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Quantitative finance
E.I. Valuation
[ E.0a ]
Valuation foundations
[ E.0a.1 ]
Value from jointly lognormal SDF and payoff
[ E.0a.2 ]
Non-linear valuation from jointly lognormal SDF and payoff
[ E.0a.3 ]
Cash-flow record date
[ E.0a.4 ]
Jump rule. Theory
[ E.0a.5 ]
Cash-flow and P&L additivity
[ E.0a.6 ]
Additivity of the reinvested cumulative cash-flow
[ E.0a.7 ]
Generalized forward cash-flow-adjusted value
[ E.0a.8 ]
Payoff and forward cash-flow adjusted value
[ E.0a.9 ]
Forward and backward cash-flow-adjusted values
[ E.0a.10 ]
Conversion of the P&L from local to base currency
[ E.0a.11 ]
Fair value in base currency
[ E.0a.12 ]
Conversion rule for linear returns from local currency to base currency
[ E.0a.13 ]
Conversion rule for linear returns as a special case of the total simple portfolio P&L
[ E.0a.14 ]
The actual exchange rate
[ E.0a.15 ]
Coupon bond dirty price
[ E.0a.16 ]
P&L of a futures
[ E.0b ]
Linear pricing theory: core
[ E.0b.1 ]
Linear pricing operator
[ E.0b.2 ]
Equivalent statements of absence of arbitrage
[ E.0b.3 ]
Law of one price, linearity and arbitrage
[ E.0b.4 ]
Stochastic discount factor and absence of arbitrage
[ E.0b.5 ]
Kernel stochastic discount factor
[ E.0b.6 ]
Minimum relative entropy numeraire measure and stochastic discount factor
[ E.0b.7 ]
Risk neutral probability measure
[ E.0b.8 ]
One period risk-neutral probability measure
[ E.0b.9 ]
Continuous time risk-neutral probability measure
[ E.0b.10 ]
Continuous rebalancing limit: risk-neutral density
[ E.0b.11 ]
Risk-neutral density in the Black-Scholes-Merton model
[ E.0b.12 ]
Derivation of the fundamental theorem of asset pricing
[ E.0b.13 ]
Maximum Sharpe Ratio portfolio
[ E.0b.14 ]
Maximum Sharpe ratio portfolio weights
[ E.0b.15 ]
Equivalent formulation of security market line in terms of Sharpe ratio
[ E.0b.16 ]
Regression model for the stochastic discount factor
[ E.0b.17 ]
About the “insurance”term appearing in the covariance principle
[ E.0b.18 ]
From Buhlmann transform to the covariance principle
[ E.0c ]
Linear pricing theory: further assumptions
[ E.0c.1 ]
Inverse of the payoff matrix of the European call option basis
[ E.0c.2 ]
Payoff-replicating portfolio with respect to the European call option basis
[ E.0c.3 ]
European payoffs as combinations of call option payoffs
[ E.0c.4 ]
Common payoffs as combinations of call payoffs: butterfly option
[ E.0c.5 ]
Common payoffs as combinations of call payoffs: strangle option
[ E.0c.6 ]
Common payoffs as combinations of call payoffs: straddle option
[ E.0c.7 ]
Common payoffs as combinations of call payoffs: condor option
[ E.0c.8 ]
Parabola payoff as combinations of call payoffs
[ E.0c.9 ]
Current value of an arbitrary payoff generated by call options
[ E.0c.10 ]
Each instrument in a complete market is a combination of Arrow-Debreu securities
[ E.0c.11 ]
Arrow-Debreu securities as butterflies
[ E.0c.12 ]
Butterfly in terms of central difference second order derivative operator
[ E.0c.13 ]
Stochastic discount factor identified by the European call option basis
[ E.0c.14 ]
Arrow-Debreu securities under continuum
[ E.0c.15 ]
Conditional stochastic discount factor
[ E.0c.16 ]
Second derivative of the call option value with respect to the strike
[ E.0c.17 ]
Stochastic discount factor in the Black-Scholes-Merton framework
[ E.0c.18 ]
Distribution of the stochastic discount factor in the Black-Scholes-Merton framework
[ E.0c.19 ]
From the security market line to the standard CAPM
[ E.0c.20 ]
Buhlmann pricing equation in terms or returns
[ E.0c.21 ]
Systematic-idiosyncratic linear factor model in a complete market
[ E.0c.22 ]
Systematic-idiosyncratic linear factor model with zero residuals
[ E.0c.23 ]
Derivation of the APT model from the stochastic discount factor model
[ E.0c.24 ]
From APT to the security market line
[ E.0c.25 ]
Pricing equation as conditional expectation: proof
[ E.0c.26 ]
Alternative formulation intertemporal pricing equation
[ E.0c.27 ]
The martingale pricing formula
[ E.0d ]
Non-linear pricing theory
[ E.0d.1 ]
Shift principles: calibration
[ E.0d.2 ]
Exponential principle: calibration
[ E.0d.3 ]
Wang distortion principle: calibration
[ E.0d.4 ]
Esscher principle: calibration
[ E.0e ]
Valuation implementation
[ E.0e.1 ]
The Gordon growth model
[ E.0e.2 ]
The value of an oil field production
[ E.0e.3 ]
Enterprise value using comparables
[ E.0e.4 ]
Value of a pure endowment contract. Theory
[ E.0e.5 ]
Value of a pure endowment contract. Application
[ E.0e.6 ]
Non-life insurance valuation in the regulatory risk framework
[ E.0e.7 ]
Non-life insurance valuation with homogeneous independent claims
[ E.0e.8 ]
Non-life insurance valuation. Application
E.II. The “Checklist”
[ E.1 ]
Risk drivers identification
[ E.1.1 ]
ODE for a perpetual American call option
[ E.1.2 ]
Solution to the ODE for a perpetual American call option
[ E.1.3 ]
Perpetual American option with arithmetic Brownian motion underlying
[ E.1.4 ]
Inverse-call transformation
[ E.1.5 ]
The forward rate
[ E.1.6 ]
Continuously compounded forward rate
[ E.1.7 ]
Instantaneous forward rate
[ E.1.8 ]
Spot rate as average of forward rates
[ E.1.9 ]
Duration-times-spread
[ E.1.10 ]
First derivative of the maximum function
[ E.1.11 ]
The canonical basis property of the Dirac delta function applied to the maximum function
[ E.1.12 ]
Common payoffs as combinations of call payoffs: put option
[ E.1.13 ]
Forward start variance swap payoff as calendar-weighted average of spot variance payoffs
[ E.1.14 ]
Value of forward variance swap
[ E.1.15 ]
Alternative formulation of the fair value of the realized variance
[ E.1.16 ]
Variance swap fair value as combination of options
[ E.1.17 ]
Rolling value versus implied volatility
[ E.1.18 ]
Parsimonious SVI parametrization (I)
[ E.1.19 ]
Parsimonious SVI parametrization (II)
[ E.1.20 ]
Parsimonious SVI parametrization (III)
[ E.1.21 ]
Parsimonious SVI parametrization (IV)
[ E.1.22 ]
SVI parameters behaving as random walks
[ E.1.23 ]
Vasicek parametrization of the yield curve
[ E.2 ]
Quest for invariance
[ E.2.1 ]
Market efficiency and random walk
[ E.2.2 ]
The Poisson process and the distribution of the waiting times in high frequency trading
[ E.2.3 ]
Moment generating function and independence
[ E.2.4 ]
Stochastic volatility model with Student t distribution
[ E.2.5 ]
Equivalence between stochastic mean/volatility and mixture of distributions
[ E.2.6 ]
Equivalence between stochastic mean/volatility and mixture of distributions (I)
[ E.2.7 ]
Equivalence between stochastic mean/volatility and mixture of distributions (II)
[ E.2.8 ]
Equivalence between stochastic mean/volatility and mixture of distributions (III)
[ E.2.9 ]
Compound probability versus uncountable mixture
[ E.2.10 ]
AR(1) as a Markov process
[ E.2.11 ]
Time homogeneity for AR(1) process
[ E.2.12 ]
Discrete mean reverting, but not stationary, risk drivers
[ E.2.13 ]
Autocorrelation of a time-homogeneous Markov chain
[ E.2.14 ]
Fast decay of time-homogeneous Markov chains autocorrelations[work in progress]
[ E.2.15 ]
Invariant of a Markov chain
[ E.2.16 ]
Calibration of a structural model
[ E.2.17 ]
Structural credit models in terms of return on equity
[ E.2.18 ]
Log-leverage, linear return, default probability under Merton’s assumptions
[ E.2.19 ]
Markov chain-structural models identification
[ E.2.20 ]
Squared volatility as a function of past squared increments in GARCH(1,1) model
[ E.2.21 ]
Stationarity of GARCH
[ E.2.22 ]
VAR(1) as special case of Kalman filter
[ E.2.23 ]
Replicating a cointegrated combination of yields with a portfolio of bonds
[ E.2.24 ]
Mixture of invariants model as hidden Markov model
[ E.3 ]
Estimation
[ E.3.1 ]
Glivenko-Cantelli theorem: theory
[ E.3.2 ]
Minimum relative entropy as best between Gaussian versus exponential kernel
[ E.3.3 ]
