The reference library
Every idea, explained properly.
127 concepts, from probability and linear algebra through derivatives, portfolio theory, and microstructure. Each is a real treatment with derivations, worked examples, and the ways it breaks, not a one-line definition.
Mathematical Foundations
Bayes' Theorem
The rule for inverting conditional probability, prior to posterior, the base-rate trap that fools intuition, conjugate updating that makes it tractable, and the sequential/log-odds forms that turn belief revision into arithmetic.
probabilitybayesianinferenceThe Central Limit Theorem
Why sums of independent shocks become Gaussian, the classical statement and its MGF proof, the Lindeberg/Lyapunov conditions that decide when it applies, the Berry–Esseen bound on its speed, and the fat-tailed reality that makes finance its most dangerous misuse.
probabilityconvergencegaussianCommon Distributions
The core distribution family every quant carries in their head, Bernoulli through Binomial, Poisson, Normal, Exponential and Lognormal, with the limiting relationships that connect them and the moment structure that decides where each one fits.
probabilitydistributionsmodellingConvex Optimization
The class of problems where every local optimum is global, convex sets and functions, the first-order optimality condition, and Lagrangian duality with the weak/strong duality gap that underlies robust portfolio construction.
optimizationconvexitydualityEigenvalues & Eigenvectors
The invariant directions of a linear map, the spectral theorem for symmetric matrices, diagonalization as a change to the natural basis, and power iteration, the algorithm that turns eigen-structure into PCA and PageRank.
linear-algebraspectral-theorypcaExpectation, Variance & Moments
Expectation as the Lebesgue integral against a distribution, LOTUS, the algebra of variance and covariance, and the higher moments (skewness, kurtosis) that decide whether a Gaussian risk model is safe or lethal.
probabilitymomentsvarianceLagrange Multipliers & KKT Conditions
The calculus of constrained optimization, Lagrange multipliers for equalities, the full Karush–Kuhn–Tucker conditions with complementary slackness for inequalities, and their application to the mean–variance efficient frontier.
optimizationkktconstraintsThe Law of Large Numbers
Why sample averages converge to the mean, the weak vs. strong versions and their modes of convergence, proof sketches via Chebyshev, and the practical ceiling it puts on Monte Carlo accuracy and any edge-based trading.
probabilityconvergencemonte-carloLinear Algebra for Quants
The vector-space machinery that every quantitative model runs on, subspaces and rank, orthogonal projection as the geometry of regression, matrix and vector norms, and the quadratic forms that encode portfolio risk.
linear-algebrafoundationsprojectionsMarkov Chains
Memoryless stochastic processes governed by a transition matrix, the stationary distribution as a left eigenvector, the ergodic theorem that makes long-run averages predictable, and hitting-time systems that price first-passage problems.
stochastic-processeslinear-algebraprobabilityMartingales
The formal model of a fair game, conditional expectation, the optional stopping theorem and why "no betting system beats a fair game", and the martingale view of arbitrage-free pricing.
stochastic-processesprobabilitypricingMoment Generating Functions
The transform $M_X(t) = \mathbb{E}[e^{tX}]$ that encodes every moment in one function, with its uniqueness theorem, the sums-of-independents multiplication rule, and cumulants, the additive cousins that linearise skewness and kurtosis.
probabilitymomentstransformsPositive Semidefinite Matrices
The matrices that make quadratic forms non-negative, equivalent tests via eigenvalues, pivots and principal minors, the Cholesky factorization that generates correlated risk, and why every valid covariance matrix must be PSD.
linear-algebracovariancecholeskyProbability Spaces
The measure-theoretic triple $(\Omega, \mathcal{F}, \mathbb{P})$ that makes probability rigorous, why we need σ-algebras, what Kolmogorov's axioms buy us, and where naive "probability = favourable/total" quietly breaks.
measure-theoryprobabilityfoundationsRandom Variables & Distributions
Random variables as measurable functions that push a probability measure forward into a distribution, CDFs, densities, the change-of-variables formula for transformations, and the joint/marginal/conditional structure that underlies every multivariate model.
probabilitydistributionsrandom-variablesSingular Value Decomposition
The universal factorization $A = U\Sigma V^\top$ that works for any matrix, its geometry as rotate-stretch-rotate, the Eckart–Young optimal low-rank approximation behind PCA and denoising, and the pseudo-inverse that solves least squares even when nothing is invertible.
