A curriculum in 9 volumes

From Calculus
to Kelly

A quantitative curriculum that starts from first-year calculus, statistics and linear algebra, and ends with statistical models, optimized portfolios and growth-optimal leverage you can build and defend yourself.

從大一的微積分、統計學與線性代數開始，一路推進到自建統計模型、投資組合統計最佳化、計算真正的風險報酬，以及在估計誤差下求最適凱利槓桿。內文以英文為主，難懂之處附中文導讀。

Scope 9 volumes · 50 learning modules Published 7 of 50 Data Taiwan & US markets Edition 0.2 License MIT

Start reading: LM1 Exponentials, Logarithms and Compounding →

What you will be able to do

Vol III–V

Build statistical models

Return distributions, regressions and factor models, GARCH volatility, cointegrated spreads — estimated, diagnosed and stated with their uncertainty.

Vol II, VI

Optimize portfolios under estimation error

Mean–variance, risk parity and Kelly portfolios with realistic constraints, shrinkage estimators, and walk-forward evaluation.

Vol IV, VII

Measure true risk and return

Geometric growth instead of arithmetic averages, confidence intervals for Sharpe ratios, Deflated Sharpe, costs and biases removed.

Vol V, VIII

Size positions with Kelly leverage

From the binary bet to multi-asset leverage, with fat tails, parameter uncertainty, margin rules and drawdown limits.

Roadmap

Each volume hands specific tools to the volumes after it. The arrows name what is handed over. Everything converges on Volume VIII, where Kelly leverage needs stochastic calculus for the growth rate, portfolio theory for the multi-asset case, and honest risk statistics for the inputs.

Prerequisite map of the nine volumes. Select a volume to jump to its modules.

How each module works

Every module follows the same architecture, built for self-study with code.

Learning Outcomes with mastery boxes you can tick; your ticks are remembered in this browser.
Learning Module Overview: the main results in a few lines.
Motivating Case: a real market situation the module resolves, with verified data.
Numbered sections, each mapped to an outcome: intuition, definition, derivation, exhibit, worked example, pitfall.
Knowledge Checks inside sections; Practice Problems and full Solutions at the end.
Glossary (English–中文) and References; a companion Jupyter lab reproduces every number.

DefinitionThe precise statement of a term, in English.

ExampleA worked problem with its full solution.

PitfallA common mistake and how to avoid it.

導讀中文導讀：用中文把最難的直覺講清楚。

Deep DiveOptional proofs and rigorous detail, collapsed by default.

Knowledge CheckA short question with a hidden answer.

ExhibitTables and charts, numbered within each module; charts show values on hover.

Notation and conventions

The same symbols mean the same thing in every module. Every approximation is marked and comes with the order of what it neglects; derivations are given in full or the result is labelled as stated without proof, with a reference.

$R_{t} = \frac{P _{t} + D _{t}}{P _{t - 1}} - 1$	simple (holding-period) return; adds across assets
$r_{t} = ln (1 + R_{t})$	log return; adds across time
$μ = E [R], σ^{2} = Var (R)$	mean and variance of simple returns
$g = E [ln (1 + R)]$	log growth rate per period (per year once annualized); what compounding delivers
$G = W_{T} / W_{0}$	growth factor over a holding period
$A$	periods per year used to annualize, always with its sample (243.1 sessions a year in the book's 0050 sample)
$f, f^{*}$	exposure, position value divided by equity (above 1 it is leverage); its growth-maximizing (Kelly) value
$SR = (μ - r_{f}) / σ$	Sharpe ratio of a return, per day or annualized (say which)
$φ, Φ$	standard normal density and distribution function
$B_{t}$	standard Brownian motion (W stays wealth)
$=, \approx, :=$	exact; approximate with the neglected order stated; defined as

A module is published after an independent review recomputes its numbers and checks its mathematics; until then it is listed as in review.

Volume ICalculus for Finance

Compounding, approximation and optimization — the language every later volume is written in.

