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Quantitative finance

[ 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 ] Proportionality constraint of the equilibrium portfolio under CAPM assumptions

[ E.0c.20 ] From the security market line to the standard CAPM

[ E.0c.21 ] Buhlmann pricing equation in terms or returns

[ E.0c.22 ] Systematic-idiosyncratic linear factor model in a complete market

[ E.0c.23 ] Systematic-idiosyncratic linear factor model with zero residuals

[ E.0c.24 ] Derivation of the APT model from the stochastic discount factor model

[ E.0c.25 ] From APT to the security market line

[ E.0c.26 ] Pricing equation as conditional expectation: proof

[ E.0c.27 ] Alternative formulation intertemporal pricing equation

[ E.0c.28 ] 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 ] Copula of Markov chain’s invariants

[ E.3.2 ] Invariants distribution from the estimated standardized invariants distribution

[ E.3.3 ] Realized and empirical variance

[ E.3.4 ] Covariance and correlation of rescaled/normalized random variables

[ E.3.5 ] 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 ] Strategic allocation from mean-variance

[ E.9a.2 ] Mean-variance dual problem: total returns

[ E.9a.3 ] Mean-variance dual problem: return in excess of non-investable risk-free return

[ E.9a.4 ] Mean-variance dual problem: return in excess of risk-free return

[ E.9a.5 ] Mean-variance dual problem: return in excess of non-replicable benchmark

[ E.9a.6 ] Mean-variance in terms of excess weights

[ E.9a.7 ] Mean-variance dual problem: return in excess of replicable benchmark

[ E.9a.8 ] Mean-variance dual optimization

[ E.9a.9 ] Extended mean-variance dual optimization

[ E.9a.10 ] Extended mean-variance dual optimization: degenerate case

[ E.9a.11 ] Mean-variance dual optimization: linear constraint

[ E.9a.12 ] Benchmark-relative efficient allocation in the plane of absolute coordinates

[ E.9a.13 ] Total return efficient allocations in the plane of benchmark-relative coordinates

[ E.9a.14 ] Lasso penalty in constrained selection

[ E.9a.15 ] Prediction in absence of signals

[ E.9a.16 ] Maximum information ratio portfolio allocation

[ E.9a.17 ] Maximum information ratio portfolio allocation

[ E.9a.18 ] Maximum mean information ratio analytical solution

[ E.9a.19 ] Information coefficient in terms of correlation

[ E.9a.20 ] Information coefficient as r-squared

[ E.9a.21 ] Mean maximum information ratio of weak signals

[ E.9b ] Construction: estimation and model risk

[ E.9c ] Construction: cross-sectional strategies

[ E.9c.1 ] Market-capitalization allocation

[ E.9c.2 ] Maximal conditional signal-to-noise (normal case)

