| ARPM Lab

Exercises »

E.I. The “Checklist”

[ E.0 ] Executive summary

[ E.1 ] Risk drivers identification

[ E.1.1 ] ODE for a perpetual American call...

[ E.1.2 ] Solution to the ODE for a perpetu...

[ E.1.3 ] Perpetual American option with ar...

[ E.1.4 ] Inverse-call transformation

[ E.1.5 ] The forward rate

[ E.1.6 ] Continuously compounded forward r...

[ E.1.7 ] Instantaneous forward rate

[ E.1.8 ] Spot rate as average of forward r...

[ E.1.9 ] Duration-times-spread

[ E.1.10 ] First derivative of the maximum f...

[ E.1.11 ] The canonical basis property of t...

[ E.1.12 ] Common payoffs as combinations of...

[ E.1.13 ] Forward start variance swap payof...

[ E.1.14 ] Value of forward variance swap

[ E.1.15 ] Alternative formulation of the fa...

[ E.1.16 ] Variance swap fair value as combi...

[ E.1.17 ] Rolling value versus implied vola...

[ E.1.18 ] Parsimonious SVI parametrization ...

[ E.1.19 ] Parsimonious SVI parametrization ...

[ E.1.20 ] Parsimonious SVI parametrization ...

[ E.1.21 ] Parsimonious SVI parametrization ...

[ E.1.22 ] SVI parameters behaving as random...

[ E.1.23 ] Vasicek parametrization of the yi...

[ E.1.24 ] Default probability of an obligor...

[ E.2 ] Quest for invariance

[ E.2.1 ] Market efficiency and random walk

[ E.2.2 ] The Poisson process and the distr...

[ E.2.3 ] Mixture of distributions (I)

[ E.2.4 ] Mixture of distributions (II)

[ E.2.5 ] Mixture of distributions (III)

[ E.2.6 ] Moment generating function and in...

[ E.2.7 ] Stochastic volatility model with ...

[ E.2.8 ] Equivalence between stochastic me...

[ E.2.9 ] Equivalence between stochastic me...

[ E.2.10 ] Equivalence between stochastic me...

[ E.2.11 ] Equivalence between stochastic me...

[ E.2.12 ] Compound probability versus uncou...

[ E.2.13 ] Fast decay of AR(1) autocorrelati...

[ E.2.14 ] Equivalent representation of an A...

[ E.2.15 ] Equivalent representation of an A...

[ E.2.16 ] Fast decay of AR(p) autocorrelati...

[ E.2.17 ] Stationarity of MA(q) processes

[ E.2.18 ] Equivalent representation of an A...

[ E.2.19 ] Fast decay of ARMA(p,q) autocorre...

[ E.2.20 ] ARIMA(1,1,1) model

[ E.2.21 ] Inversion of the linear lag polyn...

[ E.2.22 ] Stationarity of VMA(∞)

[ E.2.23 ] Explicit AR(∞) structure of ARM...

[ E.2.24 ] Explicit AR(∞) structure of ARM...

[ E.2.25 ] Discrete convolution of inverse l...

[ E.2.26 ] Explicit MA(∞) structure of ARM...

[ E.2.27 ] Explicit MA(∞) structure of ARM...

[ E.2.28 ] Stationarity of ARMA(p,q) as MA(�...

[ E.2.29 ] Equivalence of AR(1) and MA(∞)

[ E.2.30 ] Equivalence of MA(1) and AR(∞)

[ E.2.31 ] Wold theorem

[ E.2.32 ] AR(1) as a Markov process

[ E.2.33 ] Time homogeneity for AR(1) proces...

[ E.2.34 ] Non stationary Markov chain

[ E.2.35 ] Autocorrelation of a time-homogen...

[ E.2.36 ] Fast decay of time-homogeneous Ma...

[ E.2.37 ] Scenario-probability distribution...

[ E.2.38 ] Taylor expansion result

[ E.2.39 ] Equivalent formulation for fracti...

[ E.2.40 ] Power law autocorrelation decay f...

[ E.2.41 ] Squared volatility as a function ...

[ E.2.42 ] Stationarity of GARCH

[ E.2.43 ] Autocorrelation of GARCH

[ E.2.44 ] Error correction representation o...

[ E.2.45 ] VAR(1) as special case of Kalman ...

[ E.2.46 ] Matrix polynomial factorization

[ E.2.47 ] Conditional distribution of quasi...

[ E.2.48 ] Unconditional correlation of quas...

[ E.2.49 ] Copula-marginal distribution of q...

[ E.2.50 ] Mixture of invariants model as hi...

[ E.2.51 ] Independence of residuals

[ E.2.52 ] Variance of residuals

[ E.2.53 ] Uncorrelation of residuals

[ E.2.54 ] Wold decomposition

[ E.2.55 ] Expectation of ACD process

[ E.2.56 ] Quasi invariants of ACD process

[ E.2.57 ] Wold representation of ACD

[ E.2.58 ] Wold representation of GARCH(1,1)

[ E.2.59 ] Wold representation of Fractional...

[ E.2.60 ] Curve representation via pricing ...

[ E.2.61 ] Default probabilities via a logit...

[ E.3 ] Estimation

[ E.3.1 ] Glivenko-Cantelli theorem: theory

[ E.3.2 ] Minimum relative entropy as best ...

[ E.3.3 ] Generalized Glivenko-Cantelli the...

[ E.3.4 ] HFP covariance estimator

[ E.3.5 ] Minimum volume ellipsoid

[ E.3.6 ] Kernel with flexible probabilitie...

[ E.3.7 ] Regularized pdf

[ E.3.8 ] Kernel with flexible probabilitie...

[ E.3.9 ] Relative entropy vs. maximum like...

[ E.3.10 ] MLFP estimators: consistency

[ E.3.11 ] MLFP estimators for elliptical va...

[ E.3.12 ] MLFP estimators for the Student t...

[ E.3.13 ] Conditional excess distribution

[ E.3.14 ] Generalized Pareto distribution I

[ E.3.15 ] Influence function as the limit o...

[ E.3.16 ] Gateaux derivative

[ E.3.17 ] Influence function of the sample ...

[ E.3.18 ] Influence function of the sample ...

[ E.3.19 ] Influence function of maximum lik...

[ E.3.20 ] Influence function of location an...

[ E.3.21 ] Influence function of ML estimato...

[ E.3.22 ] Influence function of M-estimator...

[ E.3.23 ] M-estimators: location and disper...

[ E.3.24 ] Minimum volume ellipsoid

[ E.3.25 ] Minimum covariance determinant

[ E.3.26 ] Minimum volume ellipsoid and mini...

[ E.3.27 ] Method of moments with flexible p...

[ E.3.28 ] Equivalent formulation of conditi...

[ E.3.29 ] Maximum likelihood estimation fro...

[ E.3.30 ] Over-specified formulation of the...

[ E.3.31 ] Robust estimation of the covarian...

[ E.3.32 ] ML-estimation of transition proba...

[ E.3.33 ] MLFP estimation of transition pro...

[ E.3.34 ] Expectation-maximization with fle...

[ E.3.35 ] Maximum likelihood with flexible ...

[ E.3.36 ] Realized and empirical variance

[ E.3.37 ] MLFP estimation of GARCH(1,1) wit...

[ E.3.38 ] Exponentially weighted moving ave...

[ E.3.39 ] Dirichlet distribution

[ E.3.40 ] Invariants distribution from the ...

[ E.3.41 ] Posterior distribution of the loc...

[ E.3.42 ] Conditional distribution of two u...

[ E.3.43 ] Exponential family invariants: co...

[ E.3.44 ] Exponential family invariants: po...

[ E.3.45 ] Exponential family invariants: pr...

[ E.3.46 ] Pdf of the information set

[ E.3.47 ] A simpler model for the posterior...

