Advanced Topics in Financial Mathematics

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

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 duality Theorem. Fenchel-Moreau Theorem. The space ba, the topological dual of L^infty. Yosida-Hewitt Theorem. Penot-Volle theorem on quasi-convex lsc functions.
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, certainty equivalent.
Dual representation of coherent and convex RM by the application of the Fenchel-Moreau theorem. Equivalent conditions for the lsc of a quasi-convex monotone decreasing map ρ on L^infty. Analysis of the worst RM and of the entropic RM. Variational expression of the relative entropy.
Dual representation of monotone quasi-convex RM using the Penot-Volle theorem.
Conditional and dynamic risk measures. Regularity properties. Dual representation of conditional convex RM (Scandolo-Detlefsen).
Remarks: on the extension of Namioka Klee Theorem, on RM on Orlicz spaces, on the extension of RM from L^infty to L^p, on law invariant RM, on RM defined on distribution functions on R and on the Lambda V@R, on Scientific Research Measures.

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. The No Free Lunch with Vanishing Risk condition and the weak closure of C.

V Utility maximization
Assumptions on the utility function u and their consequences on its conjugate function. Examples. The conjugate of the integral functional. Rockafellar, Fenchel and Yosida-Hewitt theorems. The minimax theorem. 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. 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 claims and entropic penalties. The dual representation of the relative entropy and the financial interpretation 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 price for general functional and their relation with risk measures.
The indifference price and its relation with risk measures. Properties of the indifference price and dual representation.
Existence of the optimal solution to utility maximization when the budget constraint set is determined by one probability Q (complete markets). On the additional assumption on the conjugate function of u. Brief account of the existence of the optimal solution to the utility maximization problem in incomplete markets.

VI 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, uniqueness and existence of the optimal allocation, expected utility interpretation, marginal risk contribution, the exponential case.