This note builds the mathematical machinery that turns "stock prices look like noise" into a workable model. We define the Wiener process (the canonical noise), assemble the standard zoo of diffusion models driven by it, and develop Itô's lemma — the chain rule of stochastic calculus and the single most-used computational tool in the course. The payoff at the end is the closed-form solution of geometric Brownian motion, which is simultaneously the exact simulation recipe for Monte Carlo (Note 3) and the input to the Black–Scholes formula (the Black-Scholes PDE, formula and implied volatility).
Learning objectives. By the end of this note you should be able to:
- define a stochastic process and read it both as a family of random variables and as a family of paths;
- state the four defining properties of the Wiener process and the consequences used in computation (increment distribution, multiplication rules);
- write down and contrast the ABM, GBM and OU models, and justify GBM as the default stock model;
- state Itô's lemma, explain where the correction term comes from, and apply it fluently;
- solve the GBM SDE exactly and characterise the lognormal distribution of .
In this lesson
- 3.1 Why stochastic processes?
- 3.2 The Wiener process (Brownian motion)
- 3.3 The diffusion model zoo
- 3.4 Itô's lemma
- 3.5 Canonical application: solving GBM
- 3.6 Multidimensional extension (preview)
The full lesson, with worked examples and exercise solutions, is for members.
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