Mathematical Methods for Quantitative Finance

Learning modules:

1. Probability: review of laws probability; common distributions of financial mathematics; CLT, LLN, characteristic functions, asymptotics.

2. Statistics: statistical inference and hypothesis tests; time series tests and econometric analysis; regression methods

3. Time-series models: random walks and Bernoulli trials; recursive calculations for Markov processes; basic properties of linear time series models (AR(p), MA(q), GARCH(1,1)); first-passage properties; applications to forecasting and trading strategies.

4. Continuous time stochastic processes: continuous time limits of discrete processes; properties of Brownian motion; introduction to Itô calculus; solving differential equations of finance; applications to derivative pricing and risk management.

5. Linear algebra: review of axioms and operations on linear spaces; covariance and correlation matrices; applications to asset pricing.

6. Optimization: Lagrange multipliers and multivariate optimization; inequality constraints and quadratic programming; Markov decision processes and dynamic programming; variational methods; applications to portfolio construction, algorithmic trading, and best execution.|

7. Numerical methods: Monte Carlo techniques; quadratic programming