### Mathematical Finance I

The lecture covers the following topics:

**Financial markets, derivatives, arbitrage**Basic financial securities, absence of arbitrage, put-call parity.

**Finite financial market models**Financial markets, stochastic processes, filtrations, dynamic resolution of information, trading strategies, wealth process.

**Risk neutral pricing and fundamental theorems**Arbitrage, first fundamental theorem of asset pricing, risk neutral pricing, completeness, second fundamental theorem.

**Incomplete markets and superhedging**

Characterization of the set of arbitrage free prices, superhedging duality, optional decomposition.

**American options and optimal stopping**Optimal stopping problems, Snell envelope, risk neutral pricing of American options.

**Stochastic analysis of Brownian motion**

Continuous time limit of the CRR model, Brownian motion, square-intergable martingales, stochastic integration, Ito processes, Ito’s formula.

**Risk neutral pricing in the Black-Scholes model**

Black-Scholes model, absence of arbitrage, efficient strategies, option pricing, Black-Scholes formula, PDE pricing, Feynman-Kac representation.

#### Resources

- Lecture Notes: Mathematical Finance I