Going to tack on:

5) Quantitative portfolio management. This usually involves managing a portfolio of stocks, either long-only or long-short. People usually come into this from either a heavy math/tech background or from doing academic finance research. Skills include portfolio optimization using linear/quadratic/etc. programming, machine learning, simulation and backtesting. Some companies that do this include:

- Acadian Asset Management - AQR Capital - Goldman Sachs Asset Management - Janus InTech - MDT Advisers (discl: I work there) - Renaissance Technologies Institutional Equities Fund