Roadmap

A Learning Path for Mathematical Finance

Roadmap

This roadmap explains how to move through Mathematical Finance as a coherent learning sequence. The book is designed for students in applied mathematics, statistics, operations research, data science, economics, and related quantitative fields who want a mathematically clean and computationally practical introduction to modern financial modeling.

The GitHub repository for the book is:

https://github.com/wanghemath/Book-MathFinance

The computer labs are designed to open directly in Google Colab. Each lab is self-contained: it begins with the required mathematical and financial background, then develops the programming tools step by step.

Guiding philosophy

Mathematical finance is not only a collection of formulas. It is a way to translate financial questions into mathematical structures.

Typical questions include:

  • How should we measure return and risk?
  • What does no arbitrage mean mathematically?
  • Why does risk-neutral pricing work?
  • How do stochastic processes model asset prices and interest rates?
  • How can options be priced and hedged?
  • How do volatility smiles, stochastic volatility, and model risk appear in practice?
  • How do portfolios, factors, and risk measures connect to statistical modeling?

The book follows the principle:

\[ \text{financial idea} \quad \longrightarrow \quad \text{mathematical model} \quad \longrightarrow \quad \text{computational experiment} \quad \longrightarrow \quad \text{interpretation}. \]

Prerequisites

Students should be comfortable with the following topics.

Mathematics

  • single-variable and multivariable calculus;
  • linear algebra: vectors, matrices, inner products, eigenvalues;
  • probability: random variables, expectation, variance, covariance, conditional expectation;
  • basic statistics: estimation, regression, confidence intervals;
  • some differential equations are helpful but not required.

Programming

Students should know or be willing to learn:

  • Python basics;
  • NumPy arrays;
  • pandas data frames;
  • matplotlib plotting;
  • basic simulation;
  • writing simple functions.

The page python-reference.qmd gives a compact reference for the Python tools used throughout the labs.

Big picture structure

The book can be divided into six major parts.

Part Theme Main questions
I Returns and no-arbitrage How do we model prices, returns, and arbitrage-free markets?
II Derivative pricing How are options priced by replication, risk-neutral expectation, PDEs, and simulation?
III Stochastic calculus Why do Brownian motion, Ito calculus, martingales, and change of measure matter?
IV Fixed income How do discount factors, bonds, short rates, and yield curves work?
V Portfolios and risk How do we allocate capital and measure financial risk?
VI Advanced modeling How do volatility smiles, stochastic volatility, calibration, and integrated workflows fit together?

Concept dependency map

A suggested dependency sequence is:

flowchart TD
  A[Returns and Compounding] --> B[One-Period No-Arbitrage]
  B --> C[Binomial Trees]
  C --> D[Risk-Neutral Pricing]
  D --> E[Brownian Motion and GBM]
  E --> F[Black--Scholes Formula]
  F --> G[Greeks and Delta Hedging]
  F --> H[Implied Volatility]
  E --> I[Monte Carlo Pricing]
  I --> J[Variance Reduction]
  I --> K[Exotic Options]
  F --> L[Black--Scholes PDE]
  L --> M[Finite Difference Methods]
  E --> N[Ito Calculus]
  N --> O[Martingales and Change of Measure]
  O --> P[Fixed Income Models]
  P --> Q[Short-Rate Models]
  Q --> R[HJM Yield Curve Models]
  A --> S[Portfolio Optimization]
  S --> T[CAPM and Factor Models]
  T --> U[VaR and Expected Shortfall]
  H --> V[Heston Model]
  U --> W[Integrated Pricing, Hedging, Calibration, and Risk]
  V --> W
  G --> W

flowchart TD
  A[Returns and Compounding] --> B[One-Period No-Arbitrage]
  B --> C[Binomial Trees]
  C --> D[Risk-Neutral Pricing]
  D --> E[Brownian Motion and GBM]
  E --> F[Black--Scholes Formula]
  F --> G[Greeks and Delta Hedging]
  F --> H[Implied Volatility]
  E --> I[Monte Carlo Pricing]
  I --> J[Variance Reduction]
  I --> K[Exotic Options]
  F --> L[Black--Scholes PDE]
  L --> M[Finite Difference Methods]
  E --> N[Ito Calculus]
  N --> O[Martingales and Change of Measure]
  O --> P[Fixed Income Models]
  P --> Q[Short-Rate Models]
  Q --> R[HJM Yield Curve Models]
  A --> S[Portfolio Optimization]
  S --> T[CAPM and Factor Models]
  T --> U[VaR and Expected Shortfall]
  H --> V[Heston Model]
  U --> W[Integrated Pricing, Hedging, Calibration, and Risk]
  V --> W
  G --> W

Laboratory roadmap

Each lab is intended for independent study. A typical lab has four parts:

