Syllabus
1. Stochastic integration w.r.t. martingales and local martingales. Stochastic linear and bilinear equations, geometric Brownian motion. Stochastic differential equations.
2. Short rates models (Ho and Lee, Vasicek, Hull and White, CIR) , bond price.
3. Market model, portfolio value, self-financing portfolio. Risk-neutral measures, arbitrage and the 1st fundamental theorem of option pricing.
4. Girsanov theorem and risk-neutral measure in the BS model. European call option. Completeness of the market, 2nd fundamental theorem of option pricing.
5. Representation of continuous martingale by stochastic integral, hedging.
6. Feynman-Kac formula, BS equation, replication strategy for simple contingent claims. Asian and American options.
Annotation
Students are supposed to be acquainted with basics of probability theory and stochastic analysis on the level of
Lecture NMFM 408 (or a similar lecture). In the present lecture the knowledge of basic tools of stochastic analysis will be extended, taking into account the usual tools used in continuous modelling in finance mathematics - e.g. the Ito formula, Girsanov Theorem and Representation Theorems for continuous martingales. Applications to interest rate models, risk neutral measures and option pricing. Arbitrage. Fundamental Theorem of Asset Pricing.
Black-Scholes model. Hedging.