linear-algebrasvdlow-rank
Statistics & Econometrics
ARMA Models
The linear building blocks of time-series forecasting, AR, MA, and ARMA processes, their ACF/PACF signatures, the stationarity and invertibility conditions via the lag polynomial, and why returns are nearly unforecastable by them.
time-seriesforecastingarmaAutocorrelation and Serial Correlation
When regression errors are correlated across time, why it inflates or deflates naive standard errors, the Newey–West HAC estimator, the Durbin–Watson statistic, and the overlapping-returns trap that fools every backtester.
time-serieseconometricshacBayesian Inference
Treating parameters as random and updating beliefs with data, priors, posteriors, conjugacy, credible intervals, and the deep result that shrinkage and regularization are Bayesian estimation in disguise.
statisticsbayesianshrinkageBootstrap and Resampling
Estimating the sampling distribution of any statistic by resampling the data itself, the plug-in principle, why it works, the block bootstrap that preserves time-series dependence, and how to build confidence intervals for things with no formula.
statisticsresamplingbootstrapCointegration
When two non-stationary series share a common stochastic trend so a linear combination is stationary, the Engle–Granger and Johansen tests, the error-correction representation, and the econometric foundation of pairs trading.
time-seriescointegrationpairs-tradingConfidence Intervals
What a confidence interval actually claims (and what it does not), how to construct one by inverting a test, its exact duality with hypothesis testing, and why the honest interval on a Sharpe ratio is uncomfortably wide.
statisticsinferenceuncertaintyEndogeneity and Instrumental Variables
The deepest OLS failure, a regressor correlated with the error, which biases and un-fixes the estimator no matter the sample size, and the instrumental-variables / 2SLS machinery that recovers a causal effect.
econometricscausalityinstrumentsBias, Variance, and the Quality of Estimators
What makes one estimator better than another, the bias–variance decomposition of MSE, consistency, efficiency, and the Cramér–Rao bound that caps how good any unbiased estimator can be.
estimationstatisticsmseGARCH Volatility Models
How to model the one robust feature of returns, volatility clustering, from ARCH to GARCH, the persistence and stationarity conditions, forecasting the variance term structure, and the leverage and fat-tail extensions desks actually use.
time-seriesvolatilitygarchThe Gauss–Markov Theorem
The precise statement that OLS is BLUE, Best Linear Unbiased Estimator, with the assumptions it needs, a full proof that any other linear unbiased estimator has larger variance, and exactly what the theorem does and does not promise.
regressioneconometricsolsHeteroskedasticity
When error variance is not constant, why it leaves OLS unbiased but wrecks its standard errors, the White/robust sandwich covariance that fixes inference, and the tests that detect it.
regressioneconometricsrobust-seHypothesis Testing
The Neyman–Pearson framework for deciding between hypotheses under uncertainty, type I and II errors, power, the most-powerful-test lemma, and the likelihood-ratio test that generalizes it.
statisticsinferencetestingMaximum Likelihood Estimation (MLE)
The estimator that asks "what parameters make the data I saw most probable", the score equation, Fisher information, the invariance property, and the asymptotic normality that makes MLE the default engine of parametric inference.
estimationstatisticslikelihoodMethod of Moments and GMM
Estimation by matching sample moments to their theoretical counterparts, the classical method of moments, its generalization to more moment conditions than parameters (GMM), and why GMM underlies asset-pricing tests.
estimationgmmeconometricsMulticollinearity
When regressors are highly correlated, how a near-singular X'X inflates coefficient variances, the variance inflation factor that quantifies it, why signs flip and estimates go unstable, and the shrinkage cure.
regressioneconometricscollinearityOrdinary Least Squares (OLS)
The workhorse linear estimator, derived in matrix form, with its geometry, the Gauss-Markov optimality result, its sampling distribution, and the assumptions that break in financial data.
regressioneconometricsstatisticsp-values and Multiple Testing
What a p-value is and the five things it is not, why testing thousands of signals guarantees false discoveries, and the Bonferroni and Benjamini–Hochberg corrections that keep a backtesting pipeline honest.
