STOCASTIC METHODS FOR FINANCE

The pricing problem in the binomial models
Risk neutral pricing in the discrete time world
European and American options in the binomial model.

Arbitrage and risk neutral pricing in continuous time.
Pricing of contingent claims in continuous time: the Black&Scholes formula.
Black&Sholes via PDE and via Girsanov.
Hedging and completeness in the Black&Scholes framework.
Feynman-Kac formula and risk neutral pricing in continuous time.
Pur Call parity, dividends and static vs dynamic hedging.
The Greeks and the Delta-Gamma hedging. Delta-Gamma-Vega neutral portfolios.

Barrier options pricing in the Black&Scholes model.
Quanto option pricing in the Black&Scholes model.

Multi asset markets, pricing and hedging.
Exchange options pricing in the multi-asset Black&Scholes model.
Incomplete markets: quadratic hedging.

Smile and skew stylized facts.
Beyond the Black&Scholes model: stochastic volatility.
The Heston model.

Bonds and interest rates. Pre-crisis and multiple-curve frameworks.
Short rate models, Vasicek, CIR, Hull-White models, affine models.
Cap&Floor pricing in the short rate approaches.
Change of numeraire and Forward Risk Neutral measure.