Generalized Glivenko-Cantelli theorem
[ E.3.4 ]
Kernel with flexible probabilities mean estimation
[ E.3.5 ]
Regularized pdf
[ E.3.6 ]
Kernel with flexible probabilities and historical with flexible probability estimates comparison
[ E.3.7 ]
Relative entropy vs. maximum likelihood
[ E.3.8 ]
MLFP estimators for elliptical variables
[ E.3.9 ]
MLFP estimators for the Student t distribution. Theory
[ E.3.10 ]
Conditional excess distribution
[ E.3.11 ]
Generalized Pareto distribution I
[ E.3.12 ]
Influence function as the limit of sensitivity curve
[ E.3.13 ]
Influence function of the sample covariance: preliminary computation
[ E.3.14 ]
Influence function of the sample covariance
[ E.3.15 ]
Influence function of maximum likelihood estimators
[ E.3.16 ]
Influence function of location and dispersion MLFP estimators for elliptical distributions
[ E.3.17 ]
Influence function of ML estimators of location-dispersion under t
[ E.3.18 ]
Influence function of M-estimators
[ E.3.19 ]
M-estimators: location and dispersion
[ E.3.20 ]
Minimum volume ellipsoid
[ E.3.21 ]
Minimum covariance determinant
[ E.3.22 ]
Minimum volume ellipsoid and minimum covariance determinant algorithm
[ E.3.23 ]
Method of moments with flexible probabilities: reflected shifted lognormal. Theory
[ E.3.24 ]
Equivalent formulation of conditional excess distribution
[ E.3.25 ]
Maximum likelihood estimation from generalized method of moments
[ E.3.26 ]
Over-specified formulation of the generalized method of moments
[ E.3.27 ]
Robust estimation of the covariance matrix: rescaled HBFP ellipsoid
[ E.3.28 ]
ML-estimation of transition probability matrix for credit migration modeling
[ E.3.29 ]
MLFP estimation of transition probability matrix for credit migration modeling
[ E.3.30 ]
Expectation-maximization with flexible probabilities for missing values. Theory
[ E.3.31 ]
Maximum likelihood with flexible probabilities for different-length series. Theory
[ E.3.32 ]
Realized and empirical variance
[ E.3.33 ]
MLFP estimation of GARCH(1,1) with normal innovations
[ E.3.34 ]
Exponentially weighted moving average updating
[ E.3.35 ]
Dirichlet distribution
[ E.3.36 ]
Invariants distribution from the estimated standardized invariants distribution
[ E.3.37 ]
Conditional distribution of two univariate normal invariants
[ E.3.38 ]
Exponential family invariants: conjugate distribution
[ E.3.39 ]
Exponential family invariants: posterior distribution
[ E.3.40 ]
Exponential family invariants: predictive distribution
[ E.3.41 ]
Pdf of the information set
[ E.3.42 ]
Normal-inverse-Wishart location-dispersion: posterior distribution
[ E.3.43 ]
Normal-inverse-Wishart location-dispersion: predictive distribution
[ E.3.44 ]
Normal-inverse-Wishart location-dispersion: mode
[ E.3.45 ]
Normal-inverse-Wishart location-dispersion: modal dispersion
[ E.3.46 ]
Inverse-Wishart dispersion: mode
[ E.3.47 ]
Inverse-Wishart dispersion: modal dispersion
[ E.3.48 ]
Normal-inverse-Wishart location-dispersion: marginal distribution of location
[ E.3.49 ]
Independence of sample mean and covariance
[ E.3.50 ]
Distribution of the sample mean
[ E.3.51 ]
Distribution of the sample covariance
[ E.3.52 ]
The Marchenko-Pastur approximation: the general case
[ E.3.53 ]
Expectation shrinkage
[ E.3.54 ]
Singular covariance matrix
[ E.3.55 ]
Spectrum analysis
[ E.3.56 ]
Distance for sparse matrix shrinkage of correlation
[ E.3.57 ]
Distance for sparse matrix shrinkage of correlation: computations
[ E.3.58 ]
Stein’s lemma
[ E.3.59 ]
Shrinkage estimator of location
[ E.3.60 ]
Shrinkage estimator of dispersion
[ E.3.61 ]
Shrinkage estimator of dispersion: spectrum
[ E.3.62 ]
Distance matrix for correlation clustering
[ E.3.63 ]
Conditional covariance of normal variables
[ E.3.64 ]
Covariance and correlation of rescaled/normalized random variables
[ E.3.65 ]
Copula of Markov chain’s invariants
[ E.3.66 ]
Correlation of returns via GARCH residuals
[ E.4 ]
Projection
[ E.4.1 ]
Distribution of the sum of independent variables
[ E.4.2 ]
Square-root rule for a generic stochastic process
[ E.4.3 ]
Non-central moments to central moments
[ E.4.4 ]
Cumulant projection
[ E.4.5 ]
Central moments of a normal random variable
[ E.4.6 ]
Projection by averaging the historical non-overlapping distribution
[ E.4.7 ]
Hybrid Monte Carlo-historical projection: implementation
[ E.5 ]
Pricing at the horizon
[ E.5.1 ]
Dynamics and distribution of the stock value under the geometric Brownian motion assumption
[ E.5.2 ]
P&L of a forward contract
[ E.5.3 ]
Currency carry
[ E.5.4 ]
Foreign exchange carry trade
[ E.5.5 ]
Annualized carry return of a zero-coupon bond (theory)
[ E.5.6 ]
Annualized carry return of a bond (theory)
[ E.5.7 ]
Carry of a variance swap
[ E.5.8 ]
Greeks of equity P&L with stock value as risk driver
[ E.5.9 ]
Greeks of equity P&L with stock log-value as risk driver
[ E.5.10 ]
Bond Greeks
[ E.5.11 ]
Bond yield
[ E.5.12 ]
Bond convexity
[ E.5.13 ]
M-square
[ E.5.14 ]
Parallel shift of the yield curve for the Taylor approximation of the P&L of a coupon bond
[ E.5.15 ]
Equivalent definitions of effective duration and convexity
[ E.5.16 ]
Taylor approximation of the P&L of a coupon bond under parallel shifts
[ E.5.17 ]
Taylor approximation of variance swap P&L
[ E.5.18 ]
Global quadratic approximation for P&L
[ E.5.19 ]
Analytical distribution of the P&L at the horizon: MVOU drivers and Taylor approximation
[ E.5.20 ]
Elliptical risk drivers: location and dispersion of the P&L approximated at first order
[ E.5.21 ]
Analytical distribution of the joint P&L for stocks with normal compounded returns
[ E.6 ]
Aggregation
[ E.6.1 ]
Normal P&L’s imply normal returns
[ E.6.2 ]
Equally weighted portfolio
[ E.6.3 ]
Generator of ex-ante performance of elliptical risk drivers
[ E.6.4 ]
Regulatory credit framework: one-factor model
[ E.6.5 ]
CVA computation under simplifying assumptions
[ E.6.6 ]
Regulatory credit framework: conditional expectation of the portfolio P&L (II)
[ E.6.7 ]
Conditional log-characteristic function of single counterparty P&L in CreditRisk+
[ E.6.8 ]
Unconditional log-characteristic function of the portfolio P&L in CreditRisk+
[ E.6.9 ]
Minimum collateral
[ E.6.10 ]
Gross and net exposure
[ E.7 ]
Ex-ante evaluation
[ E.7.1 ]
Properties of the ex-ante performance
[ E.7.2 ]
Strong dominance implies arbitrage
[ E.7.3 ]
Estimability and monotonicity imply weak dominance consistency
[ E.7.4 ]
Translation invariance implies constancy
[ E.7.5 ]
Expected value: consistency with second order dominance
[ E.7.6 ]
Expected value: consistency with order q dominance
[ E.7.7 ]
The negative variance is concave
[ E.7.8 ]
The negative variance and the mean-variance trade-off are not consistent with order q dominance
[ E.7.9 ]
The negative variance and the mean-variance trade-off are not comonotonic additive
[ E.7.10 ]
The negative variance is not risk averse
[ E.7.11 ]
The negative variance and the mean-variance trade-off are not super-additive
[ E.7.12 ]
Expectation, variance jointly elicitable
[ E.7.13 ]
The mean-variance trade-off is translation invariant
[ E.7.14 ]
The Mean variance trade-off is joint elicitable with the variance
[ E.7.15 ]
The fundamental risk quadrangle: subquantile
[ E.7.16 ]
Certainty-equivalent: uniqueness
[ E.7.17 ]
Certainty-equivalent: estimability
[ E.7.18 ]
Certainty-equivalent: monotonicity (increasing utility)
[ E.7.19 ]
Certainty-equivalent: consistency with weak dominance (increasing utility)
[ E.7.20 ]
Certainty-equivalent: consistency with order q dominance
[ E.7.21 ]
Certainty-equivalent: constancy
[ E.7.22 ]
Certainty-equivalent: money-equivalence
[ E.7.23 ]
Certainty-equivalent: positive homogeneity of degree 1 (power utility)
[ E.7.24 ]
Certainty-equivalent: translation invariance (exponential utility)
[ E.7.25 ]
Certainty-equivalent: additivity (linear utility)
[ E.7.26 ]
