Advanced Topics in Financial Mathematics

I Brief account of the course Mathematical Finance I
The no arbitrage principle and option pricing. Complete and incomplete markets. The two fundamental theorems of asset pricing. The super replication price.

II Brief account of convex analysis
Dual spaces and weak topologies. Polar and bipolar cones and the bipolar theorem. Convex functions and their conjugate. Fenchel-Moreau Theorem. The space ba, the topological dual of L^infty. Yosida-Hewitt Theorem. The Namioka-Klee Theorem and its extension to convex monotone maps. Penot-Volle Theorem.

III Risk measures
Monetary Risk Measures (RM), coherent and convex RM. Properties and financial interpretation of RM. Cash additive property and the representation of RM in terms of acceptance sets A. Relationship among the various properties of RM, including quasiconvexity and cash subadditivity. Properties of ρ_A and A_ρ. Lipschitz continuity.
Examples: V@R, Worst RM, entropic RM.
Dual representation of coherent RM by the application of the super-replication price. Dual representation of coherent and convex RM by the application of the Fenchel-Moreau theorem. On an alternative expression for the penalty function. Equivalent conditions for the lsc of a quasi-convex monotone decreasing map ρ on L^infty. On continuity from above and from below. The Lebesgue property and the dual representation as a max. Analysis of the worst RM and of the entropic RM. Variational expression of the relative entropy.
Shortfall risk measures. Quasi-convex risk measures. The dual representation of quasi-convex risk measures and the cash additive case.
Conditional and dynamic risk measures. Regularity properties. Dual representation of conditional convex RM (Scandolo-Detlefsen). Dynamic consistency.

IV On the financial markets
On the general financial market. The cone K of replicable contingent claims and the cone C of bounded super replicable claims. Separating measures (martingale measures). The NA, NFL and NFLVR conditions.

Utility maximization
Assumptions on the utility function u and their consequences on its conjugate function. Examples. The dual of the utility maximization problem.
Utility maximization, when the budget constraint set is determined by one probability Q, on L^infty and on L^1. Measures with finite entropy. Example of the computation of (U_Q)(x) and the equality between (U_Q)(x), (U^Q)(x) and I(x,Q).
On the optimal value functional U. The minimax measures. The conjugate of the integral functional. Remarks on Rockafellar and Fenchel duality theorems. The minimax theorem.
Conditions equivalent to U(x)
The dual representation of the utility maximization problem in incomplete markets. Examples: the minimal variance, the minimal entropy, the minimal infty-norm measures.
Duality with contingent claim. The dual representation of the relative entropy.
Option pricing via minimax measures and the fair price of Davis.
The dynamic certainty equivalent and its properties.
The seller and buyer indifference price and its relation with risk measures. Properties of the indifference price and dual representation.

V Systemic risk measures