distribution of OU

[ 48.3.2 ] Stationarity and unconditional distribution of OU

[ 48.4 ] Multivariate Ornstein-Uhlenbeck

[ 48.4.1 ] Conditional distribution of MVOU

[ 48.4.2 ] Stationarity and unconditional distribution of MVOU

[ 48.4.3 ] Geometrical interpretation∗

[ 48.4.4 ] Cointegrated Ornstein-Uhlenbeck

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

[ 48.5.1 ] MVOU is VAR(1)

[ 48.5.2 ] VAR(1) is MVOU

[ 48.6 ] VAR(1)/MVOU fit

[ 49.1 ] Carry signals

[ 49.1.1 ] Fixed-income

[ 49.1.2 ] Foreign exchange

[ 49.2 ] Value signals

[ 49.2.1 ] Book

[ 49.2.2 ] Pricing

[ 49.3 ] Technical signals

[ 49.3.1 ] Momentum

[ 49.3.2 ] Filters

[ 49.3.3 ] Cointegration

[ 49.4 ] Microstructure signals

[ 49.4.1 ] Trade autocorrelation

[ 49.4.2 ] Order imbalance

[ 49.4.3 ] Price prediction

[ 49.4.4 ] Volume clustering

[ 49.5 ] Fundamental and other signals

[ 49.6 ] Signal processing

[ 49.6.1 ] Smoothing

[ 49.6.2 ] Scoring

[ 49.6.3 ] Ranking

[ 50 ] Optimization primer

[ 50.1 ] Convex programming

[ 50.1.1 ] Conic programming

[ 50.1.2 ] Semidefinite programming

[ 50.1.3 ] Second-order cone programming

[ 50.1.4 ] Quadratic programming

[ 50.1.5 ] Linear programming

[ 50.2 ] Integer ¯n-choose-k selection

[ 50.2.1 ] Naive selection

[ 50.2.2 ] Forward step-wise selection

[ 50.2.3 ] Backward step-wise selection

[ 50.2.4 ] Lasso

[ 51 ] Useful algorithms

[ 51.1 ] Factor analysis algorithms

[ 51.2 ] Matrix transpose-square-root

[ 51.2.1 ] Spectrum

[ 51.2.2 ] Riccati

[ 51.2.3 ] LDL-Cholesky

[ 51.2.4 ] Gram-Schmidt

[ 51.3 ] Markov chain Monte Carlo sampling

[ 51.3.1 ] Metropolis-Hastings

[ 51.4 ] Moment-matching scenarios

[ 51.4.1 ] Twisting scenarios

[ 51.4.2 ] Twisting probabilities

[ 51.5 ] Fast Fourier transform for projection

[ 51.5.1 ] Normalized empirical histogram

[ 51.5.2 ] Approximating the pdf and the characteristic function

[ 51.5.3 ] Using the discrete Fourier transform

[ 51.6 ] Minimum-torsion optimization algorithm

[ 52 ] Primer of linear algebra and calculus

[ 52.1 ] Inner product spaces

[ 52.2 ] Spectral theorem

[ 52.2.1 ] The eigenvalue problem

[ 52.2.2 ] Recursive solution

[ 52.2.3 ] Matrix notation

[ 52.2.4 ] The continuum limit

[ 52.3 ] Matrix algebra

[ 52.3.1 ] Key operators

[ 52.3.2 ] Useful identities

[ 52.4 ] Matrix calculus

[ 52.4.1 ] First order derivatives

[ 52.4.2 ] Second order derivatives

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