Certainty-equivalent: comonotonic additivity (linear utility)
[ E.7.27 ]
Certainty-equivalent: risk aversion, risk propensity and risk neutrality
[ E.7.28 ]
Relation between Arrow-Pratt risk aversion function and utility function
[ E.7.29 ]
Certainty-equivalent and positive affine transformations of the utility function
[ E.7.30 ]
Certainty-equivalent (quadratic normal distribution)
[ E.7.31 ]
Certainty-equivalent (elliptical distribution)
[ E.7.32 ]
The value at risk
[ E.7.33 ]
Quantile (VaR) satisfaction measure: estimability
[ E.7.34 ]
Quantile (VaR) satisfaction measure: monotonicity
[ E.7.35 ]
Quantile (VaR) satisfaction measure: consistency with weak dominance
[ E.7.36 ]
Quantile (VaR) satisfaction measure: violation of consistency with order q dominance
[ E.7.37 ]
Quantile (VaR) satisfaction measure: constancy
[ E.7.38 ]
Quantile (VaR) satisfaction measure: money-equivalence
[ E.7.39 ]
Quantile (VaR) satisfaction measure: positive homogeneity of degree 1
[ E.7.40 ]
Quantile (VaR) satisfaction measure: translation invariance
[ E.7.41 ]
Quantile (VaR) satisfaction measure: violation of super-additivity
[ E.7.42 ]
Quantile (VaR) satisfaction measure: comonotonic additivity
[ E.7.43 ]
Quantile (VaR) satisfaction measure: violation of concavity and convexity
[ E.7.44 ]
Quantile (VaR) satisfaction measure: violation of risk-aversion, risk-seeking and risk-neutrality
[ E.7.45 ]
Quantile (VaR) satisfaction measure of normally distributed ex-ante performances satisfies super-additivity
[ E.7.46 ]
Central moments of an affine transformation of a multivariate random variable
[ E.7.47 ]
Variance of an affine transformation of a multivariate random variable
[ E.7.48 ]
Expectation, standard deviation and skewness of a portfolio P&L under lognormality
[ E.7.49 ]
The expected shortfall as sub-quantile
[ E.7.50 ]
Sub-quantile satisfaction measures are monotone
[ E.7.51 ]
Sub-quantile satisfaction measures: consistency with weak dominance
[ E.7.52 ]
Sub-quantile satisfaction measures: constancy
[ E.7.53 ]
Sub-quantile satisfaction measures: money-equivalence
[ E.7.54 ]
Sub-quantile satisfaction measures: positive homogeneity of degree 1
[ E.7.55 ]
Sub-quantile satisfaction measures: translation invariance
[ E.7.56 ]
Sub-quantile satisfaction measure: super additivity
[ E.7.57 ]
Sub-quantile satisfaction measures: comonotonic additivity
[ E.7.58 ]
Sub-quantile satisfaction measures: risk-aversion
[ E.7.59 ]
Sub-quantile satisfaction measures: violation of consistency with order q dominance
[ E.7.60 ]
Spectral satisfaction measures: estimability
[ E.7.61 ]
Spectral satisfaction measures are monotone
[ E.7.62 ]
Spectral satisfaction measures: consistency with weak dominance
[ E.7.63 ]
Spectral satisfaction measures: constancy
[ E.7.64 ]
Spectral satisfaction measures: money-equivalence
[ E.7.65 ]
Spectral satisfaction measures: positive homogeneity of degree 1
[ E.7.66 ]
Spectral satisfaction measures: translation invariance
[ E.7.67 ]
Spectral satisfaction measures: comonotonic additivity
[ E.7.68 ]
Spectral satisfaction measures: violation of consistency with order q dominance
[ E.7.69 ]
Spectral satisfaction measures: violation of super-additivity
[ E.7.70 ]
Spectral satisfaction measures: violation of concavity and convexity
[ E.7.71 ]
Spectral/distortion satisfaction measure weights (scenario-probability distribution)
[ E.7.72 ]
Alternative representation of spectral satisfaction measure
[ E.7.73 ]
Quantile (VaR) satisfaction measure: distortion function
[ E.7.74 ]
Sub-quantile satisfaction measure: distortion function
[ E.7.75 ]
The Wang expectation: distortion function
[ E.7.76 ]
The Buhlmann expectation is not a distortion expectation
[ E.7.77 ]
Equivalence between spectral and distortion satisfaction measures
[ E.7.78 ]
Spectral measures as weighted averages of expected shortfalls
[ E.7.79 ]
Equivalent definitions of monotonicity
[ E.7.80 ]
The mean-lower partial moment root is coherent
[ E.7.81 ]
Dual representation of the sub-quantile satisfaction measure in the scenario-probability framework
[ E.7.82 ]
Worst possible measure for the expected shortfall in the scenario-probability framework
[ E.7.83 ]
Coherent spectral satisfaction measures as distortions
[ E.7.84 ]
Coherent spectral satisfaction measures: super-additivity
[ E.7.85 ]
Mean-lower partial moment trade-off: violation of comonotonic additivity
[ E.7.86 ]
Coherent satisfaction measures: consistency with weak dominance
[ E.7.87 ]
Coherent satisfaction measures: constancy
[ E.7.88 ]
Coherent satisfaction measures: money-equivalence
[ E.7.89 ]
Coherent representation of coherent spectral measures
[ E.7.90 ]
Coherent spectral measures: characterization
[ E.7.91 ]
Worst case representation of expectiles
[ E.7.92 ]
Convex combinations of coherent satisfaction measures
[ E.7.93 ]
The Wang distortion expectation
[ E.7.94 ]
The proportional hazards distortion expectation
[ E.7.95 ]
Cornish-Fisher approximation for spectral satisfaction measures
[ E.7.96 ]
Extreme value theory: approximation of the sub-quantile satisfaction measure
[ E.7.97 ]
Sub-quantile satisfaction measure weights (scenario-probability distribution)
[ E.7.98 ]
Derivative of an indefinite integral
[ E.7.99 ]
Relation between the omega and the kappa ratio
[ E.7.100 ]
Economic capital
[ E.7.101 ]
The Buhlmann expectation is a distortion expectation
[ E.7.102 ]
First order approximation of the Buhlmann expectation
[ E.7.103 ]
Esscher transform as minimum entropy distribution
[ E.7.104 ]
First order approximation of the Esscher expectation
[ E.7.105 ]
The Esscher expectation is neither positive homogeneous nor linear
[ E.7.106 ]
Buhlmann expectation: linearity
[ E.7.107 ]
The utility function as the cdf of a subjective distribution
[ E.7.108 ]
The certainty-equivalent as the quantile (VaR)
[ E.7.109 ]
The Arrow-Pratt approximation
[ E.7.110 ]
Esscher expectation under normality assumption
[ E.7.111 ]
Buhlmann expectation under normality assumption
[ E.8a ]
Ex-ante attribution: performance
[ E.8a.1 ]
Joint distribution factor and residual: elliptical case
[ E.8a.2 ]
Relationship between bottom-up and top-down exposures: cross-sectional instruments-level attribution
[ E.8a.3 ]
Black-Scholes-Merton delta hedging
[ E.8b ]
Ex-ante attribution: risk
[ E.8b.1 ]
Standard deviation: gradient and Euler marginal contributions
[ E.8b.2 ]
Variance: gradient and Euler marginal contributions
[ E.8b.3 ]
Certainty-equivalent: gradient and Euler marginal contributions (power utility)
[ E.8b.4 ]
Quantile (VaR): gradient and Euler marginal contributions
[ E.8b.5 ]
The spectral satisfaction measures is not differentiable in the scenario-probability framework
[ E.8b.6 ]
Quantile (VaR): gradient and Euler marginal contributions (scenario-probability)
[ E.8b.7 ]
Quantile (VaR): gradient and Euler marginal contributions (elliptical distribution)
[ E.8b.8 ]
Sub-quantile: gradient and Euler marginal contributions
[ E.8b.9 ]
Sub-quantile: gradient and Euler marginal contributions (scenario-probability)
[ E.8b.10 ]
Sub-quantile: gradient and Euler marginal contributions (elliptical distribution)
[ E.8b.11 ]
Spectral measures: gradient and Euler marginal contributions
[ E.8b.12 ]
Spectral measures: gradient and Euler marginal contributions (scenario probability)
[ E.8b.13 ]
Spectral measures: gradient and Euler marginal contributions (elliptical distribution)
[ E.8b.14 ]
Coherent measures: gradient and Euler marginal contributions
[ E.8b.15 ]
Twisted expectations and spectral measures
[ E.8b.16 ]
Computation of the marginal contributions for the Esscher expectation
[ E.8b.17 ]
Marginal risk contributions for the variance risk measure
[ E.8b.18 ]
Esscher risk contributions
[ E.8b.19 ]
The economic capital is positive homogeneous of first degree
[ E.8b.20 ]
The minimum-torsion diversification distribution
[ E.8b.21 ]
Effective number of bets
[ E.8b.22 ]