[ E.3.48 ] Normal-inverse-Wishart as the exp...

[ E.3.49 ] Normal-inverse-Wishart location-d...

[ E.3.50 ] Normal-inverse-Wishart location-d...

[ E.3.51 ] Normal-inverse-Wishart location-d...

[ E.3.52 ] Normal-inverse-Wishart location-d...

[ E.3.53 ] Inverse-Wishart dispersion: mode

[ E.3.54 ] Inverse-Wishart dispersion: modal...

[ E.3.55 ] Normal-inverse-Wishart location-d...

[ E.3.56 ] Independence of sample mean and c...

[ E.3.57 ] Distribution of the sample mean

[ E.3.58 ] Distribution of the sample covari...

[ E.3.59 ] Expectation shrinkage

[ E.3.60 ] Singular covariance matrix

[ E.3.61 ] Spectrum analysis

[ E.3.62 ] Distance for sparse matrix shrink...

[ E.3.63 ] Distance for sparse matrix shrink...

[ E.3.64 ] Stein’s lemma

[ E.3.65 ] Shrinkage estimator of location

[ E.3.66 ] Shrinkage estimator of dispersion

[ E.3.67 ] Shrinkage estimator of dispersion...

[ E.3.68 ] Distance matrix for correlation c...

[ E.3.69 ] Low-rank-diagonal structure for c...

[ E.3.70 ] Low-rank-diagonal structure on co...

[ E.3.71 ] Conditional covariance of normal ...

[ E.3.72 ] Covariance and correlation of res...

[ E.3.73 ] p-value of the sample mean: norma...

[ E.3.74 ] t-statistic of the sample mean: n...

[ E.3.75 ] Univariate testing: the z-statist...

[ E.3.76 ] Univariate testing: the square st...

[ E.3.77 ] Univariate testing: consistency o...

[ E.3.78 ] Multivariate testing: consistency...

[ E.3.79 ] Multivariate testing: the Hotelli...

[ E.3.80 ] Sample mean loss distribution

[ E.3.81 ] Error, bias, inefficiency

[ E.3.82 ] Inefficiency of the sample mean

[ E.3.83 ] Posterior expectation

[ E.3.84 ] Estimation error of the sample me...

[ E.3.85 ] Sample mean loss in a homogeneous...

[ E.3.86 ] Estimation error of the sample co...

[ E.3.87 ] The Bayes error of discrete model...

[ E.3.88 ] The worst-case error of discrete ...

[ E.3.89 ] Non-parametric classification: su...

[ E.4 ] Projection

[ E.4.1 ] Distribution of the sum of indepe...

[ E.4.2 ] Sum of two independent variables ...

[ E.4.3 ] Sum of random variables via simul...

[ E.4.4 ] Linear projection for the expecta...

[ E.4.5 ] Square-root rule for the covarian...

[ E.4.6 ] Square-root rule for a generic st...

[ E.4.7 ] Non-central moments to central mo...

[ E.4.8 ] Cumulant projection

[ E.4.9 ] Central moments of a normal rando...

[ E.4.10 ] MVOU joint distribution of a path

[ E.4.11 ] Multivariate GARCH

[ E.4.12 ] Projection of GARCH

[ E.4.13 ] Projection by averaging the histo...

[ E.4.14 ] Hybrid Monte Carlo-historical pro...

[ E.4.15 ] Principal component for multivari...

[ E.5 ] Pricing at the horizon

[ E.5.1 ] Dynamics and distribution of the ...

[ E.5.2 ] P&L of a forward contract

[ E.5.3 ] Put option P&L function

[ E.5.4 ] Currency carry

[ E.5.5 ] Foreign exchange carry trade

[ E.5.6 ] Annualized carry return of a zero...

[ E.5.7 ] Annualized carry return of a bond...

[ E.5.8 ] Carry of a variance swap

[ E.5.9 ] Numerical Greeks of P&L approxima...

[ E.5.10 ] Greeks of equity P&L with stock v...

[ E.5.11 ] Greeks of equity P&L with stock l...

[ E.5.12 ] Bond Greeks

[ E.5.13 ] Bond yield

[ E.5.14 ] Bond convexity

[ E.5.15 ] M-square

[ E.5.16 ] Parallel shift of the yield curve...

[ E.5.17 ] Taylor approximation of the P&L o...

[ E.5.18 ] Equivalent definitions of effecti...

[ E.5.19 ] Taylor approximation of variance ...

[ E.5.20 ] Global quadratic approximation fo...

[ E.5.21 ] Analytical distribution of the P&...

[ E.5.22 ] Elliptical risk drivers: location...

[ E.5.23 ] Analytical distribution of the jo...

[ E.6 ] Aggregation

[ E.6.1 ] Normal P&L’s imply normal retur...

[ E.6.2 ] Equally weighted portfolio

[ E.6.3 ] Distribution of the portfolio P&L...

[ E.6.4 ] Generator of ex-ante performance ...

[ E.6.5 ] Regulatory credit framework: one-...

[ E.6.6 ] CVA computation under simplifying...

[ E.6.7 ] Regulatory credit framework: cond...

[ E.6.8 ] Conditional log-characteristic fu...

[ E.6.9 ] Unconditional log-characteristic ...

[ E.6.10 ] Minimum collateral

[ E.6.11 ] Gross and net exposure

[ E.7 ] Ex-ante evaluation

[ E.7.1 ] Properties of the ex-ante perform...

[ E.7.2 ] Strong dominance implies arbitrag...

[ E.7.3 ] Estimability and monotonicity imp...

[ E.7.4 ] The negative variance and the mea...

[ E.7.5 ] The negative variance is concave

[ E.7.6 ] The mean-variance trade-off is tr...

[ E.7.7 ] The negative variance and the mea...

[ E.7.8 ] The negative variance and the mea...

[ E.7.9 ] The negative variance is not risk...

[ E.7.10 ] Expected value: consistency with ...

[ E.7.11 ] Expected value: consistency with ...

[ E.7.12 ] Translation invariance implies co...

[ E.7.13 ] Mean-variance trade-off: second o...

[ E.7.14 ] Certainty-equivalent: uniqueness

[ E.7.15 ] Certainty-equivalent: estimabilit...

[ E.7.16 ] Certainty-equivalent: monotonicit...

[ E.7.17 ] Certainty-equivalent: consistency...

[ E.7.18 ] Certainty-equivalent: consistency...

[ E.7.19 ] Certainty-equivalent: constancy

[ E.7.20 ] Certainty-equivalent: money-equiv...

[ E.7.21 ] Certainty-equivalent: positive ho...

[ E.7.22 ] Certainty-equivalent: Euler decom...

[ E.7.23 ] Certainty-equivalent: translation...

[ E.7.24 ] Certainty-equivalent: additivity ...

[ E.7.25 ] Certainty-equivalent: co-monotoni...

[ E.7.26 ] Certainty-equivalent: risk aversi...

[ E.7.27 ] Relation between Arrow-Pratt risk...

[ E.7.28 ] Certainty-equivalent and positive...

[ E.7.29 ] Certainty-equivalent: gradient

[ E.7.30 ] Certainty-equivalent: Hessian

[ E.7.31 ] Certainty-equivalent: gradient (s...

[ E.7.32 ] Certainty-equivalent (quadratic n...

[ E.7.33 ] Certainty-equivalent (elliptical ...

[ E.7.34 ] Certainty-equivalent: gradient (e...

[ E.7.35 ] Certainty-equivalent: Hessian (el...

[ E.7.36 ] Certainty-equivalent, gradient an...

[ E.7.37 ] The value at risk

[ E.7.38 ] The utility function as the cdf o...

[ E.7.39 ] The certainty-equivalent as the q...

[ E.7.40 ] Quantile (VaR) satisfaction measu...

