Models

• Geometric Brownian motion and the Black-Scholes assumptions it encodes
• Ornstein-Uhlenbeck and mean reversion for rates and spreads
• Jump diffusion and stochastic volatility such as Heston, and the fat tails they add

Simulation

• Euler-Maruyama and the Milstein correction, with their convergence orders
• Strong against weak convergence and which one your application needs
• Time step selection, stability and random number generation in Python

Monte Carlo

• Path simulation for pricing and the square-root convergence rate
• Variance reduction: antithetic variates and control variates
• Confidence intervals reported alongside every simulated estimate

Risk

• Value at Risk and expected shortfall, and the coherence argument between them
• Calibration to market data and the instability of fitted parameters
• Backtesting, model risk and the failures that models systematically miss