Risk attribution: principal components
[ E.8b.23 ]
The principal components diversification distribution
[ E.8b.24 ]
General solution of the minimum-torsion optimization problem
[ E.8b.25 ]
Constrained analytical solution of the minimum-torsion optimization problem
[ E.8b.26 ]
Unconstrained numerical solution of the minimum-torsion optimization problem
[ E.9a ]
Construction: portfolio optimization
[ E.9a.1 ]
Portfolio optimization problem
[ E.9b ]
Construction: estimation and model risk
[ E.9c ]
Construction: cross-sectional strategies
[ E.9c.1 ]
Market-capitalization allocation
[ E.9c.2 ]
Maximal constrained signal-to-noise ratio
[ E.9c.3 ]
Maximal conditional signal-to-noise
[ E.9c.4 ]
Maximal conditional signal-to-noise (normal case)
[ E.9c.5 ]
Fundamental law of active management (under normal assumption)
[ E.9c.6 ]
Smart beta: factor premium
[ E.9c.7 ]
Flexible and standard characteristic portfolio
[ E.9c.8 ]
Linkage matrix and signal weakness
[ E.9c.9 ]
Characteristic portfolio variance
[ E.9d ]
Construction: time series strategies
[ E.9d.1 ]
Self-financing constraint for portfolio holdings
[ E.9d.2 ]
Strategy dynamics (the general case)
[ E.9d.3 ]
Strategy dynamics (arithmetic Brownian motion)
[ E.9d.4 ]
Strategy dynamics (geometric Brownian motion)
[ E.9d.5 ]
Strategy distributions (arithmetic Brownian motion)
[ E.9d.6 ]
Strategy distributions (geometric Brownian motion)
[ E.9d.7 ]
Partial differential equation for Bachelier’s formula
[ E.9d.8 ]
Dynamic payoff replication strategy
[ E.9d.9 ]
Maximum utility for arithmetic Brownian motion
[ E.9d.10 ]
Maximum utility for geometric Brownian motion
[ E.9d.11 ]
Utility maximization versus payoff replication
[ E.9d.12 ]
Payoff function of utility maximization (exponential utility)
[ E.9d.13 ]
Payoff function of utility maximization (power utility)
[ E.9d.14 ]
PDE of power utility maximization
[ E.9d.15 ]
Solution of the power utility maximization
[ E.9d.16 ]
Cushion of the CPPI strategy
[ E.9d.17 ]
Linear time invariant filter in continuous time
[ E.9d.18 ]
The dynamic of a linear time invariant signal
[ E.9d.19 ]
Exponentially weighted moving average in continuous time
[ E.9d.20 ]
Dynamics of the exponentially weighted moving average
[ E.9d.21 ]
P&L of signal induced strategies
[ E.9d.22 ]
A simple signal induced strategy
[ E.10 ]
Execution
[ E.10.1 ]
VWAP trading strategy
[ E.10.2 ]
Meaning of one unit of volume time
[ E.10.3 ]
Interpretation of the trading speed ḣq and the daily parameter η in the Almgren-Chriss model
[ E.10.4 ]
Market impact P&L
[ E.10.5 ]
Expectation and variance of the trading P&L
[ E.10.6 ]
Normalized market impact model
[ E.10.7 ]
Expectation and variance of the market impact P&L in the Almgren-Chriss model
[ E.10.8 ]
Mean-variance optimization problem in the in the Almgren-Chriss model
[ E.10.9 ]
P&L optimization: Almgren-Chriss model
[ E.10.10 ]
The VWAP trading strategy in the Almgren-Chriss model
[ E.10.11 ]
Optimization problem in the multidimensional Almgren-Chriss model
[ E.10.12 ]
Solution of the multidimensional Almgren-Chriss model
[ E.10.13 ]
Market impact P&L under the Almgren-Chriss model
[ E.10.14 ]
Expectation and variance of the market impact P&L under a power execution strategy
[ E.10.15 ]
Transient impact: the optimization problem
[ E.10.16 ]
Transient impact: the Obizhaeva-Wang model
[ E.10.17 ]
Transient impact: the Dang model
[ E.10.18 ]
Transient impact: power law decay kernel
[ E.10.19 ]
Transient impact: logarithmic decay kernel
[ E.10.20 ]
Price manipulation
[ E.10.21 ]
Zero-intelligence model: statistical properties of the limit order book [work in progress]
E.III. Performance analysis
[ E.11 ]
Performance attribution
E.IV. Financial toolbox
[ E.12 ]
Performance definitions
[ E.12.1 ]
Computation of the internal rate of return
[ E.12.2 ]
Trading P&L (single trade)
[ E.12.3 ]
Decomposition of the total trading P&L
[ E.12.4 ]
The implementation shortfall in the total trading P&L
[ E.12.5 ]
Trading P&L (multiple trading dates)
[ E.12.6 ]
Trading P&L: opening and liquidating positions
[ E.12.7 ]
Aggregation property of linear returns (across instruments)
[ E.12.8 ]
Aggregation property of compounded returns (across time)
[ E.12.9 ]
Alternative formulation for the generalized excess return
[ E.12.10 ]
Compounded rate of return in terms of adjusted values
[ E.12.11 ]
Par swap rate as IRR of a coupon bond
[ E.12.12 ]
Generalized portfolio weights
[ E.12.13 ]
Offset cash
[ E.13 ]
Signals
[ E.13.1 ]
Equivalence of the order imbalance signal definition
[ E.14 ]
Black-Litterman
[ E.14.1 ]
Black-Litterman prior distribution
[ E.14.2 ]
Black-Litterman posterior distribution
[ E.14.3 ]
Black-Litterman: confidence level in views
Data science
E.V. Mathematics
[ E.15 ]
Linear algebra primer
[ E.15.1 ]
Linear independence
[ E.15.2 ]
Vector operations on coordinates
[ E.15.3 ]
Direct sum of vector subspaces
[ E.15.4 ]
Matrix operations
[ E.15.5 ]
Matrix basic properties
[ E.15.6 ]
Dimension of general linear group
[ E.15.7 ]
Positive semidefinite matrix
[ E.15.8 ]
Positive definiteness of block-diagonal
[ E.15.9 ]
Positive definiteness of inverse
[ E.15.10 ]
Positive definiteness of Kronecker product
[ E.15.11 ]
Linear operator as inner product
[ E.15.12 ]
Useful identities for inner product spaces
[ E.15.13 ]
Cauchy-Schwarz inequality
[ E.15.14 ]
Orthonormal sets are linearly independent
[ E.15.15 ]
Orthogonal projection over a span
[ E.15.16 ]
Orthogonal projection over direct sums
[ E.15.17 ]
p-norm is a norm
[ E.15.18 ]
Distance induced by norm
[ E.15.19 ]
Eigenvalues of symmetric matrices
[ E.15.20 ]
Eigenvalues of symmetric positive (semi)definite matrices
[ E.15.21 ]
Eigenvalues of the inverse of a matrix
[ E.15.22 ]
Relation among trace, determinant and eigenvalues
[ E.15.23 ]
The UDU-Cholesky decomposition
[ E.15.24 ]
Gramian and linear independence
[ E.15.25 ]
Finite-dimensional inner products
[ E.15.26 ]
Affine equivariance of Gram matrix
[ E.15.27 ]
Recursion for eigenvalues and eigenvectors in two dimensions
[ E.15.28 ]
PCA with repeated eigenvalues
[ E.15.29 ]
The constrained Procrustes problem
[ E.15.30 ]
Minimum torsion orthonormalization
[ E.15.31 ]
Linearity of vectorization
[ E.15.32 ]
Partitioned matrix inversion
[ E.15.33 ]
Inverse of a block-triangular matrix
[ E.15.34 ]
Inverse of an upper-triangular Toeplitz matrix
[ E.16 ]
Calculus primer
[ E.16.1 ]
Differentiability characterization
[ E.16.2 ]
Gradient of the quadratic form
[ E.16.3 ]
Gradient chain rule
[ E.16.4 ]
Chain rule for first order differential
[ E.16.5 ]
First derivative of monotonic functions
[ E.16.6 ]
Cubic function is strictly increasing
[ E.16.7 ]
Strictly monotone maps are invertible
[ E.16.8 ]
Alternative convexity criterion
[ E.16.9 ]
Convex functions have invertible gradients
[ E.17 ]
Functional analysis
[ E.17.1 ]
The Fourier integral is the most general Fourier transform form
[ E.17.2 ]
The Fourier Transform as a rescaled unitary operator
[ E.17.3 ]
Fourier transform of the Dirac delta
[ E.18 ]
Optimization primer
[ E.18.1 ]
Newton’s method
[ E.18.2 ]
Equality constraints must be affine
[ E.18.3 ]
Semi-definite cones
[ E.18.4 ]
Alternate SDP formulation
[ E.18.5 ]
Ice-cream cones of dimension ¯¯¯m
[ E.18.6 ]
QCQP as special case of SOCP
[ E.18.7 ]
Regularized regression is regularized quadratic
[ E.18.8 ]
Constrained generalized elastic net is quadratic programming
[ E.18.9 ]
Generalized lasso is lasso
[ E.18.10 ]
Lasso penalty in constrained selection
[ E.18.11 ]
Equivalent quadratic optimization for portfolio replication
E.VI. Statistics
[ E.19 ]
Distributions
[ E.19.1 ]
Pdf of an invertible function of a univariate random variable
[ E.19.2 ]
Cdf of an invertible function of a univariate random variable
[ E.19.3 ]
Quantile function and inverse cdf
[ E.19.4 ]
Quantile of an invertible function of a random variable
[ E.19.5 ]