[ E.7.41 ] Quantile (VaR) satisfaction measu...

[ E.7.42 ] Quantile (VaR) satisfaction measu...

[ E.7.43 ] Quantile (VaR) satisfaction measu...

[ E.7.44 ] Quantile (VaR) satisfaction measu...

[ E.7.45 ] Quantile (VaR) satisfaction measu...

[ E.7.46 ] Quantile (VaR) satisfaction measu...

[ E.7.47 ] Quantile (VaR) satisfaction measu...

[ E.7.48 ] Quantile (VaR) satisfaction measu...

[ E.7.49 ] Quantile (VaR) satisfaction measu...

[ E.7.50 ] Quantile (VaR) satisfaction measu...

[ E.7.51 ] Quantile (VaR) satisfaction measu...

[ E.7.52 ] Quantile (VaR) satisfaction measu...

[ E.7.53 ] Quantile (VaR) satisfaction measu...

[ E.7.54 ] Quantile (VaR) satisfaction measu...

[ E.7.55 ] Central moments of an affine tran...

[ E.7.56 ] Variance of an affine transformat...

[ E.7.57 ] Expectation, standard deviation a...

[ E.7.58 ] Economic capital

[ E.7.59 ] Coherent satisfaction measure: ri...

[ E.7.60 ] Equivalent definitions of monoton...

[ E.7.61 ] The mean-lower partial moment tra...

[ E.7.62 ] The mean-semivariance is not cohe...

[ E.7.63 ] Dual representation of the expect...

[ E.7.64 ] Worst possible measure for the ex...

[ E.7.65 ] Spectral satisfaction measures: e...

[ E.7.66 ] Spectral satisfaction measures ar...

[ E.7.67 ] Spectral satisfaction measures: c...

[ E.7.68 ] Spectral satisfaction measures: c...

[ E.7.69 ] Spectral satisfaction measures: m...

[ E.7.70 ] Spectral satisfaction measures: p...

[ E.7.71 ] Spectral satisfaction measures: t...

[ E.7.72 ] Spectral satisfaction measures: c...

[ E.7.73 ] Spectral satisfaction measures: v...

[ E.7.74 ] Spectral satisfaction measures: v...

[ E.7.75 ] Spectral satisfaction measures: v...

[ E.7.76 ] Spectral satisfaction measures: v...

[ E.7.77 ] The expected shortfall: distortio...

[ E.7.78 ] The Wang expectation: distortion ...

[ E.7.79 ] The Buhlmann expectation is not a...

[ E.7.80 ] Equivalence between spectral meas...

[ E.7.81 ] Coherent spectral satisfaction me...

[ E.7.82 ] Coherent spectral satisfaction me...

[ E.7.83 ] Alternative representation of spe...

[ E.7.84 ] Mean-lower partial moment trade-o...

[ E.7.85 ] Coherent satisfaction measures: c...

[ E.7.86 ] Coherent satisfaction measures: c...

[ E.7.87 ] Coherent satisfaction measures: m...

[ E.7.88 ] Coherent satisfaction measures: c...

[ E.7.89 ] The expected shortfall

[ E.7.90 ] The Wang distortion expectation

[ E.7.91 ] The proportional hazards distorti...

[ E.7.92 ] Spectral measures as weighted ave...

[ E.7.93 ] Spectral satisfaction measures: g...

[ E.7.94 ] Cornish-Fisher approximation for ...

[ E.7.95 ] Extreme value theory: approximati...

[ E.7.96 ] Expected shortfall weights (scena...

[ E.7.97 ] The spectral satisfaction measure...

[ E.7.98 ] Approximation of spectral satisfa...

[ E.7.99 ] Spectral satisfaction measures: g...

[ E.7.100 ] Spectral satisfaction measures: g...

[ E.7.101 ] Spectral satisfaction measures: H...

[ E.7.102 ] Quantile (VaR) satisfaction measu...

[ E.7.103 ] Expected shortfall: Hessian (elli...

[ E.7.104 ] Derivative of an indefinite integ...

[ E.7.105 ] Quantile (VaR) satisfaction measu...

[ E.7.106 ] Expected shortfall: gradient

[ E.7.107 ] Expected shortfall: gradient (ell...

[ E.7.108 ] Relation between the omega and th...

[ E.7.109 ] The Buhlmann expectation is a dis...

[ E.7.110 ] First order approximation of the ...

[ E.7.111 ] Esscher transform as minimum entr...

[ E.7.112 ] First order approximation of the ...

[ E.7.113 ] The Esscher expectation is neithe...

[ E.7.114 ] Buhlmann expectation: linearity

[ E.7.115 ] The Arrow-Pratt approximation

[ E.7.116 ] Esscher expectation under normali...

[ E.7.117 ] Buhlmann expectation under normal...

[ E.8a ] Ex-ante attribution: performance

[ E.8a.1 ] Joint distribution factor and res...

[ E.8a.2 ] Relationship between bottom-up an...

[ E.8a.3 ] Black-Scholes-Merton delta hedgin...

[ E.8a.4 ] Portfolio-based risk-attribution

[ E.8a.5 ] Risk attribution: long horizon

[ E.8b ] Ex-ante attribution: risk

[ E.8b.1 ] Euler decomposition of the certai...

[ E.8b.2 ] Euler decomposition of distortion...

[ E.8b.3 ] Risk attribution (elliptical case...

[ E.8b.4 ] Computation of the marginal contr...

[ E.8b.5 ] Computation of the Esscher expect...

[ E.8b.6 ] Computation of the marginal contr...

[ E.8b.7 ] Marginal risk contributions for t...

[ E.8b.8 ] Marginal risk contributions for t...

[ E.8b.9 ] The economic capital is positive ...

[ E.8b.10 ] The minimum-torsion diversificati...

[ E.8b.11 ] Effective number of bets

[ E.8b.12 ] Risk attribution: principal compo...

[ E.8b.13 ] The principal components diversif...

[ E.8b.14 ] Shapley risk attribution

[ E.8b.15 ] Esscher expectation risk attribut...

[ E.8b.16 ] Economic capital risk attribution

[ E.9a ] Construction: portfolio optimizat...

[ E.9a.1 ] Portfolio optimization problem

[ E.9b ] Construction: cross-sectional str...

[ E.9b.1 ] R-squared and correlation

[ E.9b.2 ] Market-capitalization allocation

[ E.9b.3 ] Fundamental law of active managem...

[ E.9b.4 ] Correlation and premium

[ E.9b.5 ] Information coefficient under nor...

[ E.9b.6 ] Signal characteristic portfolio a...

[ E.9b.7 ] Maximum information ratio: 3-vari...

[ E.9b.8 ] Information coefficient: 3-variat...

[ E.9b.9 ] Information coefficient aggregati...

[ E.9b.10 ] Information coefficient for group...

[ E.9b.11 ] Information coefficient aggregati...

[ E.9b.12 ] Maximum information ratio aggrega...

[ E.9b.13 ] Maximum information ratio for gro...

[ E.9b.14 ] Variance of weak signal: 3-variat...

[ E.9b.15 ] Maximum information ratio aggrega...

[ E.9b.16 ] Linkage matrix: 3-variate normal ...

[ E.9b.17 ] Maximum information ratio bound

[ E.9b.18 ] Signal-to-noise ratio of independ...

[ E.9b.19 ] Equivalent formulation of informa...

[ E.9b.20 ] Maximal constrained signal-to-noi...

[ E.9b.21 ] Maximal conditional signal-to-noi...

[ E.9b.22 ] Maximal conditional signal-to-noi...

[ E.9b.23 ] Fundamental law of active managem...

[ E.9b.24 ] Smart beta: factor premium

[ E.9b.25 ] Fundamental law of active managem...