LM	Module	Status
LM1	Exponentials, Logarithms and Compounding Discrete vs continuous compounding; simple vs log returns; why logs make time additive.	Available PDF
LM2	Limits and Derivatives Marginal thinking, the chain rule, elasticity; duration as a derivative.	Available PDF
LM3	Taylor Series and Approximation Local polynomial models, error bounds, volatility drag, leveraged-ETF decay, duration–convexity.	Available PDF
LM4	Optimization in One Variable First- and second-order conditions, concavity, the binary Kelly bet, Newton's method.	Available PDF
LM5	Integration Areas, accumulation and expectation as an integral; the Gaussian integral.	Available PDF
LM6	Sequences, Series and the Time Value of Money Geometric series, annuities, perpetuities, bond and dividend-discount pricing.	Available PDF
LM7	Multivariable Calculus and Lagrange Multipliers Gradients, Hessians, multivariate Taylor, constrained optimization; the two-asset minimum-variance portfolio.	Available PDF

Volume IILinear Algebra for Portfolios

Portfolios are vectors and risk is a quadratic form; this volume makes that literal.

LM	Module	Status
LM8	Vectors and Matrices Weights, returns and the dot product; portfolio return as wᵀr.	Planned
LM9	Linear Systems, Inverses and Rank Solving for weights; singular matrices as perfect collinearity.	Planned
LM10	Quadratic Forms and Covariance Matrices wᵀΣw, positive (semi)definiteness, matrix calculus.	Planned
LM11	Eigen-decomposition and PCA Eigen-portfolios of Taiwan sectors; condition numbers and why Σ⁻¹ amplifies error.	Planned
LM12	Cholesky, SVD and Least Squares Simulating correlated returns; OLS as a projection.	Planned

Volume IIIProbability

Random variables as functions, expectation as the workhorse, and the distributions markets actually produce.

LM	Module	Status
LM13	Probability Foundations and Bayes Sample spaces, conditional probability, Bayes' rule; a random variable is a function.	Planned
LM14	Random Variables, Expectation and Jensen's Inequality LOTUS, moments, E[R²] = σ² + μ², and the real reason for volatility drag.	Planned
LM15	Distributions in Finance Normal, lognormal, Student-t and Poisson, fitted to TAIEX returns.	Planned
LM16	Joint Distributions and Dependence Covariance, conditional expectation, the multivariate normal, crash correlation.	Planned
LM17	Law of Large Numbers and Central Limit Theorem Why the median path, not the mean path, is what you live through.	Planned
LM18	Monte Carlo Simulation and the Bootstrap Random generation, simulation error, variance reduction, resampling.	Planned

Volume IVStatistical Inference and Econometrics

Turning a return series into estimates you can defend, with their uncertainty attached.

LM	Module	Status
LM19	Descriptive Statistics of Returns Location, dispersion, shape; the √T rule and when it fails.	Planned
LM20	Estimation Theory and Maximum Likelihood Bias, variance, MSE; why SE(μ̂) = σ/√T ignores sampling frequency.	Planned
LM21	Hypothesis Testing and Multiple Testing p-values, power, Bonferroni and FDR; the t > 3 hurdle for strategies.	Planned
LM22	Resampling Methods Block and stationary bootstrap; confidence intervals for Sharpe and Kelly.	Planned
LM23	Linear Regression From simple to multiple OLS in matrix form; HAC errors; CAPM beta and factor models.	Planned
LM24	Bayesian Inference and Shrinkage Priors, posteriors, James–Stein; fractional Kelly as a Bayesian answer.	Planned
LM25	Time Series I: ARMA, Unit Roots and Cointegration Stationarity, mean reversion, the futures–spot basis.	Planned
LM26	Time Series II: Volatility Models EWMA, GARCH, and realized volatility from 5-second TAIEX data.	Planned

Volume VStochastic Calculus

The continuous-time tools behind GBM, Itô's lemma and the continuous Kelly result.

LM	Module	Status
LM27	From Random Walk to Brownian Motion Scaling limits, quadratic variation, and what (dB)² = dt really means.	Planned
LM28	The Itô Integral and Itô's Lemma Itô's lemma derived as a Taylor expansion that keeps its second-order term.	Planned
LM29	Geometric Brownian Motion Solving the SDE; lognormal prices; mean, median and mode.	Planned
LM30	Beyond GBM Ornstein–Uhlenbeck spreads, Merton jumps for Taiwan gap days, stochastic volatility.	Planned
LM31	Option Pricing Bridge (Optional) Black–Scholes by delta hedging; Greeks; TXO and warrants.	Planned

Volume VIPortfolio Optimization

Mean–variance theory, the convex machinery to solve it, and the estimation error that breaks it.