Expected value in terms of the quantile
[ E.19.6 ]
Sub-quantile as conditional expectation
[ E.19.7 ]
Sub-quantile of an affine transformation
[ E.19.8 ]
Multivariate Student t distribution: cumulative distribution function
[ E.19.9 ]
Chi-distribution: numerical implementation of the quantile
[ E.19.10 ]
Relation between the characteristic function and the moments
[ E.19.11 ]
Moments of the chi-squared distribution
[ E.19.12 ]
Scaling property of the gamma distribution
[ E.19.13 ]
Equivalence between gamma and chi-squared distribution
[ E.19.14 ]
Moments of the gamma distribution
[ E.19.15 ]
Expectation of the exponential of a gamma random variable
[ E.19.16 ]
Quadratic-normal distribution in terms of independent standard normal variables
[ E.19.17 ]
Log-characteristic function of quadratic-normal distribution
[ E.19.18 ]
Saddle point approximation of the quadratic-normal distribution
[ E.19.19 ]
Variance of quadratic-normal distribution
[ E.19.20 ]
Wishart and gamma distribution
[ E.19.21 ]
Marginals of a Wishart distribution
[ E.19.22 ]
Result on the joint distribution of a bivariate random variable
[ E.19.23 ]
Conditional pdf
[ E.19.24 ]
Conditional quantile and conditional characteristic function
[ E.19.25 ]
Conditional and unconditional expectation
[ E.19.26 ]
Conditional and unconditional invariance
[ E.19.27 ]
Conditional distribution between normal random variables
[ E.19.28 ]
Conditional distribution between lognormal random variables
[ E.19.29 ]
Conditional expectation of two sets of lognormal random variables
[ E.19.30 ]
Law of total variance: joint Student t
[ E.19.31 ]
Law of total variance: joint lognormal
[ E.19.32 ]
Covariances and correlations parametrizations of two sets of multivariate random variables
[ E.19.33 ]
Conditional expectation and covariance of two sets of normal random variables
[ E.19.34 ]
Marginalization cdf formula
[ E.19.35 ]
Pdf of an invertible function of a multivariate random variable
[ E.19.36 ]
Pdf of a non-invertible function of a multivariate random variable
[ E.19.37 ]
Cdf of an invertible comonotonic function of a multivariate random variable
[ E.19.38 ]
Pdf of a non-invertible affine transformation of a multivariate random variable
[ E.19.39 ]
Characteristic function of a multivariate normal random variable I
[ E.19.40 ]
Cdf of the lognormal distribution
[ E.19.41 ]
Non-central moments of a multivariate lognormal random variable
[ E.19.42 ]
Moments of the reflected shifted lognormal distribution
[ E.19.43 ]
Expectation and covariance of a multivariate lognormal random variable
[ E.19.44 ]
Expectation, standard deviation and skewness of a linear combination of multivariate shifted lognormal random vector
[ E.19.45 ]
Gradient of the pdf of a multivariate affine function
[ E.19.46 ]
Hessian of the pdf of a multivariate affine transformation
[ E.19.47 ]
Gradient of the log-pdf of a multivariate variable
[ E.19.48 ]
Hessian of the log-pdf of a multivariate variable
[ E.19.49 ]
Pdf of an inverse-Wishart random variable
[ E.19.50 ]
Equivalence between definitions of elliptical distribution
[ E.19.51 ]
Radial component and generator function of elliptical distributions
[ E.19.52 ]
Radial component of multivariate normal is chi distributed
[ E.19.53 ]
Radial component of multivariate Student t
[ E.19.54 ]
Radial component of a uniform random variable inside an ellipsoid
[ E.19.55 ]
Moments of an elliptical random variable
[ E.19.56 ]
Expectation of Mahalanobis square distance of normal random variables
[ E.19.57 ]
Expectation of Mahalanobis square distance of Student t random variables
[ E.19.58 ]
Moments of a uniform random variable inside an ellipsoid
[ E.19.59 ]
Moments of the uniform component of an elliptical distribution
[ E.19.60 ]
Elliptical distributions: formula for the generator of a univariate affine transformation
[ E.19.61 ]
Elliptical distributions: generator of the marginal distribution of a uniform inside the unit circle
[ E.19.62 ]
Normal distribution as limit of Student t distribution
[ E.19.63 ]
Normal generator as limit of Student t generator
[ E.19.64 ]
Marginal distribution of a uniform random variable inside the unit sphere
[ E.19.65 ]
Truncated quantile
[ E.19.66 ]
Stress distribution of elliptical is elliptical
[ E.19.67 ]
Stress quantile of elliptical distributions
[ E.19.68 ]
Alternative stochastic representation of elliptical random variables
[ E.19.69 ]
Gini coefficient in terms of covariance
[ E.19.70 ]
Scenario-probability distribution: cdf
[ E.19.71 ]
Scenario-probability distribution: probability density function
[ E.19.72 ]
Scenario-probability distribution: expectation
[ E.19.73 ]
Scenario-probability distribution: invariance rule
[ E.19.74 ]
Scenario-probability distribution: expectation rule
[ E.19.75 ]
Scenario-probability distribution: cdf via expectation rule
[ E.19.76 ]
Scenario-probability distribution: characteristic function
[ E.19.77 ]
Scenario-probability distribution: quantile
[ E.19.78 ]
Scenario-probability distribution: quantile for uniform flexible probabilities
[ E.19.79 ]
Smooth quantile through scenario-probability quantile
[ E.19.80 ]
Scenario-probability covariance matrix
[ E.19.81 ]
Scenario-probability correlation matrix
[ E.19.82 ]
Scenario-probability distribution: positive probabilities
[ E.19.83 ]
Conditional distribution between normal random variables in canonical parametrization
[ E.19.84 ]
Maximum partition encoder: underlying partition
[ E.19.85 ]
Multinomial logit parametrization
[ E.19.86 ]
Multinomial probit parametrization
[ E.19.87 ]
Effective number of scenarios boundedness: exponential of the entropy
[ E.19.88 ]
Effective number of scenarios boundedness: generalized exponential of the entropy
[ E.19.89 ]
Effective number of scenarios counting crisp scenarios: exponential of the entropy
[ E.19.90 ]
Effective number of scenarios counting crisp scenarios: generalized exponential of the entropy
[ E.19.91 ]
Characteristic function of exponential family distributions
[ E.19.92 ]
Exponential family distributions: expectation of the sufficient statistics
[ E.19.93 ]
Exponential family distributions: covariance of the sufficient statistics
[ E.19.94 ]
Joint mean and covariance of a mixture model
[ E.19.95 ]
Mixture probabilities
[ E.19.96 ]
Normal mixtures
[ E.19.97 ]
Abstract Bayes theorem
[ E.19.98 ]
Radon-Nikodym derivative on finite spaces
[ E.19.99 ]
Conditional expectation over elementary events
[ E.19.100 ]
Conditional pdf as L2 projection
[ E.19.101 ]
Adapted approximations
[ E.19.102 ]
Radon-Nikodym with log-normal market
[ E.19.103 ]
Conditional expectation: equivalent formulation and Radon-Nikodym
[ E.20 ]
Copulas
[ E.20.1 ]
Distribution of the grade
[ E.20.2 ]
Inverse cdf sampling
[ E.20.3 ]
Pdf of a copula
[ E.20.4 ]
Sklar’s theorem
[ E.20.5 ]
Pdf of the copula of a bivariate normal
[ E.20.6 ]
Pdf of a normal copula
[ E.20.7 ]
Cdf of a copula
[ E.20.8 ]
Comonotonic invariance of copulas
[ E.20.9 ]
Copulas of elliptical distributions
[ E.21 ]
Geometry of distributions
[ E.21.1 ]
Riemannian metric: curve length
[ E.21.2 ]
Riemannian metric: volume
[ E.21.3 ]
Fisher information metric: covariant property
[ E.21.4 ]
Fisher information metric: univariate normal distribution
[ E.21.5 ]
E-affine coordinates of univariate normal distributions
[ E.21.6 ]
M-affine coordinates of univariate normal distributions
[ E.21.7 ]
Duality of univariate normal distributions
[ E.21.8 ]
Fisher information metric: univariate normal distribution (dual parameters)
[ E.21.9 ]
Legendre dual function: Hessian matrix
[ E.21.10 ]
Legendre dual function: duality
[ E.21.11 ]
Legendre transformation
[ E.21.12 ]
Potential functions of univariate normal distributions
[ E.21.13 ]
Bregman divergence of univariate normal distributions
[ E.21.14 ]
E-affine coordinates of exponential family
[ E.21.15 ]
Geodesic of exponential family
[ E.21.16 ]
