M.3 Probability & Measure Theory

Basic Prerequisites: Elementary set theory and calculus. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Naive (high-school) probability assigns a number to outcomes and fails precisely where finance lives: on uncountable spaces and continuous information. The coin flipped infinitely often has an outcome set ${\Omega }_{\infty }$ that is uncountable (Shreve II §1.1), so no single outcome can carry positive mass - the theory must work on sets ($\sigma$-algebras), assign probability via a measure, and describe “information over time” with a filtration. Every pricing model, every risk estimator, every backtest’s lookahead bias is a statement about these objects.

Its claim is sharp: a “fair game” (martingale), the “best prediction given information” (conditional expectation), and “changing your odds without changing the impossible” (Radon–Nikodym) are the three primitives from which no-arbitrage pricing is built. This folder is the topic-folder for that toolbox.

This page is a hub: it (a) gives the fast definition & theorem lookup below (job #1), and (b) routes you to six sub-pages from raw intuition through probability spaces & filtrations, distributions & expectation, conditional expectation, martingales, and the measure-theoretic extensions.

The one-sentence essence. “Probability is the measure of sets ($\sigma$-algebras), information is a growing family of sets (filtration), the best predictor given information is the conditional expectation, a fair game is a martingale, and pricing swaps the physical measure for a martingale one via a Radon–Nikodym density.”

2. Mathematical Ground Truth & Definition Lookup

Quick-Reference Lookup (job #1). All definitions below are transcribed from Shreve Vol II Ch 1–2 (primary), cross-checked against Shreve Vol I Ch 1, 9, 11–12 and Glasserman Ch 1–2; the numbers in the check column were re-executed and reproduced exactly by the scripts in §3 and on the sub-pages.

Notation: $(\Omega ,\mathcal{F},\mathbb{P})$ probability space; ${\mathcal{F}}_t$ filtration; $\mathcal{G}\subseteq \mathcal{F}$ sub-$\sigma$-algebra; $\mathbb{E}[X\mid \mathcal{G}]$ conditional expectation; $\mathbb{P}\ll \mathbb{Q}$ absolute continuity; $Z=\frac{d\mathbb{Q}}{d\mathbb{P}}$ Radon–Nikodym derivative.

The martingale backbone (Shreve I §2.4, Ch 3; Glasserman §1.2). Under the risk-neutral measure $\tilde{\mathbb{P}}$, the discounted stock $\{S_k/(1+r{)}^k\}$ and every discounted self-financing wealth process are martingales; pricing is ${\mathbb{E}}^{\tilde{\mathbb{P}}}[V(T)/\beta (T)]$ (Glasserman eq. 1.39). The change $\mathbb{P}\to \tilde{\mathbb{P}}$ is a Radon–Nikodym reweighting (Shreve I Ch 9; Shreve II §1.6) - a measure change, not a change of opinion.

3. Computational Implementation - one stdlib sanity sweep

Reproduces the check column: LLN, CLT, conditional-expectation tower, and the martingale property in a single stdlib-only run.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure analysis lives on the sub-pages. In one line each:

1. Zero-probability conditioning. $\mathbb{E}[X\mid \{S_t=100.00\}]$ is not defined by the elementary ratio formula $P(A\cap B)/P(B)$ when $\mathbb{P}(B)=0$; only the measure-theoretic partial-averaging definition survives. See 04 · Conditional Expectation.
2. Filtration leakage (lookahead bias). A feature computed with the close of bar $t$ is ${\mathcal{F}}_{t+\Delta t}$-measurable, not ${\mathcal{F}}_t$-measurable; using it at $t$ is information from the future. See 02 · Probability Spaces & Filtrations.
3. Discounting under the wrong measure. Pricing requires expectation under the risk-neutral $\tilde{\mathbb{P}}$, not the physical $\mathbb{P}$; a $\mathbb{P}$-expectation of $e^{-rT}{V}_T$ depends on the unhedgeable drift. See 05 · Martingales.
4. Uncorrelated ≠ independent. Only jointly normal variables make zero covariance imply independence; outside Gaussianity the equivalence fails (e.g. $Y=ZX$, $Z=\pm 1$). See 03 · Distributions & Expectation.
5. Ignoring the measure’s absolute-continuity structure. $\tilde{\mathbb{P}}\ll \mathbb{P}$ breaks if $\mathbb{P}$ allows a zero-price asset while $\tilde{\mathbb{P}}$ forces $S_t>0$; the Radon–Nikodym density then blows up. See 06 · Advanced Extensions.

5. References

• Shreve, Steven E.: Stochastic Calculus for Finance II
• Shreve, Steven E.: Stochastic Calculus for Finance I
• Glasserman, Paul: Monte Carlo Methods in Financial Engineering
• Casella & Berger: Statistical Inference

6. Connected Graph Bridges

• Foundational base: Multivariable Calculus (expectations as integrals) · Linear Algebra (covariance, normal vectors)
• Sibling foundations: Stochastic Calculus & Itô (martingales, Girsanov, quadratic variation) · Econometrics & Time Series (conditional expectation as the mean model, stationarity)
• Sub-pages (in-folder): 01 From Zero · 02 Probability Spaces & Filtrations · 03 Distributions & Expectation · 04 Conditional Expectation · 05 Martingales · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05