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.