Tangent vector of multivariate normal distributions
[ E.21.17 ]
Gradient of the normal log partition function
[ E.21.18 ]
Expectation parameters of multivariate normal distributions
[ E.21.19 ]
Fisher information metric: multivariate normal distribution
[ E.21.20 ]
Transpose Jacobian of the normal distribution
[ E.21.21 ]
Relative entropy: exponential family
[ E.21.22 ]
Fisher information metric: scenario-probability distribution
[ E.22 ]
Location and dispersion
[ E.22.1 ]
Relation between z-score and signal-to-noise ratio
[ E.22.2 ]
Affine equivariance implies Mahalanobis distance invariance
[ E.22.3 ]
Mahalanobis distance invariance implies affine equivariance
[ E.22.4 ]
Absolute z-score invariance
[ E.22.5 ]
Affine property of argmax
[ E.22.6 ]
Affine equivariance of the mode
[ E.22.7 ]
Affine equivariance of the modal dispersion
[ E.22.8 ]
Affine equivariance of the median
[ E.22.9 ]
Affine equivariance of the interquantile range
[ E.22.10 ]
Affine equivariance of the expectation
[ E.22.11 ]
Affine equivariance of the standard deviation
[ E.22.12 ]
Monotonic invariance of the median
[ E.22.13 ]
Mode of a univariate lognormal
[ E.22.14 ]
Modal dispersion of a univariate lognormal
[ E.22.15 ]
Orthogonality of eigenvectors
[ E.22.16 ]
Recursion for eigenvalues and eigenvectors
[ E.22.17 ]
Points with constant Mahalanobis distance form an ellipsoid
[ E.22.18 ]
Integral of Mahalanobis distance
[ E.22.19 ]
Affine equivariance implies Mahalanobis distance invariance (multivariate case)
[ E.22.20 ]
Mahalanobis distance invariance implies affine equivariance (multivariate case)
[ E.22.21 ]
Affine equivariance of the mode (multivariate case)
[ E.22.22 ]
Affine equivariance of the modal square-dispersion
[ E.22.23 ]
Mode of a multivariate lognormal distribution
[ E.22.24 ]
Modal square-dispersion of a multivariate lognormal
[ E.22.25 ]
Compact formula for multivariate expectation
[ E.22.26 ]
Compact formula for the covariance matrix
[ E.22.27 ]
Covariance matrix as matrix-variate expectation
[ E.22.28 ]
Generalized affine equivariance of the expectation (multivariate case)
[ E.22.29 ]
Generalized affine equivariance of the covariance
[ E.22.30 ]
Bilinearity of the covariance
[ E.22.31 ]
Expectation and covariance of a multivariate shifted lognormal
[ E.22.32 ]
Affine equivariance of the cross-covariance
[ E.22.33 ]
Expectation of the sum of two variables
[ E.22.34 ]
Covariance matrix of the sum of two variables
[ E.22.35 ]
Mode of sum of two gamma distributions
[ E.22.36 ]
Expectation and variance of the gamma distribution
[ E.22.37 ]
Mode and modal square dispersion of a gamma distribution
[ E.22.38 ]
Generalized affine equivariance does not hold for modal square dispersion
[ E.22.39 ]
Alternative generalization of uncertainty band
[ E.22.40 ]
Alternative generalization of uncertainty band
[ E.22.41 ]
Multivariate uncertainty band
[ E.22.42 ]
Gradient of normal characteristic function
[ E.22.43 ]
Hessian of normal characteristic function
[ E.22.44 ]
Taylor expansion of the characteristic function
[ E.22.45 ]
Tangent box of the ellipsoid
[ E.22.46 ]
Principal directions and principal variances
[ E.22.47 ]
Property of expectation
[ E.22.48 ]
Multivariate Markov inequality
[ E.22.49 ]
Generalized Chebyshev’s inequality
[ E.22.50 ]
First order differential of the square Mahalanobis distance
[ E.22.51 ]
The Chebyshev’s inequality and most likely set
[ E.22.52 ]
Mahalanobis square distance of normal random variables
[ E.22.53 ]
Explicit expression of the error matrix
[ E.22.54 ]
Affine equivariance of linear projection and partial covariance
[ E.22.55 ]
Explicit expression of the loss matrix
[ E.22.56 ]
L2 of law of total variance
[ E.22.57 ]
Linear and non-linear projections under normality
[ E.22.58 ]
Visualization map is isometry
[ E.22.59 ]
Cauchy-Schwarz inequality
[ E.22.60 ]
Alternative formulation of multivariate inner product
[ E.22.61 ]
Expectation length and distance
[ E.22.62 ]
Relationship between non-central and central tracking errors
[ E.22.63 ]
Quantile and subquantile-deviation
[ E.22.64 ]
Variational location and dispersion are affine equivariant
[ E.22.65 ]
Bregman location-dispersion
[ E.22.66 ]
Multivariate p-quantile
[ E.22.67 ]
Lp spaces
[ E.22.68 ]
R-squared and equivalent optimization objective
[ E.22.69 ]
Inner product in terms of the covariance matrix
[ E.22.70 ]
Extension of visualization map
[ E.22.71 ]
Alternative visualization basis
[ E.22.72 ]
Best approximation: shifted orthogonal projection
[ E.22.73 ]
Best linear prediction: solution
[ E.22.74 ]
Characterization of conditional expectation
[ E.22.75 ]
Best approximation: equivalent characterization
[ E.22.76 ]
Cholesky root via Gram-Schmidt
[ E.23 ]
Correlation and generalizations
[ E.23.1 ]
Interpretation of independence
[ E.23.2 ]
Characterization of independence through copulas
[ E.23.3 ]
Cdf of uniform distribution on the unit square
[ E.23.4 ]
Cdf of an “extreme”copula (Frechet-Hoeffding bottom bound)
[ E.23.5 ]
Cdf of an “extreme”copula (Frechet-Hoeffding top bound)
[ E.23.6 ]
Frechet-Hoeffding bounds and copula of monotonic variables
[ E.23.7 ]
Schweizer-Wolff measure: equivalent expression
[ E.23.8 ]
Copulas of non comonotonic variables
[ E.23.9 ]
Regularized call option payoff
[ E.23.10 ]
Regularized put option payoff
[ E.23.11 ]
Kendall’s tau: equivalent expression
[ E.23.12 ]
Correlation: affine concordance and discordance
[ E.23.13 ]
Correlation: invariance under positive affine transformations
[ E.23.14 ]
Correlation: symmetry with affine discordance
[ E.23.15 ]
Correlation between lognormal variables
[ E.24 ]
Statistical decision theory
[ E.24.1 ]
Equivalent definition of weak dominance
[ E.24.2 ]
Strong dominance implies weak dominance
[ E.24.3 ]
Equivalent definitions of second order stochastic dominance
[ E.24.4 ]
Non-admissibility of randomized decision functions for convex decision theory problems
[ E.24.5 ]
Decision theory: the ensemble approach
[ E.25 ]
Useful algorithms
[ E.25.1 ]
Moment-matching, scenario twisting: equations proof
[ E.25.2 ]
Building Student t scenarios with a low-rank-diagonal correlation matrix
[ E.25.3 ]
Number of observations in a generic bin
[ E.25.4 ]
Normalized empirical histogram approximating the true unknown pdf
E.VII. Factor models and learning
[ E.26 ]
Linear factor models
[ E.26.1 ]
Regression LFM’s: loadings
[ E.26.2 ]
Regression LFM’s: r-squared
[ E.26.3 ]
Regression LFM’s: covariance of residuals with factors
[ E.26.4 ]
Regression LFM’s: covariance of residuals
[ E.26.5 ]
Symmetric regression: analytical solution
[ E.26.6 ]
Statistically orthogonal vectors
[ E.26.7 ]
Karhunen–Loève: covariance eigenvectors have minimum entropy
[ E.26.8 ]
Eigenvalues of 2×2 positive matrix
[ E.26.9 ]
Eigenvectors of 2×2 positive matrix
[ E.26.10 ]
Dominant-residual LFM’s: mean squared error
[ E.26.11 ]
Regression LFM’s: differential of r-squared
[ E.26.12 ]
Regression LFM’s: concavity of r-squared
[ E.26.13 ]
Regression LFM’s: independence of residuals and factors (normal case)
[ E.26.14 ]
Regression LFM’s: r-squared and residual variance (univariate normal case)
[ E.26.15 ]
Parametrization of a square-dispersion
[ E.26.16 ]
Principal-component LFM’s: differential of r-squared
[ E.26.17 ]
Eigenvectors property
[ E.26.18 ]
Eigenfunctions property
[ E.26.19 ]
Cross-sectional LFM’s: differential of r-squared
[ E.26.20 ]
Principal factors and components of a bivariate normal
[ E.26.21 ]
Principal-component LFM’s: loadings and construction matrix
[ E.26.22 ]
Principal-component LFM’s: canonical loadings and construction matrix
[ E.26.23 ]
Principal-component LFM’s: covariance of factors
[ E.26.24 ]
Principal-component LFM’s: canonical solutions via recursive approach
[ E.26.25 ]
Principal-component LFM’s: loadings matrix is full rank
[ E.26.26 ]
Principal-component LFM’s: complementary projectors
[ E.26.27 ]