LM	Module	Status
LM32	Portfolio Return and Risk N-asset portfolio mathematics in matrix form.	Planned
LM33	Mean–Variance Optimization Lagrangian derivation, global minimum variance, the frontier, tangency and the CML.	Planned
LM34	Convex Optimization in Practice Quadratic programs, KKT conditions, cvxpy; long-only, leverage and turnover limits.	Planned
LM35	Estimation Error and Robust Portfolios The 1/N puzzle, Ledoit–Wolf shrinkage, Black–Litterman, resampling.	Planned
LM36	Risk Parity, HRP and Volatility Targeting Euler risk contributions and risk-based allocation.	Planned

Volume VIIMeasuring True Risk and Return

What a track record really says once compounding, fat tails, costs and selection are accounted for.

LM	Module	Status
LM37	VaR, Expected Shortfall and Drawdown Parametric, historical and Monte Carlo VaR; ES; drawdown statistics.	Planned
LM38	The Statistics of the Sharpe Ratio Standard errors (Lo 2002), Probabilistic and Deflated Sharpe, minimum track record.	Planned
LM39	Backtesting Without Fooling Yourself Survivorship and look-ahead bias, Taiwan taxes and fees, walk-forward, PBO.	Planned
LM40	Stress Testing Taiwan crash scenarios (2008, 2015, 2020, 2024/8/5, 2025/4/7) and overnight gaps.	Planned

Volume VIIIThe Kelly Criterion and Optimal Leverage

Growth-optimal sizing from the binary bet to multi-asset leverage under uncertainty.

LM	Module	Status
LM41	Discrete Kelly Binary and general discrete bets; position size vs risk size; sensitivity to p.	Planned
LM42	Continuous-Time Kelly g(f) by Itô, f* = SR/σ, g* = r + SR²/2, and the 2f* ruin line.	Planned
LM43	Multi-Asset Kelly f* = Σ⁻¹(μ − r): the tangency portfolio, levered.	Planned
LM44	Fractional Kelly and Drawdown Control CRRA equivalence, drawdown probabilities, risk-constrained Kelly.	Planned
LM45	Kelly Under Uncertainty and Frictions Parameter uncertainty, fat tails and jumps, rebalancing, financing costs, bootstrap Kelly.	Planned
LM46	Leverage Instruments in Practice TX, MTX and TMF futures; 00631L and TQQQ; margin regimes.	Planned

Volume IXCapstone

Everything assembled into working research pipelines on Taiwan and US data.

LM	Module	Status
LM47	The Statistical Modeling Workflow Data, model, diagnostics, decision — end to end.	Planned
LM48	Capstone A: A Taiwan Core–Satellite Portfolio Mean–variance vs risk parity vs Kelly, compared walk-forward.	Planned
LM49	Capstone B: Kelly Sizing a Single Strategy From backtest returns to fractional Kelly on TX futures, uncertainty included.	Planned
LM50	Capstone C: Multi-Strategy Allocation and Monitoring Constrained multi-strategy Kelly, re-estimation and monitoring.	Planned

Labs, data and environment

Each module ships with a Jupyter notebook that rebuilds its exhibits. Labs are Polars-first and use NumPy, SciPy and Matplotlib; later volumes add statsmodels, cvxpy and arch. Taiwan data come from FinMind and US data from FinMind's US price tables. FinMind's licence does not allow its raw data to be redistributed, so the repository lists every file the book uses with a checksum: download them once with your own FinMind token, and the labs then run offline. The Python environment is managed by uv and pinned in uv.lock.

Shell

git clone https://github.com/Benjamin-Teng/Quant-math-for-beginner.git
cd Quant-math-for-beginner
uv sync                                  # Python 3.13 environment, versions pinned by uv.lock
export FINMIND_TOKEN="your-token"
uv run python data/fetch_finmind.py      # download the raw data and check every file
uv run python data/make_lab_data.py      # build each lab's data/ folder
uv run --with jupyterlab jupyter lab labs/

Edition notes

Edition 0.2 (11 October 2026). Volume I complete: LM1–LM7 with their labs. Each module passed an independent review, a blind solve of every question with a key check, a full audit of its numbers from the raw data and verified revision rounds; LM3 was revised the same way. Wide formulas and tables now fit the printed page, and equation numbers and punctuation no longer collide with formulas.

Edition 0.1 (10 October 2026). Roadmap, contents and conventions, and the sample module LM3 with its lab. Volumes are released in order, starting with Volume I.

A personal study project. Educational material only; nothing here is investment advice. Text and code are released under the MIT License. Market data: FinMind.