Mathematical Methods for Quantitative Finance

Modern finance is the science of decision making in an uncertain world, and its language is mathematics. As part of the MicroMasters® Program in Finance, this course develops the tools needed to describe financial markets, make predictions in the face of uncertainty, and find optimal solutions to business and investment decisions.

This course will help anyone seeking to confidently model risky or uncertain outcomes. Its topics are essential knowledge for applying the theory of modern finance to real-world settings. Quants, traders, risk managers, investment managers, investment advisors, developers, and engineers will all be able to apply these tools and techniques.

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