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. Penot-Volle theorem on quasi-convex lsc functions. The Namioka-Klee Theorem and its extension to convex monotone maps.

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. Properties of ρ_A and A_ρ. Lipschitz continuity. Cash subadditive property. Quasi-convex RM and their representation in terms of a family A=(A_m) of acceptance sets. Properties of ρ_A and A_ρ.
Examples: V@R, Worst RM, entropic RM.
Dual representation of monotone quasi-convex RM using the Penot-Volle theorem. 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.
Conditional and dynamic risk measures. Regularity properties. Dual representation of conditional convex RM (Scandolo-Detlefsen). Dynamic consistency. Orlicz spaces and Orlicz heart: definition and properties. RM defined on Orlicz spaces and their dual representation.

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.

A selection from the following topics:

A) 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.

B) Systemic Risk Measures
Definition of systemic risk measures and aggregation function, dual representation, optimal dual probability measure, fair probability measure, fair allocation and fair risk allocation, the optimal allocation, expected utility interpretation.

C) Robust finance and Optimal Transport
No arbitrage and super-hedging duality in a robust and pathwise setting. Brief account of the theory of Martingale Optimal Transport, application to the pathwise super-hedging duality. Extension of the Martingale Optimal Transport theory with divergence penalization.