[ E.9b.26 ] Flexible and standard characteris...

[ E.9b.27 ] Linkage matrix and signal weaknes...

[ E.9b.28 ] Characteristic portfolio for equi...

[ E.9b.29 ] Characteristic portfolio variance

[ E.9c ] Construction: time series strateg...

[ E.9c.1 ] Self-financing constraint for por...

[ E.9c.2 ] Strategy dynamics (the general ca...

[ E.9c.3 ] Strategy dynamics (arithmetic Bro...

[ E.9c.4 ] Strategy dynamics (geometric Brow...

[ E.9c.5 ] Strategy distributions (arithmeti...

[ E.9c.6 ] Strategy distributions (geometric...

[ E.9c.7 ] Partial differential equation for...

[ E.9c.8 ] Dynamic payoff replication strate...

[ E.9c.9 ] Maximum utility for arithmetic Br...

[ E.9c.10 ] Maximum utility for geometric Bro...

[ E.9c.11 ] Utility maximization versus payof...

[ E.9c.12 ] Payoff function of utility maximi...

[ E.9c.13 ] Payoff function of utility maximi...

[ E.9c.14 ] PDE of power utility maximization

[ E.9c.15 ] Solution of the power utility max...

[ E.9c.16 ] Cushion of the CPPI strategy

[ E.9c.17 ] Linear time invariant filter in c...

[ E.9c.18 ] The dynamic of a linear time inva...

[ E.9c.19 ] Exponentially weighted moving ave...

[ E.9c.20 ] Dynamics of the exponentially wei...

[ E.9c.21 ] P&L of signal induced strategies

[ E.9c.22 ] A simple signal induced strategy

[ E.9d ] Construction: estimation and mode...

[ E.10 ] Execution

[ E.10.1 ] VWAP trading strategy

[ E.10.2 ] Meaning of one unit of volume tim...

[ E.10.3 ] Interpretation of the trading spe...

[ E.10.4 ] Market impact P&L

[ E.10.5 ] Expectation and variance of the t...

[ E.10.6 ] Normalized market impact model

[ E.10.7 ] Expectation and variance of the m...

[ E.10.8 ] Mean-variance optimization proble...

[ E.10.9 ] P&L optimization: Almgren-Chriss ...

[ E.10.10 ] The VWAP trading strategy in the ...

[ E.10.11 ] Optimization problem in the multi...

[ E.10.12 ] Solution of the multidimensional ...

[ E.10.13 ] Market impact P&L under the Almgr...

[ E.10.14 ] Expectation and variance of the m...

[ E.10.15 ] Transient impact: the optimizatio...

[ E.10.16 ] Transient impact: the Obizhaeva-W...

[ E.10.17 ] Transient impact: the Dang model

[ E.10.18 ] Transient impact: power law decay...

[ E.10.19 ] Transient impact: logarithmic dec...

[ E.10.20 ] Price manipulation

[ E.10.21 ] Zero-intelligence model: statisti...

[ E.10.22 ] Zero-intelligence model: average ...

[ E.10.23 ] Zero-intelligence model: price in...

[ E.10.24 ] Zero-intelligence model: improvem...

[ E.10.25 ] The Bouchaud-Mézard-Potters mode...

[ E.10.26 ] The Bae-Jang-Park model [???work ...

[ E.10.27 ] The Ellul-Holden-Jain-Jennings mo...

[ E.10.28 ] The Gabaix-Gopikrishnan-Plerou-St...

[ E.10.29 ] The Mike-Farmer model [???work in...

[ E.10.30 ] The Hewlett model [???work in pro...

[ E.10.31 ] The Avellaneda-Stoikov model [???...

[ E.10.32 ] The Lo-Sapp model [???work in pro...

[ E.10.33 ] The Lipton-Pesavento-Sotiropoulos...

[ E.10.34 ] The Robert-Rosenbaum model [???wo...

[ E.10.35 ] The Lachapelle-Lasry-Lehalle-Lion...

[ E.10.36 ] The Burlakov-Kamal-Salvadore mode...

E.II. Factor models and learning

[ E.11 ] Executive summary

[ E.12 ] Linear factor models: theory

[ E.12.1 ] Equivalent expression of dominant...

[ E.12.2 ] Regression LFM’s: differential ...

[ E.12.3 ] Regression LFM’s: concavity of ...

[ E.12.4 ] Regression LFM’s: loadings

[ E.12.5 ] Regression LFM’s: covariance of...

[ E.12.6 ] Regression LFM’s: independence ...

[ E.12.7 ] Regression LFM’s: covariance of...

[ E.12.8 ] Regression LFM’s: r-squared

[ E.12.9 ] Regression LFM’s: target covari...

[ E.12.10 ] Parametrization of a square-dispe...

[ E.12.11 ] Eigenvalues of 2×2 positive matr...

[ E.12.12 ] Eigenvectors of 2×2 positive mat...

[ E.12.13 ] Principal factors and components ...

[ E.12.14 ] Principal-component LFM’s: diff...

[ E.12.15 ] Principal-component LFM’s: load...

[ E.12.16 ] Principal-component LFM’s: cons...

[ E.12.17 ] Principal-component LFM’s: load...

[ E.12.18 ] Principal-component LFM’s: comp...

[ E.12.19 ] Principal-component LFM’s: resc...

[ E.12.20 ] Eigenvectors property

[ E.12.21 ] Principal-component LFM’s: cova...

[ E.12.22 ] Principal-component LFM’s: cova...

[ E.12.23 ] Principal-component LFM’s: cova...

[ E.12.24 ] Principal-component LFM’s: r-sq...

[ E.12.25 ] Scaling invariance of principal c...

[ E.12.26 ] Principal-component LFM’s: zero...

[ E.12.27 ] Principal-component LFM’s: prin...

[ E.12.28 ] Principal-component LFM’s: equi...

[ E.12.29 ] Eigenfunctions property

[ E.12.30 ] Systematic-idiosyncratic LFM’s:...

[ E.12.31 ] Systematic-idiosyncratic LFM’s:...

[ E.12.32 ] Systematic-idiosyncratic LFM’s:...

[ E.12.33 ] Systematic-idiosyncratic LFM’s:...

[ E.12.34 ] Systematic-idiosyncratic LFM’s:...

[ E.12.35 ] Systematic-idiosyncratic LFM’s:...

[ E.12.36 ] Systematic-idiosyncratic LFM’s:...

[ E.12.37 ] Systematic-idiosyncratic LFM’s:...

[ E.12.38 ] Factor analysis problem: simple b...

[ E.12.39 ] Principle axis factorization: ite...

[ E.12.40 ] Rank property and positive defini...

[ E.12.41 ] Cross-sectional LFM’s: differen...

[ E.12.42 ] Cross-sectional LFM’s: concavit...

[ E.12.43 ] Cross-sectional LFM’s: construc...

[ E.12.44 ] Cross-sectional LFM’s: rank of ...

[ E.12.45 ] Cross-sectional LFM’s: r-square...

[ E.12.46 ] Cross-sectional LFM’s: covarian...

[ E.12.47 ] Cross-sectional LFM’s: regressi...

[ E.12.48 ] Cross-sectional LFM’s: compleme...

[ E.12.49 ] Cross-sectional LFM’s: r-square...

[ E.12.50 ] Cross-sectional LFM’s: minimum-...

[ E.12.51 ] Cross-sectional LFM’s: regressi...

[ E.12.52 ] Cross-sectional LFM’s: equivale...

[ E.12.53 ] Factor extraction: principal-comp...

[ E.12.54 ] Cross-sectional LFM’s: cross-se...

[ E.12.55 ] Matrix decomposition

[ E.12.56 ] Principal-component LFM’s: equi...

[ E.12.57 ] Regression LFM’s: equivalent fo...