  1. mathematical and financial background;
  2. guided Python implementation;
  3. numerical experiments;
  4. student exercises.
Lab Topic Colab
1 Prices, Returns, Compounding, and One-Period No-Arbitrage Open in Colab
2 Multi-Period Binomial Trees and Risk-Neutral Pricing Open in Colab
3 Brownian Motion, Geometric Brownian Motion, and Monte Carlo Pricing Open in Colab
4 Black–Scholes Formula, Greeks, and Delta Hedging Open in Colab
5 Implied Volatility and the Volatility Smile Open in Colab
6 Monte Carlo Variance Reduction Open in Colab
7 Exotic Options: Asian, Barrier, and Lookback Options Open in Colab
8 Finite Difference Methods for the Black–Scholes PDE Open in Colab
9 Stochastic Calculus by Simulation: Ito Integrals and Ito Formula Open in Colab
10 Martingales, Change of Measure, and Girsanov Simulation Open in Colab
11 Interest Rates, Bonds, Duration, Convexity, and Yield Curves Open in Colab
12 Short-Rate Models: Vasicek and CIR Open in Colab
13 HJM and Yield-Curve Simulation Open in Colab
14 Portfolio Optimization: Mean-Variance Theory and Constraints Open in Colab
15 CAPM, Factor Models, and Risk Decomposition Open in Colab
16 Value at Risk, Expected Shortfall, and Risk Backtesting Open in Colab
17 Stochastic Volatility: The Heston Model Open in Colab
18 Integrated Pricing, Hedging, Calibration, and Risk Open in Colab

Module 1: Prices, returns, and no-arbitrage

Recommended labs: Lab 1 and Lab 2.

This module introduces the language of asset prices, simple returns, log returns, compounding, portfolios, and arbitrage. The main mathematical insight is that absence of arbitrage imposes strong restrictions on prices.

Key ideas:

  • simple return versus log return;
  • deterministic and random payoffs;
  • portfolio value;
  • one-period no-arbitrage;
  • replication;
  • binomial trees;
  • backward induction;
  • risk-neutral probability.

After this module, students should be able to explain why derivative prices are not arbitrary market opinions. In complete models, derivative prices are forced by replication.

Module 2: Brownian motion, Black–Scholes, and Monte Carlo

Recommended labs: Lab 3, Lab 4, Lab 5, Lab 6, and Lab 7.

This module moves from discrete time to continuous time. Brownian motion becomes the fundamental source of randomness, and geometric Brownian motion becomes the basic model for stock prices.

Key ideas:

  • Brownian increments;
  • quadratic variation;
  • geometric Brownian motion;
  • Black–Scholes pricing;
  • Greeks;
  • Delta hedging;
  • implied volatility;
  • Monte Carlo standard error;
  • variance reduction;
  • path-dependent options.

Students should learn that a price is only one part of the story. Sensitivities, hedging, and model error are equally important.

Module 3: PDE and stochastic calculus foundations

Recommended labs: Lab 8, Lab 9, and Lab 10.

This module explains the deeper mathematical foundations behind continuous-time finance.

Key ideas:

  • Black–Scholes PDE;
  • terminal and boundary conditions;
  • explicit, implicit, and Crank–Nicolson finite difference methods;
  • Ito integrals;
  • Ito’s formula;
  • martingales;
  • likelihood-ratio change of measure;
  • Girsanov theorem by simulation.

A central theme is that the same option price can often be computed from several perspectives:

\[ \text{replication} \quad = \quad \text{risk-neutral expectation} \quad = \quad \text{PDE solution}. \]

Module 4: Fixed income and yield curves

Recommended labs: Lab 11, Lab 12, and Lab 13.

This module studies interest rates, bonds, and term-structure models.

Key ideas:

  • discount factors;
  • zero rates;
  • forward rates;
  • coupon bond pricing;
  • yield to maturity;
  • duration and convexity;
  • bootstrapping;
  • Vasicek model;
  • CIR model;
  • HJM no-arbitrage drift;
  • yield-curve simulation.

Fixed income is mathematically rich because the basic object is not one price process but a whole curve indexed by maturity.

Module 5: Portfolios, factors, and risk

Recommended labs: Lab 14, Lab 15, and Lab 16.

This module shifts from derivative pricing to investment and risk management.

Key ideas:

  • mean-variance optimization;
  • covariance and diversification;
  • efficient frontier;
  • tangency portfolio;
  • long-only constraints;
  • estimation risk;
  • CAPM;
  • factor exposures;
  • systematic and idiosyncratic risk;
  • Value at Risk;
  • Expected Shortfall;
  • risk backtesting.

The main statistical lesson is that portfolio theory is sensitive to estimation. The main risk-management lesson is that a risk measure must be tested, not merely computed.

Module 6: Stochastic volatility and integrated modeling

Recommended labs: Lab 17 and Lab 18.

This final module connects the whole book.

Key ideas:

  • stochastic volatility;
  • Heston model;
  • correlated Brownian motions;
  • leverage effect;
  • characteristic-function pricing;
  • implied-volatility smiles;
  • calibration;
  • hedging error;
  • full-revaluation portfolio risk;
  • Delta-Gamma-Vega approximation;
  • integrated pricing and risk workflows.

The final lab is designed as a capstone-style computational workflow.

Suggested 14-week course schedule

The following schedule assumes one semester with approximately 14 instructional weeks.

