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Advanced · Self-paced2026 Edition

Stochastic Calculus for Finance

The continuous-time mathematics behind Black-Scholes, interest-rate models, and modern derivatives pricing. Ten modules from Brownian motion to Girsanov, taught the way Steven Shreve teaches it — but with the African and emerging-market context where these models behave differently in practice.

10

Modules

~9h 55m

Reading time

Advanced

Level

Self-paced

Format

§

Syllabus

  1. 01

    From random walks to Brownian motion

    The scaling limit. Why the binomial tree of finite steps becomes continuous-time geometric Brownian motion. Donsker's theorem in finance dress.

    ~55 minModule 01
    →
  2. 02

    Brownian motion — properties and path behaviour

    Independent increments, Gaussian increments, continuous paths, non-differentiability, quadratic variation. The four facts you must own.

    ~55 minModule 02
    →
  3. 03

    The Itô integral

    Why ordinary Riemann-Stieltjes integration fails for BM. Building the Itô integral, the isometry, the martingale property.

    ~65 minModule 03
    →
  4. 04

    Itô's lemma

    The chain rule of stochastic calculus, derived from quadratic variation. The single most-used formula in derivatives mathematics.

    ~60 minModule 04
    →
  5. 05

    Stochastic differential equations and GBM

    SDEs as the continuous-time analogue of difference equations. Geometric Brownian motion, Ornstein-Uhlenbeck, mean-reverting processes.

    ~60 minModule 05
    →
  6. 06

    Girsanov's theorem and change of measure

    Switching from real-world to risk-neutral probability. The Radon-Nikodym derivative, the market price of risk, why no-arbitrage equals existence of an equivalent martingale measure.

    ~65 minModule 06
    →
  7. 07

    Deriving the Black-Scholes PDE

    Delta-hedging argument. The replicating portfolio. The PDE that prices any European derivative on a GBM underlying.

    ~65 minModule 07
    →
  8. 08

    The Black-Scholes formula and the Greeks

    Solving the BS PDE for European calls and puts. Delta, gamma, vega, theta, rho — what each tells the trader and the risk manager.

    ~55 minModule 08
    →
  9. 09

    Martingale pricing and the FTAP

    Fundamental Theorems of Asset Pricing. Pricing any payoff as the discounted risk-neutral expectation. Why this unifies BS, binomial, and Monte Carlo.

    ~55 minModule 09
    →
  10. 10

    Interest-rate models — Vasicek, CIR, HJM

    Short-rate models, the term-structure equation, no-arbitrage HJM. The trade-off between tractability and realism in EM yield-curve modelling.

    ~60 minModule 10
    →

How to use this course

Start with module 01 if the material is new; skip ahead if you have prior exposure. Each module is self-contained but the arc is sequential — the projects in the final module assume the toolkit from modules 1-11. Every module ends with key takeaways and a curated further-reading list with primary sources.