statisticsmultiple-testingfdrRidge and LASSO Regularization
Deliberately biasing a regression to cut its variance, the ridge and LASSO penalties, the L2-versus-L1 geometry that makes LASSO select and ridge shrink, and why penalization is essential in the low-signal, high-collinearity world of finance.
regressionregularizationshrinkageStationarity
The property that makes a time series learnable, strict versus weak (covariance) stationarity, why every estimator implicitly assumes it, and how transforming prices into returns is really a hunt for a stationary series.
time-serieseconometricsstationarityUnit Roots and the ADF Test
The random walk and the difference between a stochastic and deterministic trend, the augmented Dickey–Fuller test, its non-standard null distribution, and the spurious-regression trap that manufactures fake relationships between unrelated prices.
time-seriesunit-rootadf
Portfolio Construction & Risk
The Black-Litterman Model
The Bayesian fix for unstable mean-variance portfolios, reverse-optimize the market to get equilibrium returns as a prior, express subjective views with confidences, and blend them into posterior expected returns via the master formula.
portfolio-optimizationbayesianasset-allocationThe Capital Asset Pricing Model (CAPM)
The equilibrium that prices every asset by its beta to the market, derived from the tangency portfolio, giving the security market line, plus the assumptions it rests on and the anomalies that broke it.
asset-pricingbetaequilibriumCoherent Risk Measures
The Artzner et al. axioms a sensible risk measure must satisfy, monotonicity, translation invariance, positive homogeneity, and subadditivity, a proof that VaR violates subadditivity, and why expected shortfall satisfies all four.
riskaxiomstail-riskCovariance Matrix Estimation
How to estimate the covariance matrix that every portfolio optimizer inverts, the sample estimator and its bias, the curse of dimensionality when N approaches T, eigenvalue spreading and ill-conditioning, and why the naive estimate is dangerous to invert.
covarianceestimationrandom-matrix-theoryThe Efficient Frontier
The set of minimum-variance portfolios for each level of return, derived in closed form with Lagrange multipliers, giving the two-fund theorem, the global minimum-variance portfolio, the tangency portfolio, and the capital market line.
portfolio-optimizationmean-variancediversificationEqual Weighted vs Market Cap Weighted
Two common portfolio weighting schemes; equal weight often tilts toward smaller names and can improve diversification.
portfolio-optimizationweightingdiversificationExpected Shortfall (CVaR)
The average loss in the tail beyond VaR, its definition as a conditional tail expectation, why it fixes VaR's blindness to tail depth and its lack of subadditivity, the Rockafellar-Uryasev convex formulation, and its adoption by Basel.
risktail-riskcvarFactor Risk Models
Decomposing asset risk into a few systematic factors plus idiosyncratic noise, the Barra-style structure, why it cuts the covariance matrix from O(N squared) to O(NK) parameters, factor-covariance and specific-risk estimation, and risk attribution.
factor-modelsriskcovarianceHierarchical Risk Parity (López de Prado)
Portfolio construction that uses a hierarchical clustering of assets to allocate risk more evenly and robustly.
portfolio-optimizationrisk-parityclusteringThe Kelly Criterion
The bet size that maximizes long-run geometric growth, derived from log-utility, extended to continuous and multi-asset cases, with the estimation-error and drawdown reasons practitioners bet a fraction of it.
position-sizingriskprobabilityLedoit-Wolf Covariance Shrinkage
The optimal convex combination of the noisy sample covariance and a structured target, the shrinkage target, the closed-form optimal intensity that minimizes expected Frobenius loss, and why it dominates the sample matrix out of sample.
covarianceshrinkageestimationPitfalls of Mean-Variance Optimization
Why the textbook optimizer is an "error-maximizing machine", the mathematics of weight instability, extreme sensitivity to expected-return estimates, corner solutions, and the shrinkage, resampling, and constraint fixes practitioners actually use.
portfolio-optimizationestimation-errorrobustnessMPT (Harry Markowitz)
Mean–variance portfolio theory: optimal diversification by balancing expected return and variance.
portfolio-optimizationmean-variancediversificationPortfolio Capacity
The AUM ceiling beyond which a strategy stops making money, how market impact and turnover erode alpha, the square-root impact law, the capacity formula that balances gross alpha against trading costs, and why fast strategies capacity-cap first.
capacitymarket-impacttransaction-costsPosition Sizing
Rules for how much capital to allocate per trade or asset (equal weight, risk parity, Kelly, etc.).