[ E.12.58 ] Symmetric regression: analytical ...

[ E.12.59 ] Systematic-idiosyncratic LFM’s:...

[ E.12.60 ] Regression LFM’s as quadratic p...

[ E.12.61 ] Cross-sectional LFM as quadratic ...

[ E.12.62 ] Cross-sectional LFM’s: uncorrel...

[ E.13 ] Linear factor models: estimation

[ E.13.1 ] Static regression LFM’s: sample...

[ E.13.2 ] Static principal component estima...

[ E.13.3 ] Static cross-sectional LFM’s: s...

[ E.14 ] Linear factor models: pitfalls

[ E.14.1 ] Inconsistency between factor anal...

[ E.15 ] Machine learning foundations

[ E.15.1 ] Best linear approximation: soluti...

[ E.15.2 ] Best linear approximation: indepe...

[ E.15.3 ] Best linear approximation: best l...

[ E.15.4 ] Linear innovation: the lognormal ...

[ E.15.5 ] Law of total variance in ANOVA mo...

[ E.15.6 ] Quantile regression solution

[ E.15.7 ] Expectation and covariance of the...

[ E.15.8 ] Optimal generative model: mixture...

[ E.15.9 ] Discriminant classification: cond...

[ E.15.10 ] Non-parametric classification: jo...

[ E.15.11 ] Non-parametric classification: eq...

[ E.15.12 ] Non-parametric classification: co...

[ E.15.13 ] Non-parametric classification: fa...

[ E.15.14 ] Non-parametric classification: Ne...

[ E.15.15 ] Non-parametric classification: op...

[ E.15.16 ] Generative classification: quadra...

[ E.15.17 ] Non-parametric classification: RO...

[ E.15.18 ] Classification: point prediction ...

[ E.15.19 ] Supervised point predictors: fals...

[ E.15.20 ] Conditionally orthogonal linear m...

[ E.15.21 ] Independent components via condit...

[ E.15.22 ] Probabilistic factor analysis: hi...

[ E.15.23 ] Generative classification: point ...

[ E.15.24 ] Generalized linear models: probab...

[ E.15.25 ] Generalized linear models: probab...

[ E.15.26 ] Naive Bayes classifier: condition...

[ E.15.27 ] Naive Bayes models: weight of evi...

[ E.15.28 ] Naive Bayes models: weight of evi...

[ E.15.29 ] Naive Bayes models: naive Bayes c...

[ E.15.30 ] Unsupervised predictor: k-means c...

[ E.15.31 ] Computation of the conditional sc...

[ E.15.32 ] Scoring assessment measures

[ E.15.33 ] From logarithmic score to relativ...

[ E.15.34 ] Analysis of variance test

[ E.15.35 ] Mixture models: probability of th...

[ E.15.36 ] Maximum likelihood factorization:...

[ E.15.37 ] Maximum likelihood factorization:...

[ E.16 ] Estimation and assessment

[ E.16.1 ] The Bayes error of the relative e...

[ E.16.2 ] The posterior error of the relati...

[ E.17 ] Bias-variance enhancements

[ E.17.1 ] Exponential format of an arbitrar...

[ E.17.2 ] Perceptron, support vector machin...

[ E.17.3 ] Feature engineering: error deriva...

[ E.18 ] Dynamic models

[ E.18.1 ] Dominant residual DFM: equivalent...

[ E.18.2 ] Dominant residual DFM: equivalent...

[ E.18.3 ] Derivation of the dynamic regress...

[ E.18.4 ] Derivation of the dynamic princip...

[ E.18.5 ] Derivation of the dynamic princip...

[ E.19 ] Deep learning

[ E.20 ] Application: principal component ...

[ E.20.1 ] Martingale property

[ E.20.2 ] Spectral basis in the continuum

[ E.20.3 ] Yields in the fixed income market...

[ E.20.4 ] Eigenvalues integration

[ E.21 ] Application: regression

[ E.21.1 ] Regression LFM’s: maximum likel...

[ E.21.2 ] Regression LFM’s: maximum likel...

[ E.21.3 ] Ridge regression: loadings

[ E.21.4 ] Lasso as a generalization of maxi...

[ E.21.5 ] Regression LFM’s: likelihood

[ E.21.6 ] Regression LFM’s: distribution ...

[ E.21.7 ] Regression LFM’s: distribution ...

[ E.21.8 ] Regression LFM’s: posterior dis...

[ E.21.9 ] Regression LFM’s: mode of poste...

[ E.21.10 ] Regression LFM’s: modal dispers...

[ E.21.11 ] Regression LFM’s: predictive di...

[ E.21.12 ] Influence function of the OLSFP e...

[ E.21.13 ] Regression LFM’s: t-test of fac...

[ E.22 ] Application: credit default

[ E.22.1 ] Logistic regression: maximum like...

[ E.23 ] Application: clustering

E.III. Valuation

[ E.24 ] Executive summary

[ E.25 ] Background definitions

[ E.25.1 ] Value from jointly lognormal SDF ...

[ E.25.2 ] Nonlinear Valuation from jointly ...

[ E.25.3 ] Cash-flow record date

[ E.25.4 ] Jump rule. Theory

[ E.25.5 ] Cash-flow and P&L additivity

[ E.25.6 ] Additivity of the reinvested cumu...

[ E.25.7 ] Generalized forward cash-flow-adj...

[ E.25.8 ] Payoff and forward cash-flow adju...

[ E.25.9 ] Forward and backward cash-flow-ad...

[ E.25.10 ] Conversion of the P&L from local ...

[ E.25.11 ] Fair value in base currency

[ E.25.12 ] Conversion rule for linear return...

[ E.25.13 ] Conversion rule for linear return...

[ E.25.14 ] The actual exchange rate

[ E.25.15 ] Coupon bond dirty price

[ E.25.16 ] P&L of a futures

[ E.26 ] Linear pricing theory: core

[ E.26.1 ] Linearity implies the law of one ...

[ E.26.2 ] Linear pricing operator

[ E.26.3 ] Equivalent statements of absence ...

[ E.26.4 ] Kernel stochastic discount factor

[ E.26.5 ] Minimum relative entropy numerair...

[ E.26.6 ] Risk neutral probability measure

[ E.26.7 ] One period risk-neutral probabili...

[ E.26.8 ] Continuous time risk-neutral prob...

[ E.26.9 ] Continuous rebalancing limit: ris...

[ E.26.10 ] Risk-neutral density in the Black...

[ E.26.11 ] The risk-neutral density is log-n...

[ E.26.12 ] Derivation of the Black-Scholes-M...

[ E.26.13 ] From the Wang transform to the Bl...

[ E.26.14 ] Heston model: closed-form solutio...

[ E.26.15 ] Derivation of the fundamental the...

[ E.26.16 ] Risk-neutral expected return

[ E.26.17 ] Maximum Sharpe Ratio portfolio ve...

[ E.26.18 ] Invariance of maximum Sharpe rati...

[ E.26.19 ] Martingality of the Radon-Nikodym...

[ E.26.20 ] Equivalent formulation of numerai...

[ E.26.21 ] Stochastic discount factor linear...

[ E.26.22 ] CAPM-like identity from the stoch...

[ E.26.23 ] About the “insurance”term app...

[ E.26.24 ] First order conditions for the Ma...

[ E.26.25 ] CAPM-like identity on P&L’s

[ E.26.26 ] Equivalent formulation of the max...

[ E.26.27 ] Equivalent formulation of securit...

[ E.27 ] Linear pricing theory: further as...

[ E.27.1 ] Inverse of the basis payoff matri...

[ E.27.2 ] European payoffs as combinations ...

[ E.27.3 ] Common payoffs as combinations of...

[ E.27.4 ] Common payoffs as combinations of...

[ E.27.5 ] Common payoffs as combinations of...

