Probability Theory Fundamentals › Summary

This chapter covered the probability theory foundations essential for quantitative finance.

We began with sample spaces, measurable events, and the Kolmogorov axioms that define probability measures. Discrete variables use probability mass functions; absolutely continuous variables can be represented with densities, while every random variable has a CDF.

Expected value provides a probability-weighted average, representing the theoretical mean outcome. Variance measures dispersion around this mean, with standard deviation serving as a standard measure of return volatility rather than a complete measure of financial risk. Higher moments (skewness and kurtosis) capture asymmetry and tail behavior, revealing patterns that mean and variance miss.

Conditional probability quantifies how learning new information updates our beliefs. Bayes' Theorem formalizes this updating process, and gives a systematic method for incorporating evidence. The credit risk example demonstrated how new financial data can materially revise default probability estimates.

Key distributions in finance include normal models for returns and log-normal models for prices. However, many empirical return series have heavier tails than a fitted normal distribution, and some asset classes and horizons exhibit negative skewness. These departures from normality matter for risk management.

The practical simulation tied these concepts together: generating returns from probability distributions, computing sample statistics, calculating risk measures like Value at Risk, and constructing price paths. These techniques provide the computational foundation for more advanced quantitative methods.

With these probability foundations, you can tackle portfolio theory, time series analysis, and derivative pricing in the chapters ahead. Each builds directly on the concepts of distributions, expected values, and conditional probabilities introduced here.