Principal-component LFM’s: rescaled prediction
[ E.26.28 ]
Principal-component LFM’s: r-squared
[ E.26.29 ]
Principal-component LFM’s: covariance of residuals with factors
[ E.26.30 ]
Principal-component LFM’s: covariance of residuals
[ E.26.31 ]
Static principal component estimation framework
[ E.26.32 ]
Principal-component LFM’s: equivalent formulation
[ E.26.33 ]
Factor analysis LFM’s: constraints
[ E.26.34 ]
Factor-analysis LFM’s: r-squared optimization
[ E.26.35 ]
Factor analysis LFM’s: first-step optimization
[ E.26.36 ]
Factor analysis LFM’s: PAF initialization
[ E.26.37 ]
Factor analysis LFM’s: idiosyncratic variances update
[ E.26.38 ]
Factor analysis LFM’s: bivariate solution with isotropic variances
[ E.26.39 ]
Factor analysis LFM’s: general solution with isotropic variances
[ E.26.40 ]
Factor analysis LFM’s: rotated factors
[ E.26.41 ]
Factor analysis LFM’s: regression factors
[ E.26.42 ]
Cross-sectional LFM’s: construction matrix
[ E.26.43 ]
Cross-sectional LFM’s: concavity of r-squared
[ E.26.44 ]
Rank property and positive definiteness for products
[ E.26.45 ]
Cross-sectional LFM’s: rank of construction matrix
[ E.26.46 ]
Cross-sectional LFM’s: complementary projectors
[ E.26.47 ]
Cross-sectional LFM’s: r-squared
[ E.26.48 ]
Cross-sectional LFM’s: r-squared under natural scatter specification
[ E.26.49 ]
Cross-sectional LFM’s: regression loadings under natural scatter specification
[ E.26.50 ]
Cross-sectional LFM’s: covariance of residuals with factors under natural scatter specification
[ E.26.51 ]
Cross-sectional LFM’s: minimum-variance portfolio
[ E.26.52 ]
Cross-sectional LFM’s: equivalent pseudo inverses
[ E.26.53 ]
Cross-sectional LFM’s: regression loadings under systematic-idiosyncratic assumption
[ E.26.54 ]
Cross-sectional LFM’s: regression factor replication
[ E.26.55 ]
Inconsistency between factor analysis and LFM’s with hidden factors
[ E.26.56 ]
Static cross-sectional LFM’s: sample r-squared maximization
[ E.26.57 ]
Affine equivariance of factor loadings and shift
[ E.26.58 ]
Conditional principal component analysis by iterating the classical PCA
[ E.26.59 ]
Eigenvalues of multiplication
[ E.26.60 ]
Transpose-square-root via CPCA
[ E.27 ]
Machine learning foundations
[ E.27.1 ]
Conditionally orthogonal linear model: relationship with systematic-idiosyncratic linear factor models
[ E.27.2 ]
Loss-implied scoring rules
[ E.27.3 ]
Discriminant model as generative model
[ E.27.4 ]
Cross entropy and relative entropy
[ E.27.5 ]
Scoring rule divergence as regret
[ E.27.6 ]
Loss-implied scoring rule as proper scoring rule
[ E.28 ]
Supervised learning: regression
[ E.28.1 ]
Regression LFM and linear least-squares regression
[ E.28.2 ]
Law of total variance in ANOVA models
[ E.28.3 ]
Linear normal regression gradient
[ E.28.4 ]
Linear discriminant regression model with affine features
[ E.28.5 ]
Linear discriminant regression model with affine features: generative embedding
[ E.28.6 ]
Cross-entropy minimization of a normal model
[ E.28.7 ]
Non-linear normal regression gradient
[ E.28.8 ]
Generalized linear models: optimum predictor
[ E.28.9 ]
Non-linear generalized models
[ E.29 ]
Supervised learning: classification
[ E.29.1 ]
Non-parametric classification: equivalent minimizations for binary classification
[ E.29.2 ]
Non-parametric classification: equivalent minimizations for multiple classification
[ E.29.3 ]
Non-parametric classification: conditional probability and weight of evidence
[ E.29.4 ]
Binary point classification: theoretical optimum via Neyman-Pearson lemma
[ E.29.5 ]
Binary point classification: likelihood ratio invariance
[ E.29.6 ]
Binary point classification: ROC curvature
[ E.29.7 ]
Non-parametric classification: false and true positive rates (normal case)
[ E.29.8 ]
Non-parametric classification: optimal predictor (normal case)
[ E.29.9 ]
Non-parametric classification: ROC function (normal case)
[ E.29.10 ]
Binary classification: alternative optimal cutoff
[ E.29.11 ]
Supervised point predictors: false positive and negative rates
[ E.29.12 ]
General case of 0-1 loss
[ E.29.13 ]
Expected loss
[ E.29.14 ]
Perceptron error
[ E.29.15 ]
Multinomial classification: binary loss generalization
[ E.29.16 ]
Multinomial probit regression
[ E.29.17 ]
Probabilistic misclassification: score and error
[ E.29.18 ]
Non-parametric classification: joint and marginal distribution of inputs and output
[ E.29.19 ]
Softmax optimization
[ E.30 ]
Unsupervised learning
[ E.30.1 ]
Statistical minimum-torsion optimization
[ E.30.2 ]
Unsupervised predictor: k-means clustering
[ E.30.3 ]
Partial orthogonality in systematic-idiosyncratic linear factor models
[ E.30.4 ]
Naive Bayes models: weight of evidence
[ E.30.5 ]
Naive Bayes models: weight of evidence (normal case)
[ E.30.6 ]
Kernel principal component analysis
[ E.30.7 ]
From logarithmic score to relative entropy
[ E.30.8 ]
Graphical models: probabilistic principal component
[ E.31 ]
Generalized probabilistic inference
[ E.31.1 ]
Minimum relative entropy and exponential family
[ E.31.2 ]
Distributional views updated: analytical formula
[ E.31.3 ]
Point views updated: analytical formula
[ E.31.4 ]
Multiplicative opinion pooling as minimum relative entropy updated
[ E.31.5 ]
Minimum relative entropy and exponential family: view parameters range
[ E.31.6 ]
Conditioning between normal variables
[ E.31.7 ]
Gradient of relative entropy
[ E.31.8 ]
Hessian of relative entropy
[ E.31.9 ]
Extremeness of the views
[ E.31.10 ]
Sensitivity to the views
[ E.31.11 ]
Convexity of relative entropy
[ E.31.12 ]
Minimum relative entropy via analytical implementation: updated distribution
[ E.31.13 ]
Minimum relative entropy via analytical implementation: updated distribution via projectors
[ E.31.14 ]
Minimum relative entropy via scenario-probability implementation: updated distribution
[ E.31.15 ]
Minimum relative entropy with scenario-probability implementation: gradient and Hessian of dual Lagrangian
[ E.31.16 ]
Minimum relative entropy with scenario-probability implementation: views on conditional value at risk
[ E.31.17 ]
Partial views of the exponential family: view on standard deviation
[ E.31.18 ]
Partial views of the exponential family: view on correlation
[ E.31.19 ]
Gradient of relative entropy with low-rank-diagonal covariance
[ E.31.20 ]
Degrees of freedom of a low-rank-diagonal matrix
[ E.31.21 ]
Chain rule for second derivatives
[ E.31.22 ]
Hessian of relative entropy with low-rank-diagonal covariance
[ E.31.23 ]
Gradient of constraint function on signal
[ E.31.24 ]
Hessian of constraint function on signal
[ E.31.25 ]
Views on joint and conditional distributions
[ E.31.26 ]
Views on ex-ante signal-to-noise ratios: formula
[ E.31.27 ]
Copula opinion pooling - Choice of the rotation matrix
[ E.31.28 ]
Distance minimum
[ E.31.29 ]
Distance equivalence
[ E.31.30 ]
Generalized shrinkage for covariance: sparse eigenvector rotation
[ E.31.31 ]
Generalized shrinkage for correlation: homogeneous clusters
[ E.31.32 ]
Generalized shrinkage for correlation: Markov networks
[ E.32 ]
Dynamic and spatial models
[ E.32.1 ]
Dominant residual DFM: equivalent formulation I
[ E.32.2 ]
Dominant residual DFM: equivalent formulation II
[ E.32.3 ]
Derivation of the dynamic regression filter
E.VIII. Stochastic processes
[ E.33 ]
Stochastic processes primer
[ E.33.1 ]
Finite-dimensional distributions of AR(1) with normal shocks
[ E.33.2 ]
Finite-dimensional distributions of AR(1) with Student t shocks
[ E.33.3 ]
Conditional distributions of AR(1) with normal shocks
[ E.33.4 ]
Conditional distributions of AR(1) with Student t shocks
[ E.33.5 ]
Conditional expectation of the stochastic process is coherent with the conditional expectation with respect to random variables
[ E.33.6 ]
Stochastic processes adapted to a filtration: fundamental property
[ E.33.7 ]
Conditional probabilities of adapted processes