[ E.9c.3 ] Smart beta: factor premium

[ E.9c.4 ] Flexible and standard characteristic portfolio

[ E.9c.5 ] Linkage matrix and signal weakness

[ E.9c.6 ] 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

Data science

E.V. Mathematics

[ E.14 ] Linear algebra primer

[ E.14.1 ] Linear independence

[ E.14.2 ] Vector operations on coordinates

[ E.14.3 ] Direct sum of vector subspaces

[ E.14.4 ] Matrix operations

[ E.14.5 ] Matrix basic properties

[ E.14.6 ] Dimension of general linear group

[ E.14.7 ] Positive semidefinite matrix

[ E.14.8 ] Positive definiteness of block-diagonal

[ E.14.9 ] Positive definiteness of inverse

[ E.14.10 ] Positive definiteness of Kronecker product

[ E.14.11 ] Linear operator as inner product

[ E.14.12 ] Useful identities for inner product spaces

[ E.14.13 ] Cauchy-Schwarz inequality

[ E.14.14 ] Orthonormal sets are linearly independent

[ E.14.15 ] Orthogonal projection over a span

[ E.14.16 ] Orthogonal projection over direct sums

[ E.14.17 ] p-norm is a norm

[ E.14.18 ] Distance induced by norm

[ E.14.19 ] Eigenvalues of symmetric matrices

[ E.14.20 ] Eigenvalues of symmetric positive (semi)definite matrices

[ E.14.21 ] Eigenvalues of the inverse of a matrix

[ E.14.22 ] Relation among trace, determinant and eigenvalues

[ E.14.23 ] The UDU-Cholesky decomposition

[ E.14.24 ] Gramian and linear independence

[ E.14.25 ] Finite-dimensional inner products

[ E.14.26 ] Affine equivariance of Gram matrix

[ E.14.27 ] Recursion for eigenvalues and eigenvectors in two dimensions

[ E.14.28 ] PCA with repeated eigenvalues

[ E.14.29 ] The constrained Procrustes problem

[ E.14.30 ] Minimum torsion orthonormalization

[ E.14.31 ] Linearity of vectorization

[ E.14.32 ] Partitioned matrix inversion

[ E.14.33 ] Inverse of a block-triangular matrix

[ E.14.34 ] Inverse of an upper-triangular Toeplitz matrix

[ E.15 ] Calculus primer

[ E.15.1 ] Differentiability characterization

[ E.15.2 ] Gradient of the quadratic form

[ E.15.3 ] Gradient chain rule

[ E.15.4 ] Chain rule for first order differential

[ E.15.5 ] First derivative of monotonic functions

[ E.15.6 ] Cubic function is strictly increasing

[ E.15.7 ] Strictly monotone maps are invertible

[ E.15.8 ] Alternative convexity criterion

[ E.15.9 ] Convex functions have invertible gradients

[ E.16 ] Functional analysis

[ E.16.1 ] The Fourier integral is the most general Fourier transform form

[ E.16.2 ] The Fourier Transform as a rescaled unitary operator

[ E.16.3 ] Fourier transform of the Dirac delta

[ E.17 ] Optimization primer

[ E.17.1 ] Newton’s method

[ E.17.2 ] Equality constraints must be affine

[ E.17.3 ] Semi-definite cones

[ E.17.4 ] Alternate SDP formulation

[ E.17.5 ] Ice-cream cones of dimension ¯¯¯m

[ E.17.6 ] QCQP as special case of SOCP

[ E.17.7 ] Regularized regression is regularized quadratic

[ E.17.8 ] Constrained generalized elastic net is quadratic programming

[ E.17.9 ] Generalized lasso is lasso

[ E.17.10 ] Equivalent quadratic optimization for portfolio replication

E.VI. Statistics

[ E.18 ] Distributions

[ E.18.1 ] Pdf of an invertible function of a univariate random variable

[ E.18.2 ] Cdf of an invertible function of a univariate random variable

[ E.18.3 ] Quantile function and inverse cdf

[ E.18.4 ] Quantile of an invertible function of a random variable

[ E.18.5 ] Expected value in terms of the quantile

[ E.18.6 ] Sub-quantile as conditional expectation

[ E.18.7 ] Sub-quantile of an affine transformation

[ E.18.8 ] Multivariate Student t distribution: cumulative distribution function

[ E.18.9 ] Chi-distribution: numerical implementation of the quantile

[ E.18.10 ] Relation between the characteristic function and the moments

[ E.18.11 ] Moments of the chi-squared distribution

[ E.18.12 ] Scaling property of the gamma distribution

[ E.18.13 ] Equivalence between gamma and chi-squared distribution

[ E.18.14 ] Moments of the gamma distribution

[ E.18.15 ] Expectation of the exponential of a gamma random variable

[ E.18.16 ] Quadratic-normal distribution in terms of independent standard normal variables

[ E.18.17 ] Log-characteristic function of quadratic-normal distribution

[ E.18.18 ] Saddle point approximation of the quadratic-normal distribution

[ E.18.19 ] Variance of quadratic-normal distribution

[ E.18.20 ] Wishart and gamma distribution

[ E.18.21 ] Marginals of a Wishart distribution

[ E.18.22 ] Result on the joint distribution of a bivariate random variable

[ E.18.23 ] Conditional pdf

[ E.18.24 ] Conditional quantile and conditional characteristic function

[ E.18.25 ] Conditional and unconditional expectation

[ E.18.26 ] Conditional and unconditional invariance

[ E.18.27 ] Conditional distribution between normal random variables

[ E.18.28 ] Conditional distribution between lognormal random variables

[ E.18.29 ] Conditional expectation of two sets of lognormal random variables

[ E.18.30 ] Bayes’ rule

[ E.18.31 ] Law of total variance: joint Student t

[ E.18.32 ] Law of total variance: joint lognormal

[ E.18.33 ] Covariances and correlations parametrizations of two sets of multivariate random variables

[ E.18.34 ] Conditional expectation and covariance of two sets of normal random variables

[ E.18.35 ] Marginalization cdf formula

[ E.18.36 ] Pdf of an invertible function of a multivariate random variable

[ E.18.37 ] Pdf of a non-invertible function of a multivariate random variable

[ E.18.38 ] Cdf of an invertible comonotonic function of a multivariate random variable

[ E.18.39 ] Pdf of a non-invertible affine transformation of a multivariate random variable

[ E.18.40 ] Characteristic function of a multivariate normal random variable I

[ E.18.41 ] Cdf of the lognormal distribution

[ E.18.42 ] Non-central moments of a multivariate lognormal random variable

[ E.18.43 ] Moments of the reflected shifted lognormal distribution

[ E.18.44 ] Expectation and covariance of a multivariate lognormal random variable

[ E.18.45 ] Expectation, standard deviation and skewness of a linear combination of multivariate shifted lognormal random vector

[ E.18.46 ] Gradient of the pdf of a multivariate affine function

[ E.18.47 ] Hessian of the pdf of a multivariate affine transformation

[ E.18.48 ] Gradient of the log-pdf of a multivariate variable

[ E.18.49 ] Hessian of the log-pdf of a multivariate variable

[ E.18.50 ] Pdf of an inverse-Wishart random variable

[ E.18.51 ] Equivalence between definitions of elliptical distribution

[ E.18.52 ] Radial component and generator function of elliptical distributions

[ E.18.53 ] Radial component of multivariate normal is chi distributed

[ E.18.54 ] Radial component of multivariate Student t

[ E.18.55 ] Building Student t scenarios with a low-rank-diagonal correlation matrix

[ E.18.56 ] Radial component of a uniform random variable inside an ellipsoid

[ E.18.57 ] Moments of an elliptical random variable

[ E.18.58 ] Expectation of Mahalanobis square distance of normal random variables

[ E.18.59 ] Expectation of Mahalanobis square distance of Student t random variables

[ E.18.60 ] Moments of a uniform random variable inside an ellipsoid

[ E.18.61 ] Moments of the uniform component of an elliptical distribution

[ E.18.62 ] Elliptical distributions: formula for the generator of a univariate affine transformation

[ E.18.63 ] Elliptical distributions: generator of the marginal distribution of a uniform inside the unit circle

[ E.18.64 ] Normal distribution as limit of Student t distribution

[ E.18.65 ] Normal generator as limit of Student t generator

[ E.18.66 ] Marginal distribution of a uniform random variable inside the unit sphere

[ E.18.67 ] Truncated quantile

[ E.18.68 ] Stress distribution of elliptical is elliptical

[ E.18.69 ] Stress quantile of elliptical distributions

[ E.18.70 ] Alternative stochastic representation of elliptical random variables

[ E.18.71 ] Gini coefficient in terms of covariance

[ E.18.72 ] Scenario-probability distribution: cdf

[ E.18.73 ] Scenario-probability distribution: probability density function

[ E.18.74 ] Scenario-probability distribution: expectation

[ E.18.75 ] Scenario-probability distribution: invariance rule

[ E.18.76 ] Scenario-probability distribution: expectation rule

[ E.18.77 ] Scenario-probability distribution: cdf via expectation rule

[ E.18.78 ] Scenario-probability distribution: characteristic function

[ E.18.79 ] Scenario-probability distribution: quantile

[ E.18.80 ] Scenario-probability distribution: quantile for uniform flexible probabilities

[ E.18.81 ] Smooth quantile through scenario-probability quantile

[ E.18.82 ] Scenario-probability covariance matrix

[ E.18.83 ] Scenario-probability correlation matrix

[ E.18.84 ] Scenario-probability distribution: positive probabilities

[ E.18.85 ] Conditional distribution between normal random variables in canonical parametrization