Week Topics Suggested labs
1 Prices, returns, compounding, one-period no-arbitrage Lab 1
2 Binomial trees, replication, risk-neutral pricing Lab 2
3 Brownian motion, GBM, Monte Carlo pricing Lab 3
4 Black–Scholes formula, Greeks, Delta hedging Lab 4
5 Implied volatility, volatility smile, variance reduction Labs 5–6
6 Exotic options and path dependence Lab 7
7 Black–Scholes PDE and finite differences Lab 8
8 Ito integral and Ito formula Lab 9
9 Martingales, change of measure, Girsanov theorem Lab 10
10 Bonds, duration, convexity, yield curves Lab 11
11 Short-rate models: Vasicek and CIR Lab 12
12 HJM and yield-curve simulation Lab 13
13 Portfolio optimization, CAPM, factor models Labs 14–15
14 VaR, ES, Heston, integrated capstone Labs 16–18

Suggested shorter paths

Path A: Derivative pricing focus

For a course focused mainly on option pricing:

  1. Lab 1: returns and no-arbitrage;
  2. Lab 2: binomial trees;
  3. Lab 3: GBM and Monte Carlo;
  4. Lab 4: Black–Scholes and Greeks;
  5. Lab 5: implied volatility;
  6. Lab 6: variance reduction;
  7. Lab 7: exotic options;
  8. Lab 8: finite differences;
  9. Lab 17: Heston model;
  10. Lab 18: integrated pricing and hedging.

Path B: Stochastic-processes focus

For a mathematically deeper stochastic-processes route:

  1. Lab 3: Brownian motion and GBM;
  2. Lab 4: Black–Scholes;
  3. Lab 8: PDE methods;
  4. Lab 9: Ito calculus;
  5. Lab 10: martingales and Girsanov;
  6. Lab 12: Vasicek and CIR;
  7. Lab 13: HJM;
  8. Lab 17: Heston.

Path C: Portfolio and risk focus

For students interested in asset management and risk:

  1. Lab 1: returns and compounding;
  2. Lab 11: bonds and duration;
  3. Lab 14: portfolio optimization;
  4. Lab 15: CAPM and factor models;
  5. Lab 16: VaR and Expected Shortfall;
  6. Lab 18: integrated pricing, hedging, calibration, and risk.

Assessment design

A balanced course could use:

Component Weight Description
Lab notebooks 35% Students complete selected lab exercises and submit notebooks.
Problem sets 25% Mathematical derivations and conceptual questions.
Midterm project 15% A focused computational study, such as binomial convergence or Monte Carlo variance reduction.
Final project 25% An integrated modeling project with written interpretation.

Suggested final projects

Possible final projects include:

  • compare binomial, Monte Carlo, and finite-difference prices for the same option;
  • study Delta hedging error under volatility misspecification;
  • calibrate a volatility smile to synthetic or real option quotes;
  • price and risk-manage a small portfolio of options;
  • compare Vasicek and CIR yield curves under different parameter regimes;
  • construct a factor model and decompose portfolio risk;
  • backtest VaR and Expected Shortfall models;
  • simulate Heston prices and explain the resulting implied-volatility skew.

Milestones for students

By the end of the book, students should be able to:

  • compute and interpret simple and log returns;
  • detect simple arbitrage opportunities in one-period models;
  • price options using binomial trees;
  • simulate Brownian motion and geometric Brownian motion;
  • implement Black–Scholes prices and Greeks;
  • estimate implied volatility from option prices;
  • run Monte Carlo pricing with confidence intervals;
  • apply variance reduction methods;
  • price path-dependent options;
  • solve the Black–Scholes PDE numerically;
  • explain Ito’s formula and its role in finance;
  • simulate change of measure with likelihood ratios;
  • price bonds and compute duration and convexity;
  • simulate Vasicek, CIR, and HJM models;
  • solve basic portfolio optimization problems;
  • estimate CAPM and factor-model exposures;
  • compute and backtest VaR and Expected Shortfall;
  • simulate stochastic volatility and interpret implied-volatility smiles.

How to use the labs in Colab

Each lab can be opened from the GitHub repository using the pattern:

https://colab.research.google.com/github/wanghemath/Book-MathFinance/blob/main/labs/lab-XX-file-name.ipynb

Students should use this workflow:

  1. open the notebook in Colab;
  2. choose File -> Save a copy in Drive;
  3. read the background sections before running code;
  4. run all cells once from top to bottom;
  5. change parameters and observe what changes;
  6. complete exercises in new cells;
  7. submit the completed notebook if required.

Instructor notes

The labs are intentionally designed to be self-contained. This makes them useful for:

  • in-class demonstrations;
  • weekly homework;
  • flipped classroom preparation;
  • independent student study;
  • project scaffolding;
  • review before exams.

A good rhythm is:

\[ \text{theory in lecture} \quad + \quad \text{guided computation in lab} \quad + \quad \text{interpretation in writing}. \]

Closing perspective

The goal of this book is not only to teach students formulas. It is to help them understand how mathematical structures create financial meaning.

A successful student should leave the course able to move fluently between:

\[ \text{market data}, \quad \text{models}, \quad \text{simulation}, \quad \text{pricing}, \quad \text{hedging}, \quad \text{risk}. \]

That fluency is the central skill of mathematical finance.