Part XVIII — Applications to Quantitative Finance

Where the appendix meets the desk: applications of the probability and statistics above to sizing, risk, market microstructure, and regime detection. These pages connect the formal results to the course's trading content.

Topics

Topic	Focus
Kelly Criterion	Bet sizing that maximizes long-run log wealth
Probability of Ruin	The chance a betting or trading process ever hits zero
Drawdown Probabilities	Distributions of peak-to-trough losses
Hitting Times	The time a stochastic process first reaches a target level
First Passage Times	Distributions of first crossings for random walks and diffusions
Queue Models	Birth–death queues and waiting-time behavior
Order Arrival Processes	Modeling order flow as point processes
Bayesian Signal Updating	Updating trade signals as evidence accumulates
Monte Carlo Option Pricing	Pricing derivatives by simulating risk-neutral paths
Portfolio Risk Simulation	Simulating portfolio P&L distributions
Value at Risk	Quantile-based loss thresholds and their estimation
Expected Shortfall	Average loss beyond the VaR threshold
Heavy-Tailed Returns	Power-law tails and why normal assumptions fail
Extreme Value Theory	Limit laws for maxima and tail estimation
Copulas	Separating marginal distributions from dependence structure
Regime Detection	Identifying market states from observable data
Hidden-State Models	Latent-state approaches beyond the basic HMM