[ E.18.86 ] Maximum partition encoder: underlying partition

[ E.18.87 ] Multinomial logit parametrization

[ E.18.88 ] Multinomial probit parametrization

[ E.18.89 ] Effective number of scenarios boundedness: exponential of the entropy

[ E.18.90 ] Effective number of scenarios boundedness: generalized exponential of the entropy

[ E.18.91 ] Effective number of scenarios counting crisp scenarios: exponential of the entropy

[ E.18.92 ] Effective number of scenarios counting crisp scenarios: generalized exponential of the entropy

[ E.18.93 ] Characteristic function of exponential family distributions

[ E.18.94 ] Exponential family distributions: expectation of the sufficient statistics

[ E.18.95 ] Exponential family distributions: covariance of the sufficient statistics

[ E.18.96 ] Joint mean and covariance of a mixture model

[ E.18.97 ] Mixture probabilities

[ E.18.98 ] Normal mixtures

[ E.18.99 ] Abstract Bayes theorem

[ E.18.100 ] Radon-Nikodym derivative on finite spaces

[ E.18.101 ] Conditional expectation over elementary events

[ E.18.102 ] Conditional pdf as L2 projection

[ E.18.103 ] Adapted approximations

[ E.18.104 ] Radon-Nikodym with log-normal market

[ E.18.105 ] Conditional expectation: equivalent formulation and Radon-Nikodym

[ E.19 ] Copulas

[ E.19.1 ] Distribution of the grade

[ E.19.2 ] Inverse cdf sampling

[ E.19.3 ] Pdf of a copula

[ E.19.4 ] Sklar’s theorem

[ E.19.5 ] Pdf of the copula of a bivariate normal

[ E.19.6 ] Pdf of a normal copula

[ E.19.7 ] Cdf of a copula

[ E.19.8 ] Comonotonic invariance of copulas

[ E.19.9 ] Copulas of elliptical distributions

[ E.20 ] Geometry of distributions

[ E.20.1 ] Riemannian metric: curve length

[ E.20.2 ] Riemannian metric: volume

[ E.20.3 ] Fisher information metric: covariant property

[ E.20.4 ] Fisher information metric: univariate normal distribution

[ E.20.5 ] E-affine coordinates of univariate normal distributions

[ E.20.6 ] M-affine coordinates of univariate normal distributions

[ E.20.7 ] Duality of univariate normal distributions

[ E.20.8 ] Fisher information metric: univariate normal distribution (dual parameters)

[ E.20.9 ] Legendre dual function: Hessian matrix

[ E.20.10 ] Legendre dual function: duality

[ E.20.11 ] Legendre transformation

[ E.20.12 ] Potential functions of univariate normal distributions

[ E.20.13 ] Bregman divergence of univariate normal distributions

[ E.20.14 ] E-affine coordinates of exponential family

[ E.20.15 ] Geodesic of exponential family

[ E.20.16 ] Tangent vector of multivariate normal distributions

[ E.20.17 ] Gradient of the normal log partition function

[ E.20.18 ] Expectation parameters of multivariate normal distributions

[ E.20.19 ] Fisher information metric: multivariate normal distribution

[ E.20.20 ] Transpose Jacobian of the normal distribution

[ E.20.21 ] Relative entropy: exponential family

[ E.20.22 ] Fisher information metric: scenario-probability distribution

[ E.21 ] Location and dispersion

[ E.21.1 ] Relation between z-score and signal-to-noise ratio

[ E.21.2 ] Affine equivariance implies Mahalanobis distance invariance

[ E.21.3 ] Mahalanobis distance invariance implies affine equivariance

[ E.21.4 ] Absolute z-score invariance

[ E.21.5 ] Affine property of argmax

[ E.21.6 ] Affine equivariance of the mode

[ E.21.7 ] Affine equivariance of the modal dispersion

[ E.21.8 ] Affine equivariance of the median

[ E.21.9 ] Affine equivariance of the interquantile range

[ E.21.10 ] Affine equivariance of the expectation

[ E.21.11 ] Affine equivariance of the standard deviation

[ E.21.12 ] Monotonic invariance of the median

[ E.21.13 ] Mode of a univariate lognormal

[ E.21.14 ] Modal dispersion of a univariate lognormal

[ E.21.15 ] Orthogonality of eigenvectors

[ E.21.16 ] Recursion for eigenvalues and eigenvectors

[ E.21.17 ] Points with constant Mahalanobis distance form an ellipsoid

[ E.21.18 ] Integral of Mahalanobis distance

[ E.21.19 ] Affine equivariance implies Mahalanobis distance invariance (multivariate case)

[ E.21.20 ] Mahalanobis distance invariance implies affine equivariance (multivariate case)

[ E.21.21 ] Affine equivariance of the mode (multivariate case)

[ E.21.22 ] Affine equivariance of the modal square-dispersion

[ E.21.23 ] Mode of a multivariate lognormal distribution

[ E.21.24 ] Modal square-dispersion of a multivariate lognormal

[ E.21.25 ] Compact formula for multivariate expectation

[ E.21.26 ] Compact formula for the covariance matrix

[ E.21.27 ] Covariance matrix as matrix-variate expectation

[ E.21.28 ] Generalized affine equivariance of the expectation (multivariate case)

[ E.21.29 ] Generalized affine equivariance of the covariance

[ E.21.30 ] Bilinearity of the covariance

[ E.21.31 ] Expectation and covariance of a multivariate shifted lognormal

[ E.21.32 ] Affine equivariance of the cross-covariance

[ E.21.33 ] Expectation of the sum of two variables

[ E.21.34 ] Covariance matrix of the sum of two variables

[ E.21.35 ] Mode of sum of two gamma distributions

[ E.21.36 ] Expectation and variance of the gamma distribution

[ E.21.37 ] Mode and modal square dispersion of a gamma distribution

[ E.21.38 ] Generalized affine equivariance does not hold for modal square dispersion