riskportfolio-constructionportfolio-optimizationRisk Parity
Allocating so that every asset contributes equal risk rather than equal capital, derived from Euler's decomposition of portfolio volatility, the equal-risk-contribution condition, the role of leverage, and the comparison to 60/40.
portfolio-optimizationrisk-parityrisk-budgetingValue at Risk (VaR)
The loss quantile that dominates risk reporting and Basel capital, its definition, the three estimation methods (historical, parametric, Monte Carlo), how to backtest it with the Kupiec and Christoffersen tests, and its fatal flaws.
riskvartail-risk
Risk & Performance Ratios
Alpha (α)
Excess return over a benchmark after accounting for beta; a measure of skill or edge in backtesting.
risk-ratiosbacktestingfactorBeta (β)
Sensitivity of strategy returns to a benchmark; measures systematic (market) risk in backtesting.
risk-ratiosbacktestingfactorCalmar Ratio
Annualized return divided by maximum drawdown; emphasizes drawdown risk in backtesting.
risk-ratiosbacktestingdrawdownInformation Ratio
Active return per unit of tracking error; measures excess performance vs a benchmark in backtesting.
risk-ratiosbacktestingactive-managementMax Drawdown
Largest peak-to-trough decline in cumulative equity; a key risk metric for backtesting and live performance.
risk-ratiosbacktestingdrawdownSharpe Ratio
Return per unit of total risk (volatility); the standard risk-adjusted performance metric for backtesting.
risk-ratiosbacktestingperformanceSortino Ratio
Return per unit of downside volatility; penalizes only bad volatility in backtesting.
risk-ratiosbacktestingdownside-riskVolatility
Standard deviation of returns; the standard measure of dispersion and risk in backtesting.
risk-ratiosbacktestingrisk
Derivatives & Volatility
The Black-Scholes Model
The founding model of option pricing, the PDE derived two ways (delta-hedging and risk-neutral expectation), the closed-form call price with N(d₁) and N(d₂), the meaning of those two terms, and the assumptions that the volatility smile later broke.
optionspricingpdeBrownian Motion
The Wiener process, the continuous-time random walk that drives every diffusion model in finance, defined by its four axioms, with its quadratic variation, martingale and scaling properties, and the nowhere-differentiability that forces us into Itô calculus.
stochastic-calculusprobabilitydiffusionDelta-Hedging P&L
The fundamental P&L equation of a delta-hedged option, why a hedged position earns gamma times the difference between realized and implied variance, derived step by step, with the discrete-hedging error and its variance.
optionshedginggammaExotic Options
Options beyond the vanilla payoff, barriers, Asians, digitals, and lookbacks, the closed forms that exist (reflection principle, digital = call spread) and the path dependence that forces Monte Carlo or PDE methods, plus the discontinuous Greeks that make them hard to hedge.
optionsexoticsbarriersThe Feynman-Kac Theorem
The bridge between PDEs and expectations, why the solution of a parabolic PDE equals a discounted expectation of a diffusion's terminal payoff, derived by making a discounted process a martingale, and why it makes the Black-Scholes PDE and the pricing integral the same thing.
stochastic-calculuspdeexpectationGirsanov's Theorem
The change-of-measure engine behind risk-neutral pricing, how the Radon-Nikodym derivative reweights probabilities to remove a drift, why volatility is invariant, the Novikov condition, and how choosing the market price of risk turns P into Q.
stochastic-calculuschange-of-measureradon-nikodymImplied Volatility
The volatility that makes Black-Scholes match a market price, why the inversion is unique, how it is computed, the ATM approximation, and the crucial distinction between implied, realized, and the variance risk premium between them.
optionsvolatilityimplied-volItô's Lemma
The chain rule of stochastic calculus, how a smooth function of a diffusion evolves, why the second-order (dW)²=dt term survives, derived from a Taylor expansion, and applied to geometric Brownian motion to get the lognormal solution.
stochastic-calculusdiffusionsdeLocal Volatility and Dupire's Formula
The unique deterministic volatility function that reprices the entire option surface, derived via Breeden-Litzenberger and Dupire's forward equation, why it fits every vanilla exactly, and why its wrong smile dynamics make it misprice exotics.