[ E.33.8 ]
Conditional expectation process at each time t is adapted to the information set at time t
[ E.33.9 ]
Paths of conditional expectation process
[ E.33.10 ]
Law of the iterated expectations
[ E.33.11 ]
Radon-Nikodym process
[ E.33.12 ]
Adapted abstract Bayes theorem
[ E.33.13 ]
Distribution of price process under change of measures
[ E.33.14 ]
The price process is a martingale under Q
[ E.33.15 ]
Radon-Nikodym derivative process
[ E.34 ]
Covariance stationary processes
[ E.34.1 ]
About prediction as orthogonal projection with respect to infinite-dimensional subspaces
[ E.34.2 ]
Symmetry of the autocovariance function
[ E.34.3 ]
The sinusoid process
[ E.34.4 ]
The eigenvectors of the covariance matrix of a cov. stationary process are trigonometric waves
[ E.34.5 ]
Spectral theorem for univariate processes
[ E.34.6 ]
An example of process with purely singular integral power spectrum
[ E.34.7 ]
Symmetry of the spectral density
[ E.34.8 ]
The diagonal elements of the spectral density are real-valued and positive
[ E.34.9 ]
Some notes on the stochastic integral appearing in Cramer’s decomposition
[ E.34.10 ]
Equivalent formulations of the Cramer’s decomposition
[ E.34.11 ]
LTI filter of covariance stationary processes: properties
[ E.34.12 ]
Equivalence between frequency response and impulse response representations
[ E.34.13 ]
Impulse response function of the composition of LTI filters
[ E.34.14 ]
Composition of causal LTI filters is casual
[ E.34.15 ]
Impulse response function of the inverse of a polynomial shift filter
[ E.34.16 ]
Non causal VAR(1)
[ E.34.17 ]
One-step prediction error matrix and covariance stationarity
[ E.34.18 ]
The partial Wold decomposition
[ E.34.19 ]
Convergence of the partial Wold decomposition
[ E.34.20 ]
Wold representation and rotations
[ E.35 ]
Common mean-covariance processes
[ E.35.1 ]
Expectation and autocovariance function of the AR(1) process
[ E.35.2 ]
Half-life of AR(1) process
[ E.35.3 ]
Linear prediction of the AR(1) processes
[ E.35.4 ]
Spectral density of AR(1) processes
[ E.35.5 ]
Expectation and autocovariance function of the VAR(1) process
[ E.35.6 ]
Linear prediction of the VAR(1) process
[ E.35.7 ]
Linear prediction of VAR(1): case unit eigenvalues
[ E.35.8 ]
Error correction representation of unit-root VAR(1) process
[ E.35.9 ]
Linear prediction of cointegrated bivariate VAR(1)
[ E.35.10 ]
VARMA as linear state-space model
[ E.35.11 ]
Spectral density of VAR(1) processes
[ E.36 ]
Invariance tests
[ E.37 ]
Continuous time processes
[ E.37.1 ]
Characteristic function of standard Poisson process
[ E.37.2 ]
The compound Poisson process is a continuous combination of Poisson processes
[ E.37.3 ]
Characteristic function of continuous combination of Poisson processes
[ E.37.4 ]
Characteristic function of Poisson process on grid
[ E.37.5 ]
Characteristic function of compound Poisson process
[ E.37.6 ]
Characteristic function of standard Brownian motion
[ E.37.7 ]
Characteristic function of arithmetic Brownian motion with drift
[ E.37.8 ]
The Lévy-Khintchine representation of Lévy processes
[ E.37.9 ]
Δt-step location and dispersion parameters of Cauchy random walk
[ E.37.10 ]
Variance gamma parametrizations
[ E.37.11 ]
Relationship between CIR and Ornstein-Uhlenbeck processes
[ E.37.12 ]
Projection of a Markov chain: generator
[ E.37.13 ]
Fractional Brownian motion
[ E.37.14 ]
Distribution of the Δt-step and of the Δt-step shock of the OU process
[ E.37.15 ]
Conditional distribution and moments of OU
[ E.37.16 ]
Unconditional distribution of stationary OU
[ E.37.17 ]
OU process and the Brownian motion
[ E.37.18 ]
VAR(1) is MVOU
[ E.37.19 ]
Deterministic linear dynamic system
[ E.37.20 ]
Dynamics and distribution of MVOU process Zt
[ E.37.21 ]
Distribution of the Δt-step and Δt-step shock of the MVOU process
[ E.37.22 ]
Conditional distribution and moments of MVOU
[ E.37.23 ]
MVOU process and the Brownian motion
[ E.37.24 ]
Unconditional distribution of stationary MVOU
[ E.37.25 ]
MVOU (auto)covariances
E.IX. Estimation theory
[ E.38 ]
Probabilistic estimation and inference techniques
[ E.38.1 ]
Maximum likelihood parameters of multivariate normal
[ E.38.2 ]
Free energy of the posterior
[ E.38.3 ]
Minimum of free energy
[ E.38.4 ]
Bayes’ rule
[ E.38.5 ]
Normal distribution with fixed variance: prior, posterior, and posterior predictive
[ E.38.6 ]
Posterior distribution of exponential family distributions
[ E.38.7 ]
Predictive distribution of exponential family distributions
[ E.38.8 ]
The EM algorithm
[ E.38.9 ]
EM algorithm for i.i.d. processes
[ E.38.10 ]
Maximum likelihood for longitudinal panels of data
[ E.38.11 ]
Smoothing/nowcasting of the hidden variables for i.i.d. processes
[ E.38.12 ]
EM algorithm for state-space processes
[ E.38.13 ]
Conditioning as IM projection
[ E.38.14 ]
The ELBO is a lower bound for the evidence of data
[ E.38.15 ]
The EM algorithm in population
[ E.39 ]
Estimation and assessment
[ E.39.1 ]
The posterior error of the relative entropy loss
[ E.39.2 ]
p-value of the sample mean: normal invariants, known variance
[ E.39.3 ]
Sample mean loss distribution
[ E.39.4 ]
Error, bias, inefficiency
[ E.39.5 ]
Inefficiency of the sample mean
[ E.39.6 ]
Posterior expectation
[ E.39.7 ]
Estimation error of the sample mean
[ E.39.8 ]
Sample mean loss in a homogeneously correlated market
[ E.39.9 ]
Estimation error of the sample covariance
[ E.40 ]
Bias reduction
[ E.40.1 ]
Functional gradient descent
[ E.40.2 ]
Gradient boosting
[ E.40.3 ]
Solution of linear least-squares regression
[ E.40.4 ]
Polynomial features
[ E.40.5 ]
Kernel trick
[ E.40.6 ]
Equivalent expression for piecewise linear functions
[ E.41 ]
Estimation and regularization
[ E.41.1 ]
Regression LFM’s: equivalent formulation
[ E.41.2 ]
Matrix decomposition
[ E.41.3 ]
Regression LFM’s as quadratic programming
[ E.41.4 ]
Cross-sectional LFM as quadratic programming
[ E.41.5 ]
Feature engineering: error derivative
[ E.41.6 ]
Exponential format of an arbitrary pdf
[ E.42 ]
Hypothesis testing
[ E.42.1 ]
t-statistic of the sample mean: normal invariants, unknown variance
[ E.42.2 ]
Univariate testing: consistency of the sample variance
[ E.42.3 ]
Univariate testing: the square standard error
[ E.42.4 ]
Univariate testing: the z-statistic
[ E.42.5 ]
Multivariate testing: consistency of the sample covariance
[ E.42.6 ]
Multivariate testing: the Hotelling statistic
Featured case studies
E.X. Quantitative finance: the "Checklist"
[ E.43 ]
Historical Checklist
[ E.44 ]
Monte Carlo Checklist
E.XI. Data science: factor models and learning
[ E.45 ]
Principal component analysis of the yield curve
[ E.45.1 ]
Martingale property
[ E.45.2 ]
Spectral basis in the continuum
[ E.45.3 ]
Eigenvalues integration
[ E.46 ]
Machine learning for hedging
[ E.46.1 ]
Machine learning for hedging: CART predictor as a portfolio of digital options
[ E.47 ]
Regression in the stock market
[ E.47.1 ]
Regression LFM’s: maximum likelihood with flexible probabilities estimates of factor loadings and residual covariance
[ E.47.2 ]
Regression LFM’s: maximum likelihood with flexible probabilities estimates under t-conditional residuals
[ E.47.3 ]
Regression LFM’s: distribution of least square estimates
[ E.47.4 ]
Regression LFM’s: likelihood
[ E.47.5 ]
Regression LFM’s: distribution of loadings under NIW assumption
[ E.47.6 ]
Regression LFM’s: posterior distribution under NIW assumption
[ E.47.7 ]
Regression LFM’s: mode of posterior under NIW
[ E.47.8 ]
Regression LFM’s: modal dispersion of posterior under NIW
[ E.47.9 ]
Regression LFM’s: predictive distribution
[ E.47.10 ]
Lasso as a generalization of maximum likelihood with flexible probabilities
[ E.47.11 ]
Inputs standardization in lasso regression
[ E.48 ]
Credit default classification
[ E.49 ]
Clustering for the stock market
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