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Advanced Topics in Financial Mathematics

A.Y. 2017/2018

Learning objectives

Two central topics of Mathematical Finance: the theory of risk measures and the valuation of contingent claims in incomplete markets by utility maximization and indifference pricing.

Expected learning outcomes

Methods of convex analysis and optimization. Pricing and hedging of financial instruments.

**Lesson period:**
First semester

**Assessment methods:**

**Assessment result:**

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

**Course syllabus**

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)

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.

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.

SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES - University credits: 6

Lessons: 42 hours

Professor:
Frittelli Marco

Professor(s)

Reception:

on appointment

Office 1043, first floor, Math. Dept., Via Saldini 50.