volatilitylocal-voldupireThe Option Greeks
The sensitivities of an option's value, delta, gamma, vega, theta, rho and the key second-order Greeks, with their Black-Scholes formulas, sign intuition, and the gamma-theta tradeoff that the pricing PDE encodes.
optionsgreekshedgingPut-Call Parity
The model-free arbitrage relationship linking a European call, a put, the stock, and a bond, derived purely from replication with no distributional assumptions, plus its consequences for synthetic positions and implied-vol consistency.
optionsarbitrageno-arbitrageRisk-Neutral Pricing
The fundamental theorem of asset pricing, why no-arbitrage is equivalent to the existence of an equivalent martingale measure, why under it every asset drifts at the risk-free rate, and why prices are discounted expectations of payoffs.
arbitragemartingale-measurepricingThe SABR Model
The stochastic-alpha-beta-rho model that dominates rates and FX smile-fitting, its CEV-plus-lognormal-vol SDEs, the meaning of β, ρ and ν, Hagan's implied-vol expansion, and why its closed-form smile made it the market standard for interpolation.
volatilitysabrsmileStochastic Volatility and the Heston Model
Making volatility itself random, the Heston SDEs with mean-reverting CIR variance, the roles of vol-of-vol and correlation in shaping the smile, the semi-closed-form characteristic-function solution, and why stochastic vol beats local vol on dynamics.
volatilitystochastic-volhestonThe Term Structure of Volatility
Implied volatility as a function of maturity, contango versus backwardation, the variance-additivity that defines forward volatility, the no-calendar-arbitrage constraint, and how calendar spreads trade the slope.
volatilityterm-structureforward-volVariance Swaps
The clean instrument for trading realized variance, its payoff, the log-contract derivation that shows realized variance replicates via a static strip of options weighted 1/K², and the model-free fair strike that underlies the VIX.
volatilityvariance-swapreplicationThe VIX Index
The market's fear gauge as model-free implied variance, the CBOE replication formula that is a discretized variance swap, why it is a 30-day risk-neutral vol expectation, and the futures term structure and roll that make VIX products behave.
volatilityvixvarianceThe Volatility Smile and Skew
Why implied volatility varies by strike, the direct contradiction of Black-Scholes, the equity skew versus the FX smile, the risk-reversal and butterfly that parametrize it, and the fat-tail and leverage stories that generate it.
optionsvolatilitysmile
Systematic Strategies & Alpha
Alpha Decay
Why signals lose their edge over time, the half-life of predictive power, post-publication crowding, and the turnover-versus-decay tradeoff that determines how fast you must trade a signal before transaction costs eat it.
alpha-decaycrowdingturnoverCarry
The return you earn if prices do not move, defined consistently across currencies, rates, and commodities, why it is best understood as a risk premium for bearing crash and liquidity risk, and how the carry trade blows up.
carryrisk-premiafxCross-Sectional vs. Time-Series Strategies
The two fundamental ways to build a systematic strategy, ranking assets against each other (cross-sectional) versus each asset against its own history (time-series), how demeaning determines market-neutrality, and the algebraic identity linking them.
cross-sectionaltime-seriesmarket-neutralFactor Investing
The idea that expected returns are earned by exposure to a small set of systematic factors, the distinction between risk premia and anomalies, how factor portfolios are constructed, and the "factor zoo" multiple-testing critique that haunts the field.
factorsrisk-premiacross-sectionalThe Fama-French Factor Models
The three- and five-factor models that replaced the CAPM as the empirical benchmark, the construction of SMB, HML, RMW, and CMA, the time-series regressions used to test them, and the GRS test for whether alphas are jointly zero.
factorsasset-pricingfama-frenchThe Low-Volatility Anomaly
The empirical fact that low-risk stocks earn higher risk-adjusted returns than high-risk stocks, a direct contradiction of the CAPM, and the leverage-constraint (betting-against-beta) explanation that resolves it.
low-volatilityfactorsanomalyMarket Efficiency (The EMH)
The efficient-market hypothesis in its weak, semi-strong, and strong forms, why it is untestable in isolation (the joint-hypothesis problem), why it cannot be literally true (Grossman-Stiglitz), and what "efficiency" actually means for a systematic trader.
