Python for Finance, 3rd Edition

Appendix B. Probability, Statistics, and Stochastic Calculus Essentials

This appendix summarizes key concepts from probability theory, statistics, and stochastic calculus that underpin the modeling work in the main part of the book. The emphasis is practical: each concept is stated mathematically and then connected to a direct implementation in NumPy, SciPy, or SymPy.

This appendix links uncertainty, estimation, and continuous-time modeling to concrete Python workflows. Probability distributions describe uncertain returns and state variables, estimators turn samples into parameters and standard errors, covariance and correlation summarize joint variation, Brownian motion provides the continuous-time noise source for diffusion models, and Ito calculus explains how functions of stochastic processes evolve. The examples connect those ideas to reproducible simulation and estimation code.

Try This in 60 Seconds

Run stats.norm(loc=0.0, scale=1.0).cdf(1.96) after importing scipy.stats as stats. The result is about 0.975, which connects the normal distribution table, confidence intervals, and value-at-risk style quantiles.

Random Variables and Distributions

A random variable maps uncertain outcomes to numbers. Its distribution describes which values are possible and how likely they are. For a continuous random variable $X$ with density $f$, probabilities are integrals:

$\mathbb{P}(a\le X\le b)={\int }_a^{b}f(x)dx.$

The normal distribution is not a perfect model for financial returns, but it is a useful reference ...