Mathematics of Trading: Introduction
A Practical Guide to Understanding Markets Through Randomness
Financial markets look noisy and chaotic. Prices swing around, drift for a while, reverse sharply, and sometimes make extreme moves that seem impossible in the moment.
But there is a deeper structure beneath the noise, not a structure of predictability, but a structure of probability, randomness, and uncertainty.
These notes are my attempt to understand markets through that lens.
They are not about finding a trading signal or predicting the next price. They are about building a mental model that treats markets as probabilistic systems rather than puzzles to solve.
The emphasis throughout this series is on:
- uncertainty
- expected value
- edge
- distributions
- variance & volatility
- drawdowns
- risk of ruin
- randomness
- Brownian motion
- geometric Brownian motion
- Monte Carlo simulations
The goal is simple:
Learn to think clearly about uncertainty so that decisions under risk become structured instead of emotional.
The reason why I’m writing these notes is because when most people begin trading, they start with:
- indicators
- chart patterns
- intuitive setups
- predictions about direction
But the more I study markets, the more it becomes obvious that:
- one outcome doesn’t matter
- price is not predictable
- returns come from repeated decisions, not one trade
- risk compounds faster than intuition expects
- randomness creates both opportunity and danger
This series is a way of organizing the ideas that actually matter for long-term decision-making.
What These Notes Cover
There are six essential modules, each building on the previous one.
1. Uncertainty
Why markets are inherently unpredictable and why thinking in terms of possible futures is the only logical approach.
2. Expected Value
The foundation of rational decision-making. Why EV matters more than win rate, accuracy, or intuition.
3. Edge
Expected value describes a single bet. Edge describes a repeatable advantage.
This module explores:
- why tiny edges are powerful
- why costs and variance can destroy edges
- why detecting edge requires math and patience
4. Probability Distributions
Markets don’t follow single numbers, they follow shapes.
In this module:
- normal vs fat-tailed distributions
- skew and kurtosis
- why extreme events happen more often than intuition expects
5. Variance, Volatility, Drawdowns, and Risk of Ruin
This is where theoretical advantage meets real-world emotion.
You’ll see how:
- variance creates losing streaks
- losing streaks create drawdowns
- drawdowns create risk of ruin
- position size determines survival more than strategy quality
6. Modeling Randomness
(Brownian Motion → GBM → Monte Carlo)**
Once the conceptual pieces are in place, these notes move into modeling:
- Brownian motion as a continuous random walk
- geometric Brownian motion (GBM) as a model for stock prices
- volatility and drift as the drivers of uncertainty
- Monte Carlo simulations to explore thousands of alternate futures
How These Modules Fit Together
The series moves from the most abstract idea, uncertainty to the most practical, simulation.
The goal is not to predict prices. The goal is to understand the structure of randomness well enough to reason about decisions.