[ E.21.39 ] Alternative generalization of uncertainty band

[ E.21.40 ] Alternative generalization of uncertainty band

[ E.21.41 ] Multivariate uncertainty band

[ E.21.42 ] Gradient of normal characteristic function

[ E.21.43 ] Hessian of normal characteristic function

[ E.21.44 ] Taylor expansion of the characteristic function

[ E.21.45 ] Tangent box of the ellipsoid

[ E.21.46 ] Principal directions and principal variances

[ E.21.47 ] Property of expectation

[ E.21.48 ] Multivariate Markov inequality

[ E.21.49 ] Generalized Chebyshev’s inequality

[ E.21.50 ] First order differential of the square Mahalanobis distance

[ E.21.51 ] The Chebyshev’s inequality and most likely set

[ E.21.52 ] Mahalanobis square distance of normal random variables

[ E.21.53 ] Explicit expression of the error matrix

[ E.21.54 ] Affine equivariance of linear projection and partial covariance

[ E.21.55 ] Explicit expression of the loss matrix

[ E.21.56 ] L2 of law of total variance

[ E.21.57 ] Linear and non-linear projections under normality

[ E.21.58 ] Visualization map is isometry

[ E.21.59 ] Cauchy-Schwarz inequality

[ E.21.60 ] Alternative formulation of multivariate inner product

[ E.21.61 ] Expectation length and distance

[ E.21.62 ] Relationship between non-central and central tracking errors

[ E.21.63 ] Quantile and subquantile-deviation

[ E.21.64 ] Variational location and dispersion are affine equivariant

[ E.21.65 ] Bregman location-dispersion

[ E.21.66 ] Multivariate p-quantile

[ E.21.67 ] Lp spaces

[ E.21.68 ] R-squared and equivalent optimization objective

[ E.21.69 ] Inner product in terms of the covariance matrix

[ E.21.70 ] Extension of visualization map

[ E.21.71 ] Alternative visualization basis

[ E.21.72 ] Best approximation: shifted orthogonal projection

[ E.21.73 ] Best linear prediction: solution

[ E.21.74 ] Characterization of conditional expectation

[ E.21.75 ] Best approximation: equivalent characterization

[ E.21.76 ] Cholesky root via Gram-Schmidt

[ E.22 ] Correlation and generalizations

[ E.22.1 ] Interpretation of independence

[ E.22.2 ] Characterization of independence through copulas

[ E.22.3 ] Cdf of uniform distribution on the unit square

[ E.22.4 ] Cdf of an “extreme”copula (Frechet-Hoeffding bottom bound)

[ E.22.5 ] Cdf of an “extreme”copula (Frechet-Hoeffding top bound)

[ E.22.6 ] Frechet-Hoeffding bounds and copula of monotonic variables

[ E.22.7 ] Schweizer-Wolff measure: equivalent expression

[ E.22.8 ] Copulas of non comonotonic variables

[ E.22.9 ] Regularized call option payoff

[ E.22.10 ] Regularized put option payoff

[ E.22.11 ] Kendall’s tau: equivalent expression

[ E.22.12 ] Correlation: affine concordance and discordance

[ E.22.13 ] Correlation: invariance under positive affine transformations

[ E.22.14 ] Correlation: symmetry with affine discordance

[ E.22.15 ] Correlation between lognormal variables

[ E.23 ] Statistical decision theory

[ E.23.1 ] Regret and optimal action

[ E.23.2 ] Equivalent definition of weak dominance

[ E.23.3 ] Strong dominance implies weak dominance

[ E.23.4 ] Equivalent definitions of second order stochastic dominance

[ E.23.5 ] Decision theory: the frequentist way

[ E.23.6 ] Decision theory: the Bayesian way

[ E.23.7 ] Decision theory: the frequentist way for randomized decision

[ E.23.8 ] Non-admissibility of randomized decision functions for convex decision theory problems

[ E.23.9 ] Decision theory: the ensemble approach

[ E.24 ] Useful algorithms

[ E.24.1 ] Moment-matching, scenario twisting: equations proof

[ E.24.2 ] Number of observations in a generic bin

[ E.24.3 ] Normalized empirical histogram approximating the true unknown pdf

E.VII. Factor models and learning

[ E.25 ] Linear factor models

[ E.25.1 ] Regression LFM’s: loadings

[ E.25.2 ] Regression LFM’s: r-squared

[ E.25.3 ] Regression LFM’s: covariance of residuals with factors

[ E.25.4 ] Regression LFM’s: covariance of residuals

[ E.25.5 ] Symmetric regression: analytical solution

[ E.25.6 ] Statistically orthogonal vectors

[ E.25.7 ] Karhunen–Loève: covariance eigenvectors have minimum entropy

[ E.25.8 ] Eigenvalues of 2×2 positive matrix

[ E.25.9 ] Eigenvectors of 2×2 positive matrix

[ E.25.10 ] Dominant-residual LFM’s: mean squared error

[ E.25.11 ] Regression LFM’s: differential of r-squared

[ E.25.12 ] Regression LFM’s: concavity of r-squared

[ E.25.13 ] Regression LFM’s: independence of residuals and factors (normal case)

[ E.25.14 ] Regression LFM’s: r-squared and residual variance (univariate normal case)

[ E.25.15 ] Parametrization of a square-dispersion

[ E.25.16 ] Principal-component LFM’s: differential of r-squared

[ E.25.17 ] Eigenvectors property

[ E.25.18 ] Eigenfunctions property

[ E.25.19 ] Cross-sectional LFM’s: differential of r-squared

[ E.25.20 ] Principal factors and components of a bivariate normal

[ E.25.21 ] Principal-component LFM’s: loadings and construction matrix

[ E.25.22 ] Principal-component LFM’s: canonical loadings and construction matrix

[ E.25.23 ] Principal-component LFM’s: covariance of factors

[ E.25.24 ] Principal-component LFM’s: canonical solutions via recursive approach