[ E.27.6 ] Common payoffs as combinations of...

[ E.27.7 ] Parabola payoff as combinations o...

[ E.27.8 ] Current value of an arbitrary pay...

[ E.27.9 ] Each instrument in a complete mar...

[ E.27.10 ] Arrow-Debreu securities as butter...

[ E.27.11 ] Butterfly in terms of second-orde...

[ E.27.12 ] Arrow-Debreu securities under con...

[ E.27.13 ] Conditional stochastic discount f...

[ E.27.14 ] Second derivative of the call opt...

[ E.27.15 ] Stochastic discount factor in the...

[ E.27.16 ] Distribution of the stochastic di...

[ E.27.17 ] From the security market line to ...

[ E.27.18 ] From Buhlmann transform to the co...

[ E.27.19 ] Systematic-idiosyncratic linear f...

[ E.27.20 ] Systematic-idiosyncratic linear f...

[ E.27.21 ] Factor-replicating portfolios

[ E.27.22 ] Derivation of the APT model from ...

[ E.27.23 ] From APT to CAPM-like identity

[ E.27.24 ] Binomial tree for geometric Brown...

[ E.27.25 ] Intertemporal consistency of the ...

[ E.27.26 ] Risk-neutral probability of binom...

[ E.27.27 ] The abstract Bayes theorem

[ E.28 ] Non-linear pricing theory

[ E.28.1 ] Equivalence between the distortio...

[ E.28.2 ] Shift principles: calibration

[ E.28.3 ] Exponential principle: calibratio...

[ E.28.4 ] Wang distortion principle: calibr...

[ E.28.5 ] Esscher principle: calibration

[ E.29 ] Valuation implementation

[ E.29.1 ] The Gordon growth model

[ E.29.2 ] The value of an oil field product...

[ E.29.3 ] Enterprise value using comparable...

[ E.29.4 ] Value of a pure endowment contrac...

[ E.29.5 ] Value of a pure endowment contrac...

[ E.29.6 ] Non-life insurance valuation in t...

[ E.29.7 ] Non-life insurance valuation with...

[ E.29.8 ] Non-life insurance valuation. App...

E.IV. Performance analysis

[ E.30 ] Executive summary

[ E.31 ] Performance definitions

[ E.31.1 ] Computation of the internal rate ...

[ E.31.2 ] The P&L linearity

[ E.31.3 ] Trading P&L (single trade)

[ E.31.4 ] Decomposition of the total tradin...

[ E.31.5 ] The implementation shortfall in t...

[ E.31.6 ] Trading P&L (multiple trading dat...

[ E.31.7 ] Trading P&L: opening and liquidat...

[ E.31.8 ] Aggregation property of linear re...

[ E.31.9 ] Aggregation property of compounde...

[ E.31.10 ] Alternative formulation for the g...

[ E.31.11 ] Compounded rate of return in term...

[ E.31.12 ] Par swap rate as IRR of a coupon ...

[ E.31.13 ] Generalized portfolio weights

[ E.31.14 ] Offset cash

[ E.32 ] Performance attribution

E.V. Quant toolbox

[ E.33 ] Summary

[ E.34 ] Multivariate distributions

[ E.34.1 ] Pdf of an invertible function of ...

[ E.34.2 ] Cdf of an invertible function of ...

[ E.34.3 ] Quantile function and inverse cdf

[ E.34.4 ] Quantile of an invertible functio...

[ E.34.5 ] Expected value in terms of the qu...

[ E.34.6 ] Pdf, cdf and quantile function of...

[ E.34.7 ] Equivalence between Gamma and chi...

[ E.34.8 ] Expectation of the exponential of...

[ E.34.9 ] Wishart and gamma distribution

[ E.34.10 ] Marginals of a Wishart distributi...

[ E.34.11 ] Result on the joint distribution ...

[ E.34.12 ] Conditional pdf

[ E.34.13 ] Conditional quantile and conditio...

[ E.34.14 ] Conditional and unconditional exp...

[ E.34.15 ] Conditional and unconditional inv...

[ E.34.16 ] Conditional distribution between ...

[ E.34.17 ] Conditional distribution between ...

[ E.34.18 ] Conditional signal-to-noise ratio...

[ E.34.19 ] Covariances and correlations para...

[ E.34.20 ] Conditional expectation and covar...

[ E.34.21 ] Characterization of conditional e...

[ E.34.22 ] Best approximation: equivalent ch...

[ E.34.23 ] Best approximation: solution

[ E.34.24 ] Innovation: decoupling problem an...

[ E.34.25 ] Best approximation: quantile repr...

[ E.34.26 ] Innovation: the lognormal case, r...

[ E.34.27 ] Co-monotonic invariance of innova...

[ E.34.28 ] Pdf of an invertible function of ...

[ E.34.29 ] Cdf of an invertible co-monotonic...

[ E.34.30 ] Characteristic function of an aff...

[ E.34.31 ] Pdf of a non-invertible affine tr...

[ E.34.32 ] Relation between the characterist...

[ E.34.33 ] Characteristic function of a mult...

[ E.34.34 ] Non-central moments of a multivar...

[ E.34.35 ] Moments of the reflected shifted ...

[ E.34.36 ] Covariances of a multivariate log...

[ E.34.37 ] Expectation, standard deviation a...

[ E.34.38 ] Gradient of the pdf of a multivar...

[ E.34.39 ] Hessian of the pdf of a multivari...

[ E.34.40 ] Gradient of the log-pdf of a mult...

[ E.34.41 ] Hessian of the log-pdf of a multi...

[ E.34.42 ] Pdf of an inverse-Wishart random ...

[ E.34.43 ] Equivalence between definitions o...

[ E.34.44 ] Radial component and generator fu...

[ E.34.45 ] Radial component of multivariate ...

[ E.34.46 ] Radial component of multivariate ...

[ E.34.47 ] Radial component of a uniform ran...

[ E.34.48 ] Moments of an elliptical random v...

[ E.34.49 ] Expectation of Mahalanobis square...

[ E.34.50 ] Moments of a uniform random varia...

[ E.34.51 ] Moments of the uniform component ...

[ E.34.52 ] Elliptical distributions: formula...

[ E.34.53 ] Elliptical distributions: generat...

[ E.34.54 ] Marginal distribution of a unifor...

[ E.34.55 ] Conditioned radial component of m...

[ E.34.56 ] Truncated quantile

[ E.34.57 ] Stress distribution of elliptical...

[ E.34.58 ] Stress quantile of elliptical dis...

[ E.34.59 ] Alternative stochastic representa...

[ E.34.60 ] Scenario-probability framework: c...

[ E.34.61 ] Scenario-probability framework: p...

[ E.34.62 ] Scenario-probability framework: p...

[ E.34.63 ] Scenario-probability distribution...

[ E.34.64 ] Scenario-probability distribution...

[ E.34.65 ] Scenario-probability distribution...

[ E.34.66 ] Scenario-probability distribution...

[ E.34.67 ] Scenario-probability distribution...

[ E.34.68 ] Scenario-probability distribution...

[ E.34.69 ] Scenario-probability distribution...

[ E.34.70 ] Smooth quantile through scenario-...

[ E.34.71 ] Scenario-probability covariance m...

[ E.34.72 ] Scenario-probability correlation ...

[ E.34.73 ] Scenario-probability distribution...

[ E.34.74 ] Scenario-probability as exponenti...

[ E.34.75 ] Effective number of scenarios: ex...

[ E.34.76 ] Effective number of scenarios: ge...

[ E.34.77 ] Effective number of scenarios: ex...

[ E.34.78 ] Effective number of scenarios: ge...

[ E.34.79 ] Characteristic function of expone...

[ E.34.80 ] Exponential family distributions:...