market-efficiencyasset-pricinganomaliesPairs Trading
The original convergence trade, find two cointegrated assets, trade the mean-reverting spread with z-score entry and exit rules, size by the half-life of reversion, and understand why the relationship eventually breaks.
pairs-tradingcointegrationmean-reversionThe Quality Factor
Paying up for good businesses, profitable, growing, safe, well-managed firms, and why they earn higher returns than junk despite being "better." The QMJ framework, its link to the dividend-discount identity, and its role as the complement to value.
qualityfactorsprofitabilitySignal Construction
Turning a raw predictor into a tradeable signal, winsorizing, z-scoring and ranking, neutralizing unwanted exposures, and measuring its quality with the information coefficient and the fundamental law of active management, IR = IC·√breadth.
signalsinformation-coefficientportfolio-constructionStatistical Arbitrage
Cross-sectional mean reversion at scale, strip out common factors with PCA, model the idiosyncratic residual as a mean-reverting process, and trade its s-score across hundreds of names. The Avellaneda-Lee framework and why breadth is the whole game.
statistical-arbitragemean-reversionpcaTrend Following
Time-series momentum traded across dozens of futures markets, the CTA strategy that buys what's going up and sells what's going down, why trends persist, and the long-volatility convexity that makes it "crisis alpha."
trend-followingtime-series-momentumctaThe Value Factor
Buying cheap and selling expensive stocks, the book-to-market signal, the risk-based versus behavioral explanations for why value earns a premium, and the anatomy of value's decade-long drawdown from 2007 to 2020.
valuefactorshml
Backtesting & ML in Finance
Backtest Design
What separates a historical simulation you can trust from a curve-fit, point-in-time data, realistic costs and slippage, and the crucial distinction between a backtest and a forecast.
backtestingsimulationevaluationBacktest Overfitting
The probability that the best-performing backtest is in-sample-optimal but out-of-sample worthless, measured by the PBO via combinatorially-symmetric cross-validation, and bounded by the minimum backtest length.
backtestingoverfittingmultiple-testingCross-Validation for Time Series
Why ordinary k-fold cross-validation is invalid on financial time series, the iid assumption fails, autocorrelation and overlapping labels leak across folds, and shuffling trains on the future.
machine-learningcross-validationbacktestingData-Snooping Bias
The multiple-testing problem in strategy research, why repeatedly testing ideas on the same data guarantees false discoveries, and how family-wise error and false-discovery-rate corrections raise the bar.
backtestingmultiple-testingstatisticsThe Deflated Sharpe Ratio
Bailey & López de Prado's correction that adjusts an observed Sharpe ratio for the number of trials, non-normal returns, and sample length, deflating it against the expected maximum Sharpe achievable by luck alone.
backtestingmultiple-testingsharpe-ratioFeature Engineering for Finance
Building predictive features from financial data without destroying signal, the stationarity-versus-memory tradeoff, fractional differentiation, and detecting structural breaks.
machine-learningfeature-engineeringtime-seriesLook-Ahead Bias
The most common way backtests lie, leaking information into the simulation that was not knowable at decision time, through restated fundamentals, same-bar execution, or full-sample preprocessing.
backtestingbiasdata-integrityPurged & Embargoed Cross-Validation
López de Prado's fix for validating models on labeled financial data, purge training observations whose label windows overlap the test set, then embargo a buffer after it to kill serial-correlation leakage.
machine-learningcross-validationbacktestingSurvivorship Bias
The upward distortion in backtests that use only the securities or funds that survived to today, quantifying the bias, why delisted names matter, and how point-in-time universes fix it.
backtestingbiasdata-integrityTree Ensembles in Finance
Random forests and gradient boosting on financial data, why their flexibility overfits low-signal, non-stationary markets, and why their feature-importance measures mislead.
machine-learningtree-ensemblesfeature-importanceTriple-Barrier Labeling
López de Prado's method for turning price paths into supervised-learning labels, a profit-taking barrier, a stop-loss barrier, and a time barrier, plus meta-labeling to separate the side of a bet from its size.
machine-learninglabelingbacktesting
Market Microstructure & Execution
Adverse Selection
The market maker's winner's curse, because informed traders trade only when they have an edge, the flow that hits your quotes is systematically toxic, and the spread is the price you charge to survive it.