[ E.25.25 ] Principal-component LFM’s: loadings matrix is full rank

[ E.25.26 ] Principal-component LFM’s: complementary projectors

[ E.25.27 ] Principal-component LFM’s: rescaled prediction

[ E.25.28 ] Principal-component LFM’s: r-squared

[ E.25.29 ] Principal-component LFM’s: covariance of residuals with factors

[ E.25.30 ] Principal-component LFM’s: covariance of residuals

[ E.25.31 ] Static principal component estimation framework

[ E.25.32 ] Principal-component LFM’s: equivalent formulation

[ E.25.33 ] Factor analysis LFM’s: constraints

[ E.25.34 ] Factor-analysis LFM’s: r-squared optimization

[ E.25.35 ] Factor analysis LFM’s: first-step optimization

[ E.25.36 ] Factor analysis LFM’s: PAF initialization

[ E.25.37 ] Factor analysis LFM’s: idiosyncratic variances update

[ E.25.38 ] Factor analysis LFM’s: bivariate solution with isotropic variances

[ E.25.39 ] Factor analysis LFM’s: general solution with isotropic variances

[ E.25.40 ] Factor analysis LFM’s: rotated factors

[ E.25.41 ] Factor analysis LFM’s: regression factors

[ E.25.42 ] Cross-sectional LFM’s: construction matrix

[ E.25.43 ] Cross-sectional LFM’s: concavity of r-squared

[ E.25.44 ] Rank property and positive definiteness for products

[ E.25.45 ] Cross-sectional LFM’s: rank of construction matrix

[ E.25.46 ] Cross-sectional LFM’s: complementary projectors

[ E.25.47 ] Cross-sectional LFM’s: r-squared

[ E.25.48 ] Cross-sectional LFM’s: r-squared under natural scatter specification

[ E.25.49 ] Cross-sectional LFM’s: regression loadings under natural scatter specification

[ E.25.50 ] Cross-sectional LFM’s: covariance of residuals with factors under natural scatter specification

[ E.25.51 ] Cross-sectional LFM’s: minimum-variance portfolio

[ E.25.52 ] Cross-sectional LFM’s: equivalent pseudo inverses

[ E.25.53 ] Cross-sectional LFM’s: regression loadings under systematic-idiosyncratic assumption

[ E.25.54 ] Cross-sectional LFM’s: regression factor replication

[ E.25.55 ] Inconsistency between factor analysis and LFM’s with hidden factors

[ E.25.56 ] Static cross-sectional LFM’s: sample r-squared maximization

[ E.25.57 ] Affine equivariance of factor loadings and shift

[ E.25.58 ] Conditional principal component analysis by iterating the classical PCA

[ E.25.59 ] Eigenvalues of multiplication

[ E.25.60 ] Transpose-square-root via CPCA

[ E.26 ] Machine learning foundations

[ E.26.1 ] Conditionally orthogonal linear model: relationship with systematic-idiosyncratic linear factor models

[ E.26.2 ] Loss-implied scoring rules

[ E.26.3 ] Discriminant model as generative model

[ E.26.4 ] Cross entropy and relative entropy

[ E.26.5 ] Scoring rule divergence as regret

[ E.26.6 ] Loss-implied scoring rule as proper scoring rule

[ E.27 ] Supervised learning: regression

[ E.27.1 ] Regression LFM and linear least-squares regression

[ E.27.2 ] Law of total variance in ANOVA models

[ E.27.3 ] Linear normal regression gradient

[ E.27.4 ] Functional derivative of mean-squared error

[ E.27.5 ] Linear discriminant regression model with affine features

[ E.27.6 ] Linear discriminant regression model with affine features: generative embedding

[ E.27.7 ] Cross-entropy minimization of a normal model

[ E.27.8 ] Non-linear normal regression gradient

[ E.27.9 ] Generalized linear models: optimum predictor

[ E.27.10 ] Non-linear generalized models

[ E.28 ] Supervised learning: classification

[ E.28.1 ] Non-parametric classification: equivalent minimizations for binary classification

[ E.28.2 ] Non-parametric classification: equivalent minimizations for multiple classification

[ E.28.3 ] Non-parametric classification: conditional probability and weight of evidence

[ E.28.4 ] Binary point classification: theoretical optimum via Neyman-Pearson lemma

[ E.28.5 ] Binary point classification: likelihood ratio invariance

[ E.28.6 ] Binary point classification: ROC curvature

[ E.28.7 ] Linear discriminative regression model with affine features: interactions

[ E.28.8 ] Non-parametric classification: false and true positive rates (normal case)

[ E.28.9 ] Non-parametric classification: optimal predictor (normal case)

[ E.28.10 ] Non-parametric classification: ROC function (normal case)

[ E.28.11 ] Binary classification: alternative optimal cutoff

[ E.28.12 ] Supervised point predictors: false positive and negative rates

[ E.28.13 ] General case of 0-1 loss

[ E.28.14 ] Expected loss

[ E.28.15 ] Perceptron error

[ E.28.16 ] Multinomial classification: binary loss generalization

[ E.28.17 ] Discriminant classification: cross-entropy

[ E.28.18 ] Multinomial probit regression

[ E.28.19 ] Probabilistic misclassification: score and error

[ E.28.20 ] Non-parametric classification: joint and marginal distribution of inputs and output

[ E.28.21 ] LDA as generative embbeding

[ E.28.22 ] QDA as generative embbeding

[ E.28.23 ] Multinomial logistic regression: generalized linear model

[ E.28.24 ] Binary logistic regression: error

[ E.28.25 ] Multinomial logistic regression: error

[ E.28.26 ] Binary probit regression: error

[ E.28.27 ] Multinomial probit regression: error

[ E.28.28 ] Functional derivative of cross-entropy

[ E.29 ] Unsupervised learning

[ E.29.1 ] Statistical minimum-torsion optimization

[ E.29.2 ] Unsupervised predictor: k-means clustering

[ E.29.3 ] Partial orthogonality in systematic-idiosyncratic linear factor models

[ E.29.4 ] Probabilistic factor analysis: consistency with factor analysis models

[ E.29.5 ] Naive Bayes models: weight of evidence

[ E.29.6 ] Naive Bayes models: weight of evidence (normal case)

[ E.29.7 ] Kernel principal component analysis

[ E.29.8 ] Probabilistic factor analysis as maximum likelihood factorization

[ E.29.9 ] Probabilistic factor analysis: equivalence with factor analysis model

[ E.29.10 ] Probabilistic factor analysis: rotated factors

[ E.29.11 ] Probabilistic factor analysis: inference

[ E.29.12 ] Probabilistic factor analysis: recovered factors

[ E.29.13 ] Edges of Gaussian Markov networks

[ E.29.14 ] Graphical models: probabilistic principal component

[ E.29.15 ] Bayes networks: equivalent specification of the local Markov property