[ E.34.81 ] Exponential family distributions:...

[ E.35 ] Estimation techniques

[ E.35.1 ] Conditional path distribution

[ E.35.2 ] Maximum likelihood parameters of ...

[ E.35.3 ] Free energy of the posterior

[ E.35.4 ] Minimum of free energy

[ E.35.5 ] Bayes’ rule

[ E.35.6 ] Normal distribution with fixed va...

[ E.35.7 ] Posterior distribution of exponen...

[ E.35.8 ] Predictive distribution of expone...

[ E.35.9 ] Normal distribution with fixed va...

[ E.36 ] Views processing

[ E.36.1 ] Point views manifold

[ E.36.2 ] Point views regression plane

[ E.36.3 ] Pdf of view variables

[ E.36.4 ] Distributional views as a special...

[ E.36.5 ] Minimum relative entropy and expo...

[ E.36.6 ] Minimum relative entropy and expo...

[ E.36.7 ] Minimum relative entropy and expo...

[ E.36.8 ] Partial views of the exponential ...

[ E.36.9 ] Distributional views updated: ana...

[ E.36.10 ] Point views updated: analytical f...

[ E.36.11 ] Multiplicative opinion pooling as...

[ E.36.12 ] Minimum relative entropy and expo...

[ E.36.13 ] Conditioning between normal varia...

[ E.36.14 ] Gradient of relative entropy

[ E.36.15 ] Hessian of relative entropy

[ E.36.16 ] Extremeness of the views

[ E.36.17 ] Sensitivity to the views

[ E.36.18 ] Positive definiteness of block-di...

[ E.36.19 ] Positive definiteness of inverse

[ E.36.20 ] Positive definiteness of Kronecke...

[ E.36.21 ] Convexity of relative entropy

[ E.36.22 ] Minimum relative entropy via anal...

[ E.36.23 ] Minimum relative entropy via anal...

[ E.36.24 ] Minimum relative entropy via scen...

[ E.36.25 ] Minimum relative entropy with sce...

[ E.36.26 ] Minimum relative entropy with sce...

[ E.36.27 ] Minimum relative entropy with sce...

[ E.36.28 ] Minimum relative entropy with sce...

[ E.36.29 ] Partial views of the exponential ...

[ E.36.30 ] Partial views of the exponential ...

[ E.36.31 ] Gradient of relative entropy with...

[ E.36.32 ] Degrees of freedom of a low-rank-...

[ E.36.33 ] Chain rule for first derivatives

[ E.36.34 ] Chain rule for first order differ...

[ E.36.35 ] Chain rule for second derivatives

[ E.36.36 ] Hessian of relative entropy with ...

[ E.36.37 ] Gradient of constraint function o...

[ E.36.38 ] Hessian of constraint function on...

[ E.36.39 ] Views on joint and conditional di...

[ E.36.40 ] Views on ex-ante signal-to-noise ...

[ E.36.41 ] Copula opinion pooling - Choice o...

[ E.36.42 ] Copula opinion pooling - Skew t d...

[ E.36.43 ] Copula opinion pooling - Skew t d...

[ E.36.44 ] Distance minimum

[ E.36.45 ] Distance equivalence

[ E.36.46 ] Generalized shrinkage for covaria...

[ E.36.47 ] Generalized shrinkage for correla...

[ E.36.48 ] Generalized shrinkage for correla...

[ E.37 ] Black-Litterman

[ E.37.1 ] Black-Litterman prior distributio...

[ E.37.2 ] Black-Litterman posterior distrib...

[ E.37.3 ] Equilibrium market portfolio

[ E.37.4 ] Black-Litterman: confidence level...

[ E.38 ] Geometry of random variables

[ E.38.1 ] Cauchy-Schwartz inequality

[ E.38.2 ] Alternative formulation of multiv...

[ E.38.3 ] Non-central tracking error and ex...

[ E.38.4 ] Relationship between non-central ...

[ E.38.5 ] Lp spaces

[ E.38.6 ] Lp spaces as Hilbert spaces

[ E.38.7 ] Inclusion relationships between L...

[ E.38.8 ] Sample r-squared in terms of samp...

[ E.38.9 ] R-squared and generalization

[ E.38.10 ] R-squared and equivalent optimiza...

[ E.38.11 ] Inner product in terms of the cov...

[ E.38.12 ] Extension of visualization map

[ E.38.13 ] Alternative visualization basis

[ E.38.14 ] Best approximation: shifted ortho...

[ E.38.15 ] Strictly linear approximation

[ E.39 ] Geometry of distributions

[ E.39.1 ] Riemannian metric: curve length

[ E.39.2 ] Riemannian metric: volume

[ E.39.3 ] Fisher information metric: covari...

[ E.39.4 ] Fisher information metric: univar...

[ E.39.5 ] E-affine coordinates of univariat...

[ E.39.6 ] M-affine coordinates of univariat...

[ E.39.7 ] Duality of univariate normal dist...

[ E.39.8 ] Fisher information metric: univar...

[ E.39.9 ] Legendre dual function: Hessian m...

[ E.39.10 ] Legendre dual function: duality

[ E.39.11 ] Legendre transformation

[ E.39.12 ] Potential functions of univariate...

[ E.39.13 ] Bregman divergence of univariate ...

[ E.39.14 ] E-affine coordinates of exponenti...

[ E.39.15 ] Geodesic of exponential family

[ E.39.16 ] Tangent vector of multivariate no...

[ E.39.17 ] Gradient of the normal log partit...

[ E.39.18 ] Expectation parameters of multiva...

[ E.39.19 ] Fisher information metric: multiv...

[ E.39.20 ] Transpose Jacobian of the normal ...

[ E.39.21 ] Relative entropy: exponential fam...

[ E.39.22 ] Relative entropy: scenario-probab...

[ E.39.23 ] Fisher information metric: scenar...

[ E.39.24 ] E-geodesic of scenario-probabilit...

[ E.40 ] Decision theory with model uncert...

[ E.40.1 ] Decision theory: the frequentist ...

[ E.40.2 ] Decision theory: the Bayesian way

[ E.40.3 ] Decision theory: the model risk

[ E.40.4 ] Decision theory: the ensemble app...

[ E.40.5 ] Decision theory: the robust appro...

[ E.40.6 ] Maximum a posteriori classical eq...

[ E.41 ] Copulas

[ E.41.1 ] Distribution of the grade

[ E.41.2 ] Inverse cdf sampling

[ E.41.3 ] Pdf of a copula

[ E.41.4 ] Sklar’s theorem

[ E.41.5 ] Pdf of the copula of a bivariate ...

[ E.41.6 ] Pdf of a normal copula

[ E.41.7 ] Cdf of a copula

[ E.41.8 ] Co-monotonic invariance of copula...

[ E.41.9 ] Copulas of elliptical distributio...

[ E.42 ] Location and dispersion

[ E.42.1 ] Relation between z-score and sign...

[ E.42.2 ] Affine equivariance implies Mahal...

[ E.42.3 ] Mahalanobis distance invariance i...

[ E.42.4 ] Affine property of argmax

[ E.42.5 ] Affine equivariance of the mode

[ E.42.6 ] Affine equivariance of the modal ...

[ E.42.7 ] Affine equivariance of the median

[ E.42.8 ] Affine equivariance of the interq...

[ E.42.9 ] Affine equivariance of the expect...

[ E.42.10 ] Affine equivariance of the standa...

[ E.42.11 ] Monotonic invariance of the media...

[ E.42.12 ] Mode of a univariate lognormal

[ E.42.13 ] Modal dispersion of a univariate ...

[ E.42.14 ] Quantile of a univariate lognorma...

[ E.42.15 ] Recursion for eigenvalues and eig...

[ E.42.16 ] Relation among trace, determinant...

