Mathematical Finance 1

I Introduction to financial markets and options
Simple and compound interest. Preference relations and their numerical representation via utility functions. Risk aversion and certainty equivalent.
One period markets: binomial and trinomial models. Definitions and properties of options. The No Arbitrage principle. Replicable contingent claims. Risk neutral measures. Complete and incomplete markets. Option pricing and hedging. Put-Call parity.

II Brief introduction to stochastic processes
Probability spaces and L^p spaces. Conditional expected value and its properties. Stochastic processes, adapted and predictable processes. Natural filtration. Discrete time martingales. Supermartingales and Doob decomposition. Submartingale property of the square of a martingale. Equivalent martingale measures. Martingale property of the elementary stochastic integral.

III Discrete time models
Multi-period markets. Hedging strategies and replicable contingent claims. The value process. Equivalent martingale measures and no arbitrage pricing. The first fundamental theorem of asset pricing: NA is equivalent to the existence of an equivalent martingale measure. The pricing formula of replicable claims.
Pricing and hedging in the binomial model. Pricing formulae and hedging strategies by backward induction. Examples and pricing of particular contingent claims: Lookback, Asian, Chooser, Compound options.

IV Complete and Incomplete Markets
Brief introduction to convex analysis. Dual spaces and weak topologies. Polar and bipolar of a convex cone.
Admissible trading strategies. The set K and the cone C of super replicable claims. The proof of the I FTAP in the one period conditional case. The proof in the multiperiod case. The NA, NFL and NFLVR conditions. The weak closure of C. Equivalence between NFL and M^e not empty. The second fundamental theorem of asset pricing: complete markets. Characterization of the set M of martingale measures, as the (normalized) polar set of C. Density of the set of equivalent martingale measures. Characterization of the cone C of bounded super replicable claims as the polar of M^a. Characterization of replicable claims. Proof of the II fundamental theorem of asset pricing.
Incomplete markets: the pricing problem. The super replication cost and the no arbitrage interval. Duality theorem for the super replication cost under the assumption that C is weakly closed. Maximization of expected utility. Definition of the "Fair Price". The associated equivalent martingale measure. The case of exponential utility. Duality between utility maximization and relative entropy minimization.

V American contingent claim in complete markets
American claims, stopping times and exercise strategies, Doob decomposition. Stopped process, Doob's stopping theorem for martingales and supermartingales.
Seller approach, the Snell envelope and properties.
Buyer approach, optimal stopping problem, the Snell envelope is the solution of the optimal stopping problem. Two characterizations of optimal stopping times. Stopping times min e max. P*-a.s. hedging with a P* optimal stopping time.
Comparison between European and American call and put: theory and interpretation.

VI American contingent claim in incomplete markets
American claims, arbitrage free prices, upper and lower points of the arbitrage free interval. Attainable american claims, characterization of attainable claims.

VII Superhedging american contingent claims
P-Supermertingale, uniform Doob decomposition. Upper Snell envelope with properties. Equivalent formulation of the upper Snell envelope. Superhedging of american contingent claims.
Superhedging strategies, minimal amount needed to superhedge.

VIII On alternative pricing methods
Quantile hedging. Shortfall risk minimization. Super replicazion and risk measures. Local risk minimization.