E.VIII. Stochastic processes

[ E.31 ] Stochastic processes primer

[ E.31.1 ] Finite-dimensional distributions of AR(1) with normal shocks

[ E.31.2 ] Finite-dimensional distributions of AR(1) with Student t shocks

[ E.31.3 ] Conditional distributions of AR(1) with normal shocks

[ E.31.4 ] Conditional distributions of AR(1) with Student t shocks

[ E.31.5 ] Conditional expectation of the stochastic process is coherent with the conditional expectation with respect to random variables

[ E.31.6 ] Linear prediction is a mean-covariance concept

[ E.31.7 ] Stochastic processes adapted to a filtration: fundamental property

[ E.31.8 ] Conditional probabilities of adapted processes

[ E.31.9 ] Conditional expectation process at each time t is adapted to the information set at time t

[ E.31.10 ] Paths of conditional expectation process

[ E.31.11 ] Law of the iterated expectations

[ E.31.12 ] Radon-Nikodym process

[ E.31.13 ] Adapted abstract Bayes theorem

[ E.31.14 ] Distribution of price process under change of measures

[ E.31.15 ] The price process is a martingale under Q

[ E.31.16 ] Radon-Nikodym derivative process

[ E.32 ] Covariance stationary processes

[ E.32.1 ] Linear prediction as direct minimization: the AR(1) case

[ E.32.2 ] About prediction as orthogonal projection with respect to infinite-dimensional subspaces

[ E.32.3 ] Spectral decomposition in terms of rescaled eigenvectors

[ E.32.4 ] The eigenvectors of the covariance matrix of a cov. stationary process are trigonometric waves

[ E.32.5 ] Principal factors re-indexing and rescaling

[ E.32.6 ] Spectral density of the sum of an AR(1) process and a sinusoid process

[ E.32.7 ] Symmetry of the spectral density

[ E.32.8 ] The diagonal elements of the spectral density are real-valued and positive

[ E.32.9 ] Some notes on the stochastic integral appearing in Cramer’s decomposition

[ E.32.10 ] Equivalent formulations of the Cramer’s decomposition

[ E.32.11 ] Cramer decomposition refined

[ E.32.12 ] Expectation and autocovariance functions of the filtered process

[ E.32.13 ] Conditions for the finiteness of covariance of the filtered process

[ E.32.14 ] Cross-spectral density between the two input processes

[ E.32.15 ] Impulse response function of the composition of LTI filters

[ E.32.16 ] Composition of causal LTI filters is causal

[ E.32.17 ] Autocovariance of the inverse filter

[ E.32.18 ] VAR(1) and Granger causality[???Work in progress]

[ E.32.19 ] About “perfect predictability”of the sinusoid component

[ E.32.20 ] Wold decomposition derivation (technical points)

[ E.32.21 ] Identifying the Wold coefficients (univariate case)

[ E.32.22 ] Identifying the Wold coefficients (multivariate case)

[ E.32.23 ] Endogenous Wiener-Kolmogorov: the AR(1) example

[ E.32.24 ] Endogenous Wiener-Kolmogorov filter asymptotics and truncations

[ E.32.25 ] Exogenous causal Wiener-Kolmogorov filter derivation

[ E.32.26 ] Exogenous Wiener-Kolmogorov: the bivariate VAR(1) example

[ E.32.27 ] Exogenouss Wiener-Kolmogorov filter for large prediction horizons

[ E.33 ] Common mean-covariance processes

[ E.33.1 ] Structural VAR(1)

[ E.33.2 ] Expectation and autocovariance function of the AR(1) process

[ E.33.3 ] Spectral density of AR(1) processes

[ E.33.4 ] Half-life of AR(1) process

[ E.33.5 ] Expectation and autocovariance function of the VAR(1) process

[ E.33.6 ] Bivariate VAR(1) process: autocovariance function, autocovariance function, and spectral density

[ E.33.7 ] Spectral density of VAR(1) processes

[ E.33.8 ] Non causal VAR(1)

[ E.33.9 ] Error correction representation of unit-root VAR(1) process

[ E.33.10 ] Linear prediction of cointegrated bivariate VAR(1)

[ E.33.11 ] Linear prediction of the VAR(1) process

[ E.33.12 ] Linear prediction of the VAR(1) process: random walk and cointegration limit

[ E.33.13 ] Causal/invertible VARMA(p,q) processes

[ E.33.14 ] Autocovariance and autocorrelation function and autocorrelation function of ARMA(p,q) processes

[ E.33.15 ] Autocorrelation function of an ARMA(p,q) process (complex roots)

[ E.33.16 ] Characterization of VMA(q) processes

[ E.33.17 ] Spectral density of causal ARMA(p,q) processes

[ E.33.18 ] Spectral density of causal VARMA(p,q) processes

[ E.33.19 ] Cointegrated VARMA

[ E.33.20 ] Linear prediction and the prediction error matrix of causal and invertible VARMA(p,q) processes

[ E.33.21 ] ARMA(2,1) process properties

[ E.33.22 ] VARMA as linear state-space model

[ E.33.23 ] Identification of the hidden process in linear state-space models

[ E.33.24 ] Prediction of linear state-space models

[ E.33.25 ] Derivation of the Kalman filter

[ E.33.26 ] The Kalman filter (static case)

[ E.33.27 ] Expectation and autocovariance function of a sinusoid process

[ E.33.28 ] Spectral density of the sinusoid process

[ E.33.29 ] Harmonic process as linear combination of its past

[ E.33.30 ] Equivalent formulation for the optimization behind dynamic factor models

[ E.33.31 ] Dynamic regression model

[ E.34 ] Probabilistic discrete time stochastic processes

[ E.35 ] Invariance tests

[ E.36 ] Continuous time processes

[ E.36.1 ] Characteristic function of standard Poisson process

[ E.36.2 ] The compound Poisson process is a continuous combination of Poisson processes

[ E.36.3 ] Characteristic function of continuous combination of Poisson processes

