Random walks, Brownian motion, stochastic calculus, and continuous-time models for quantitative finance.
Discrete-time random walks as the foundation for continuous models
Counting processes, exponential inter-arrivals, and compound Poisson models
Wiener process, properties, and simulation
Modeling stock prices with drift and volatility
The chain rule for stochastic calculus
Ornstein-Uhlenbeck, Vasicek model, and pairs trading
Itô integrals, quadratic variation, and stochastic differential equations
Merton's model, Poisson jumps, fat tails, and volatility smile
Change of measure, Girsanov theorem, and the fundamental theorem of asset pricing
Delta-hedging derivation, theta-gamma tradeoff, and Feynman-Kac connection
Heston model, leverage effect, and volatility surface dynamics