[ E.42.17 ] Points with constant Mahalanobis ...

[ E.42.18 ] Integral of Mahalanobis distance

[ E.42.19 ] Affine equivariance implies Mahal...

[ E.42.20 ] Mahalanobis distance invariance i...

[ E.42.21 ] Affine equivariance of the mode (...

[ E.42.22 ] Affine equivariance of the modal ...

[ E.42.23 ] Mode of a multivariate lognormal ...

[ E.42.24 ] Modal square-dispersion of a mult...

[ E.42.25 ] Compact formula for multivariate ...

[ E.42.26 ] Compact formula for the covarianc...

[ E.42.27 ] Covariance matrix as matrix-varia...

[ E.42.28 ] Generalized affine equivariance o...

[ E.42.29 ] Generalized affine equivariance o...

[ E.42.30 ] Generalized affine equivariance o...

[ E.42.31 ] Bilinearity of the covariance

[ E.42.32 ] Expectation of a multivariate log...

[ E.42.33 ] Covariance of a multivariate logn...

[ E.42.34 ] Expectation and covariance of a m...

[ E.42.35 ] Generalized bilinearity of the co...

[ E.42.36 ] Expectation of the sum of two var...

[ E.42.37 ] Covariance matrix of the sum of t...

[ E.42.38 ] Mode of sum of two gamma distribu...

[ E.42.39 ] Expectation and variance of the g...

[ E.42.40 ] Alternative generalization of unc...

[ E.42.41 ] Alternative generalization of unc...

[ E.42.42 ] Multivariate uncertainty band

[ E.42.43 ] Gradient of normal characteristic...

[ E.42.44 ] Hessian of normal characteristic ...

[ E.42.45 ] Taylor expansion of the character...

[ E.42.46 ] Tangent box of the ellipsoid

[ E.42.47 ] Property of expectation

[ E.42.48 ] Generalized Chebyshev’s inequal...

[ E.42.49 ] First order differential of the s...

[ E.42.50 ] First order differential of the d...

[ E.42.51 ] The Chebyshev’s inequality and ...

[ E.42.52 ] Mahalanobis square distance of no...

[ E.43 ] Correlation and generalizations

[ E.43.1 ] Characterization of independence ...

[ E.43.2 ] Cdf of uniform distribution on th...

[ E.43.3 ] Cdf of an “extreme”copula (Fr...

[ E.43.4 ] Cdf of an “extreme”copula (Fr...

[ E.43.5 ] Frechet-Hoeffding bounds and copu...

[ E.43.6 ] Schweizer-Wolff measure: equivale...

[ E.43.7 ] Copulas of non co-monotonic varia...

[ E.43.8 ] Regularized call option payoff

[ E.43.9 ] Regularized put option payoff

[ E.43.10 ] Kendall’s tau: equivalent expre...

[ E.43.11 ] Correlation: affine concordance a...

[ E.43.12 ] Correlation: invariance under pos...

[ E.43.13 ] Correlation: symmetry with affine...

[ E.43.14 ] Correlation between lognormal var...

[ E.44 ] Invariance tests

[ E.45 ] Stochastic processes cheat sheet

[ E.45.1 ] Conditional distribution of a pro...

[ E.45.2 ] AR(1) is an ergodic process

[ E.46 ] Continuous time processes

[ E.46.1 ] Characteristic function of standa...

[ E.46.2 ] The compound Poisson process is a...

[ E.46.3 ] Characteristic function of contin...

[ E.46.4 ] Characteristic function of Poisso...

[ E.46.5 ] Characteristic function of compou...

[ E.46.6 ] Characteristic function of standa...

[ E.46.7 ] Characteristic function of arithm...

[ E.46.8 ] The Lévy-Khintchine representati...

[ E.46.9 ] Lévy-Khintchine formula: correct...

[ E.46.10 ] Δt-step location and dispersion ...

[ E.46.11 ] Variance gamma parametrizations

[ E.46.12 ] Relationship between CIR and Orns...

[ E.46.13 ] Projection of a Markov chain: gen...

[ E.46.14 ] Fractional Brownian motion

[ E.47 ] First order autoregresion

[ E.47.1 ] Conditional moments of the AR(1) ...

[ E.47.2 ] Unconditional moments of the AR(1...

[ E.47.3 ] Half-life of AR(1) process

[ E.47.4 ] Conditional moments of the VAR(1)...

[ E.47.5 ] Distribution of stable τ-step sh...

[ E.47.6 ] Distribution of the Δt-step and ...

[ E.47.7 ] Conditional distribution and mome...

[ E.47.8 ] Unconditional distribution of the...

[ E.47.9 ] Unconditional distribution of the...

[ E.47.10 ] Short-term approximation and squa...

[ E.47.11 ] Unconditional moments of the VAR(...

[ E.47.12 ] Stationary VAR(1) is VMA(∞)

[ E.47.13 ] Spectral density of VAR(1) proces...

[ E.47.14 ] MVOU is VAR(1)

[ E.47.15 ] VAR(1) is MVOU

[ E.47.16 ] Deterministic linear dynamic syst...

[ E.47.17 ] Dynamics and distribution of MVOU...

[ E.47.18 ] Distribution of the Δt-step and ...

[ E.47.19 ] Distribution of the MVOU process:...

[ E.47.20 ] Conditional distribution and mome...

[ E.47.21 ] MVOU (auto)covariances

[ E.48 ] Spectral analysis

[ E.48.1 ] The Fourier integral is the most ...

[ E.48.2 ] Symmetry of autocovariances for c...

[ E.48.3 ] Autocovariance of AR(1) processes...

[ E.48.4 ] Symmetry of cross-spectral densit...

[ E.48.5 ] The diagonal elements of cross sp...

[ E.48.6 ] Integral of the diagonal elements...

[ E.48.7 ] Cross-spectral density of AR(1) p...

[ E.48.8 ] Spectral representation for multi...

[ E.48.9 ] Spectral representation of AR(1) ...

[ E.48.10 ] Fourier transform of the zero-inc...

[ E.48.11 ] Spectral density of linear combin...

[ E.48.12 ] Spectral density of multi-variate...

[ E.49 ] Signals

[ E.49.1 ] Momentum signal as AR(1)

[ E.49.2 ] Equivalence of the order imbalanc...

[ E.50 ] Stochastic dominance

[ E.50.1 ] Strong dominance implies weak dom...

[ E.50.2 ] Equivalent definitions of second ...

[ E.51 ] Optimization primer

[ E.51.1 ] Ice-cream cones of dimension ¯m

[ E.51.2 ] Semi-definite cones

[ E.51.3 ] QCQP as special case of SOCP

[ E.52 ] Useful algorithms

[ E.52.1 ] Transpose-square-root via CPCA

[ E.52.2 ] Cholesky decomposition of the cov...

[ E.52.3 ] PCA decomposition of the covarian...

[ E.52.4 ] Conditional principal component a...

[ E.52.5 ] Eigenvalues of multiplication

[ E.52.6 ] Moment-matching, scenario twistin...

[ E.52.7 ] Building Student t scenarios with...

[ E.52.8 ] Quadratic-normal distribution in ...

[ E.52.9 ] Log-characteristic function of qu...

[ E.52.10 ] Saddle point approximation of the...

[ E.52.11 ] Variance of quadratic-normal dist...

[ E.52.12 ] The Fourier Transform as a rescal...

[ E.52.13 ] Number of observations in a gener...

[ E.52.14 ] Normalized empirical histogram ap...

[ E.52.15 ] General solution of the minimum-t...

[ E.52.16 ] Constrained analytical solution o...

[ E.52.17 ] Unconstrained numerical solution ...

[ E.52.18 ] The constrained Procrustes proble...

[ E.53 ] Matrix manipulations

[ E.53.1 ] Partitioned matrix inversion

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