[ E.36.4 ] Characteristic function of Poisson process on grid

[ E.36.5 ] Characteristic function of compound Poisson process

[ E.36.6 ] Characteristic function of standard Brownian motion

[ E.36.7 ] Characteristic function of arithmetic Brownian motion with drift

[ E.36.8 ] The Lévy-Khintchine representation of Lévy processes

[ E.36.9 ] Δt-step location and dispersion parameters of Cauchy random walk

[ E.36.10 ] Variance gamma parametrizations

[ E.36.11 ] Relationship between CIR and Ornstein-Uhlenbeck processes

[ E.36.12 ] Projection of a Markov chain: generator

[ E.36.13 ] Fractional Brownian motion

[ E.36.14 ] Distribution of the Δt-step and of the Δt-step shock of the OU process

[ E.36.15 ] Conditional distribution and moments of OU

[ E.36.16 ] Unconditional distribution of stationary OU

[ E.36.17 ] OU process and the Brownian motion

[ E.36.18 ] VAR(1) is MVOU

[ E.36.19 ] Deterministic linear dynamic system

[ E.36.20 ] Dynamics and distribution of MVOU process Zt

[ E.36.21 ] Distribution of the Δt-step and Δt-step shock of the MVOU process

[ E.36.22 ] Conditional distribution and moments of MVOU

[ E.36.23 ] MVOU process and the Brownian motion

[ E.36.24 ] Unconditional distribution of stationary MVOU

[ E.36.25 ] MVOU (auto)covariances

E.IX. Estimation

[ E.37 ] Estimation primer

[ E.37.1 ] Minimum relative entropy as best between Gaussian versus exponential kernel

[ E.37.2 ] Glivenko-Cantelli theorem: theory

[ E.37.3 ] Generalized Glivenko-Cantelli theorem

[ E.37.4 ] Kernel with flexible probabilities mean estimation

[ E.37.5 ] Regularized pdf

[ E.37.6 ] Kernel with flexible probabilities and historical with flexible probability estimates comparison

[ E.37.7 ] Relative entropy vs. maximum likelihood

[ E.37.8 ] Conditional excess distribution

[ E.37.9 ] Equivalent formulation of conditional excess distribution

[ E.37.10 ] Generalized Pareto distribution I

[ E.37.11 ] Influence function as the limit of sensitivity curve

[ E.37.12 ] Influence function of the sample covariance: preliminary computation

[ E.37.13 ] Influence function of the sample covariance

[ E.37.14 ] Influence function of maximum likelihood estimators

[ E.37.15 ] Influence function of location and dispersion MLFP estimators for elliptical distributions

[ E.37.16 ] Influence function of ML estimators of location-dispersion under t

[ E.37.17 ] Influence function of M-estimators

[ E.37.18 ] M-estimators: location and dispersion

[ E.37.19 ] Method of moments with flexible probabilities: reflected shifted lognormal. Theory

[ E.37.20 ] Maximum likelihood estimation from generalized method of moments

[ E.37.21 ] Over-specified formulation of the generalized method of moments

[ E.37.22 ] ML-estimation of transition probability matrix for credit migration modeling

[ E.37.23 ] MLFP estimation of transition probability matrix for credit migration modeling

[ E.37.24 ] Expectation-maximization with flexible probabilities for missing values. Theory

[ E.37.25 ] MLFP estimation of GARCH(1,1) with normal innovations

[ E.37.26 ] Exponentially weighted moving average updating

[ E.37.27 ] Conditional distribution of two univariate normal invariants

[ E.37.28 ] Exponential family invariants: conjugate distribution

[ E.37.29 ] Exponential family invariants: posterior distribution

[ E.37.30 ] Exponential family invariants: predictive distribution

[ E.37.31 ] Pdf of the information set

[ E.37.32 ] Maximum likelihood parameters of multivariate normal

[ E.37.33 ] Normal distribution with fixed variance: prior, posterior, and posterior predictive

[ E.37.34 ] Posterior distribution of exponential family distributions

[ E.37.35 ] Predictive distribution of exponential family distributions

[ E.37.36 ] The EM algorithm

[ E.37.37 ] EM algorithm for i.i.d. processes

[ E.37.38 ] Maximum likelihood for longitudinal panels of data

[ E.37.39 ] Smoothing/nowcasting of the hidden variables for i.i.d. processes

[ E.37.40 ] EM algorithm for state-space processes

[ E.38 ] Estimation: location and dispersion

[ E.38.1 ] HFP covariance estimator

[ E.38.2 ] MLFP estimators for elliptical variables

[ E.38.3 ] MLFP estimators for the Student t distribution. Theory

[ E.38.4 ] Minimum volume ellipsoid

[ E.38.5 ] Minimum covariance determinant

[ E.38.6 ] Minimum volume ellipsoid and minimum covariance determinant algorithm

[ E.38.7 ] Robust estimation of the covariance matrix: rescaled HBFP ellipsoid

[ E.38.8 ] Maximum likelihood with flexible probabilities for different-length series. Theory

[ E.38.9 ] Normal-inverse-Wishart location-dispersion: posterior distribution

[ E.38.10 ] Normal-inverse-Wishart location-dispersion: predictive distribution

[ E.38.11 ] Normal-inverse-Wishart location-dispersion: mode

[ E.38.12 ] Normal-inverse-Wishart location-dispersion: modal dispersion

[ E.38.13 ] Inverse-Wishart dispersion: mode

[ E.38.14 ] Inverse-Wishart dispersion: modal dispersion

[ E.38.15 ] Normal-inverse-Wishart location-dispersion: marginal distribution of location

[ E.38.16 ] Independence of sample mean and covariance

[ E.38.17 ] Distribution of the sample mean

[ E.38.18 ] Distribution of the sample covariance

[ E.38.19 ] The Marchenko-Pastur approximation: the general case

[ E.38.20 ] Expectation shrinkage

[ E.38.21 ] Singular covariance matrix

[ E.38.22 ] Spectrum analysis

[ E.38.23 ] Distance for sparse matrix shrinkage of correlation

[ E.38.24 ] Distance for sparse matrix shrinkage of correlation: computations

[ E.38.25 ] Stein’s lemma

[ E.38.26 ] Shrinkage estimator of location

[ E.38.27 ] Shrinkage estimator of dispersion

[ E.38.28 ] Shrinkage estimator of dispersion: spectrum

[ E.38.29 ] Distance matrix for correlation clustering

[ E.38.30 ] Conditional covariance of normal variables

[ E.39 ] Estimation: regression loadings

[ E.39.1 ] Regression LFM’s: maximum likelihood with flexible probabilities estimates of factor loadings and residual covariance