microstructuremarket-makinginformationThe Almgren-Chriss Model
The foundational optimal-execution framework, trade a position off to minimize expected impact cost plus a risk penalty on the leftover exposure, yielding a closed-form hyperbolic-sine trajectory and the efficient frontier of execution.
microstructureexecutionoptimizationThe Avellaneda-Stoikov Model
The canonical high-frequency market-making model, an inventory-averse quoter derives a reservation price that skews with position, then sets optimal bid/ask offsets balancing spread capture against inventory risk, with a closed-form optimal spread.
microstructuremarket-makinginventoryBid-Ask Spread Decomposition
The spread is compensation for three distinct costs, order processing, inventory, and adverse selection, plus the Roll model that recovers the effective spread from the autocovariance of transaction prices alone.
microstructurespreadmarket-makingThe Glosten-Milgrom Model
The canonical model of how a spread arises from pure adverse selection, competitive quotes set as Bayesian conditional expectations, updated trade by trade, with the bid and ask derived explicitly.
microstructuremarket-makingadverse-selectionThe Kyle Model
The strategic-informed-trader model, a single insider hides in noise-trader flow, the market maker prices linearly off aggregate order flow, and the equilibrium delivers Kyle's lambda, the price-impact coefficient λ = σ_v / (2σ_u).
microstructuremarket-impactinformationLatency & High-Frequency Trading
Why microseconds are worth millions, co-location and the speed race, queue position as the prize, latency arbitrage and the sniping of stale quotes, and the Budish-Cramton-Shim argument that the continuous limit order book manufactures an arms race by design.
microstructurehftlatencyMarket Impact
Why your own trading moves the price against you, the split into permanent (information) and temporary (liquidity) impact, the propagator that ties them together, and why impact is the dominant cost for institutional size.
microstructureexecutionimpactMarket vs. Limit Orders
The fundamental execution tradeoff, pay the spread for certainty with a market order, or post a limit order that is a free option you write to the market, bearing non-execution and adverse-selection risk.
microstructureexecutionorder-typesOptimal Execution
The trader's problem of converting a decision into fills at least cost, implementation shortfall as the objective, the impact-versus-timing-risk dilemma, and the extensions (adaptive, transient-impact, multi-asset) beyond Almgren-Chriss.
microstructureexecutionimplementation-shortfallOrder Book Mechanics
How a limit order book actually works, the two-sided queue of resting orders, price-time priority, matching, and why your queue position is an asset with real option value.
microstructurelimit-order-bookexecutionThe Square-Root Impact Law
The strikingly universal empirical finding that a metaorder's price impact grows as the square root of its size relative to volume, its functional form, the propagator and latent-liquidity arguments for why, and how to use and abuse it.
microstructureimpactexecutionTWAP, VWAP & POV
The three benchmark execution algorithms, time-weighted, volume-weighted, and percentage-of-volume, their scheduling math, when each is appropriate, and how each can be gamed.
microstructureexecutionalgos
Other concepts
Correlation Clustering
Grouping assets or signals by correlation structure to manage diversification and risk.
riskdiversificationclusteringDrawdown
The peak-to-trough decline in portfolio value; key risk metric for live and backtest.
riskevaluationmonitoringMean Reversion
The tendency of prices or returns to revert to a long-run average after deviations.
signalsequitystat-arbMomentum
The tendency of assets that have performed well (poorly) to continue performing well (poorly) over the next period.
signalstrendcross-sectionalOverfitting
Fitting a model or strategy so closely to historical data that it fails out-of-sample.
backtestingevaluationmachine-learningPCA (Principal Component Analysis)
Dimensionality reduction by projecting data onto orthogonal directions of maximum variance.
factor-modelsdimensionality-reductionriskRegime Detection
Identifying and adapting to different market states (trending vs mean-reverting, risk-on vs risk-off).
risksignalsadaptationShrinkage
Pulling estimates (e.g. mean, covariance) toward a prior or global average to reduce estimation error.
estimationcovarianceportfolio-constructionTransaction Costs
Slippage, spread, commission, and market impact that reduce strategy returns in practice.
implementationbacktestingliquidityVol Targeting
Scaling portfolio exposure so that realized volatility stays near a target (e.g. 10% annual).
riskposition-sizingleverageWalk-Forward Analysis
Out-of-sample testing by rolling a training window forward and evaluating on the next period repeatedly.
backtestingoverfittingevaluation