- Review of probabilistic notation (probability spaces; product spaces; physical, subjective and synthetic probability measures; filtrations; trees; atoms; transition probabilities)
- Definition of adapted and predictable stochastic processes
- Bank account, numéraire
- Stock price processes, discounting
- Trading strategies, self-financing property
- Set of discounted trading gains
- Definition of arbitrage
- Localization of arbitrage in time
- Price functionals
- Conditional expectations (definition and basic properties)
- Martingales, submartingales, supermartingales (definition and basic properties)
- Stopping times and their sigma-algebras
- Bayes' formula in connection with conditional expectations
- Equivalent martingale measures (with bounded density)
- Theorem of Dalang, Morton and Willinger (including proof in the general case)
- Minimal and maximal prices of contingent claims
- Markov processes (definition)
- Existence of martingale measures preserving the Markov property
- Call and put options in the binomial model
- Digression on weak convergence
- Passage to to limit in a scaled binomial model
- Derivation of the Black-Scholes formula
- Call-put parity
- Complete markets and uniqueness of the equivalent martingale measure
- American options, Snell envelopes
- Forward contracts, hedge
- Futures and their relation to forwards
- Fixed income securities (zero-coupon bonds, bonds and their decomposition, fixed and floating rate loans)
- Interest rate caps and floors, decomposition into captions and floorlets
- Interest rate swaps, swap rate, hedge
- Swaptions

- Filtrations, stochastic processes and stopping times
- Martingales and their properties
- Brownian motion and its properties
- Brownian motion with drift, distribution of certain hitting times
- Introduction to stochastic integration
- Itô's lemma, Girsanov-Maruyama theorem
- Stochastic differential equations
- Black-Scholes option-pricing theory
- European and American options

Please send comments and suggestions to Uwe Schmock, email: schmock@fam.tuwien.ac.at. Last update: June 14, 2003 |