[ E.39.2 ] Regression LFM’s: maximum likelihood with flexible probabilities estimates under t-conditional residuals

[ E.39.3 ] Regression LFM’s: distribution of least square estimates

[ E.39.4 ] Regression LFM’s: likelihood

[ E.39.5 ] Regression LFM’s: distribution of loadings under NIW assumption

[ E.39.6 ] Regression LFM’s: posterior distribution under NIW assumption

[ E.39.7 ] Regression LFM’s: mode of posterior under NIW

[ E.39.8 ] Regression LFM’s: modal dispersion of posterior under NIW

[ E.39.9 ] Regression LFM’s: predictive distribution

[ E.39.10 ] Lasso as a generalization of maximum likelihood with flexible probabilities

[ E.39.11 ] Inputs standardization in lasso regression

[ E.40 ] Estimation and assessment

[ E.40.1 ] The posterior risk of the relative entropy loss

[ E.40.2 ] p-value of the sample mean: normal invariants, known variance

[ E.40.3 ] Sample mean loss distribution

[ E.40.4 ] Risk, bias, inefficiency

[ E.40.5 ] Inefficiency of the sample mean

[ E.40.6 ] Posterior expectation

[ E.40.7 ] Estimation risk of the sample mean

[ E.40.8 ] Sample mean loss in a homogeneously correlated market

[ E.40.9 ] Estimation risk of the sample covariance

[ E.41 ] Bias reduction

[ E.41.1 ] Functional gradient descent

[ E.41.2 ] Gradient boosting

[ E.41.3 ] Solution of linear least-squares regression

[ E.41.4 ] Interactions as linear transformations

[ E.41.5 ] Polynomial features

[ E.41.6 ] Kernel trick

[ E.41.7 ] Equivalent expression for piecewise linear functions

[ E.42 ] Estimation and regularization

[ E.42.1 ] Regression LFM’s: equivalent formulation

[ E.42.2 ] Matrix decomposition

[ E.42.3 ] Regression LFM’s as quadratic programming

[ E.42.4 ] Cross-sectional LFM as quadratic programming

[ E.42.5 ] Feature engineering: error derivative

[ E.42.6 ] Exponential format of an arbitrary pdf

[ E.42.7 ] Dirichlet distribution

[ E.43 ] Black-Litterman

[ E.43.1 ] Black-Litterman prior distribution

[ E.43.2 ] Black-Litterman active views distribution

[ E.43.3 ] Black-Litterman posterior distribution

[ E.43.4 ] Black-Litterman: confidence level in views

[ E.44 ] Generalized probabilistic inference

[ E.44.1 ] Minimum relative entropy and exponential family

[ E.44.2 ] Distributional views updated: analytical formula

[ E.44.3 ] Point views updated: analytical formula

[ E.44.4 ] Multiplicative opinion pooling as minimum relative entropy updated

[ E.44.5 ] Minimum relative entropy and exponential family: view parameters range

[ E.44.6 ] Conditioning between normal variables

[ E.44.7 ] Gradient of relative entropy

[ E.44.8 ] Hessian of relative entropy

[ E.44.9 ] Extremeness of the views

[ E.44.10 ] Sensitivity to the views

[ E.44.11 ] Convexity of relative entropy

[ E.44.12 ] Minimum relative entropy via analytical implementation: updated distribution

[ E.44.13 ] Minimum relative entropy via analytical implementation: updated distribution via projectors

[ E.44.14 ] Minimum relative entropy via scenario-probability implementation: updated distribution

[ E.44.15 ] Minimum relative entropy with scenario-probability implementation: gradient and Hessian of dual Lagrangian

[ E.44.16 ] Minimum relative entropy with scenario-probability implementation: views on conditional value at risk

[ E.44.17 ] Partial views of the exponential family: view on standard deviation

[ E.44.18 ] Partial views of the exponential family: view on correlation

[ E.44.19 ] Gradient of relative entropy with low-rank-diagonal covariance

[ E.44.20 ] Degrees of freedom of a low-rank-diagonal matrix

[ E.44.21 ] Chain rule for second derivatives

[ E.44.22 ] Hessian of relative entropy with low-rank-diagonal covariance

[ E.44.23 ] Gradient of constraint function on signal

[ E.44.24 ] Hessian of constraint function on signal

[ E.44.25 ] Views on joint and conditional distributions

[ E.44.26 ] Views on ex-ante signal-to-noise ratios: formula

[ E.44.27 ] Copula opinion pooling - Choice of the rotation matrix

[ E.44.28 ] Distance minimum

[ E.44.29 ] Distance equivalence

[ E.44.30 ] Generalized shrinkage for covariance: sparse eigenvector rotation

[ E.44.31 ] Generalized shrinkage for correlation: homogeneous clusters

[ E.44.32 ] Generalized shrinkage for correlation: Markov networks

[ E.45 ] Inference via Monte Carlo and variational techniques

[ E.45.1 ] Conditioning as IM projection

[ E.45.2 ] The ELBO is a lower bound for the evidence of data

[ E.45.3 ] The EM algorithm in population

[ E.45.4 ] Free energy of the posterior

[ E.45.5 ] Minimum of free energy

[ E.46 ] Hypothesis testing

[ E.46.1 ] t-statistic of the sample mean: normal invariants, unknown variance

[ E.46.2 ] Univariate testing: consistency of the sample variance

[ E.46.3 ] Univariate testing: the square standard error

[ E.46.4 ] Univariate testing: the z-statistic

[ E.46.5 ] Multivariate testing: consistency of the sample covariance

[ E.46.6 ] Multivariate testing: the Hotelling statistic

Featured case studies

E.XI. Quantitative finance: the "Checklist"

[ E.47 ] Historical Checklist

[ E.48 ] Monte Carlo Checklist

E.XII. Data science: factor models and learning

[ E.49 ] Principal component analysis of the yield curve

[ E.49.1 ] Martingale property

[ E.49.2 ] Spectral basis in the continuum

[ E.49.3 ] Eigenvalues integration

[ E.50 ] Machine learning for hedging

[ E.50.1 ] Machine learning for hedging: CART predictor as a portfolio of digital options

[ E.51 ] Credit default classification

[ E.52 ] Clustering for the stock market

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