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Advanced Topics in Financial Mathematics

A.Y. 2020/2021

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
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

I will provide the lectures on Zoom and Microsoft Teams in synchronous and asynchronous mode. The synchronous lectures will be delivered as scheduled in the timetable of the department lectures. In case of asynchronous mode, the registration of the lectures will be posted on ARIEL - except for ad hoc agreements with the students - at the scheduled time.

The overall program, the notes and textbook remain the same, but it is possible a partial reduction of the program that will be determined during the course.

The exam will be oral: either on Zoom or, whenever possible according to the regulation restriction concerning the Covid pandemic, at the university.

In the future, if possible according to the regulation restriction, and previous disclosure to the students, some lectures will be delivered in the university classrooms. In any case, each student will have the opportunity to take the lectures on line.

The overall program, the notes and textbook remain the same, but it is possible a partial reduction of the program that will be determined during the course.

The exam will be oral: either on Zoom or, whenever possible according to the regulation restriction concerning the Covid pandemic, at the university.

In the future, if possible according to the regulation restriction, and previous disclosure to the students, some lectures will be delivered in the university classrooms. In any case, each student will have the opportunity to take the lectures on line.

**Course syllabus**

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)

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.

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.

**Prerequisites for admission**

Mathematical Finance I, Probability Theory

**Teaching methods**

Lectures at the blackboard (or in the way described in the emergency phase paragraph).

**Teaching Resources**

Notes written by the teacher.

H. Follmer, A. Schied: "Stochastic Finance", 3rd Edition, de Gruyter, 2010.

C. Aliprantis, K. Border: "Infinite Dimensional Analysis", 3rd Edition, Springer 2006.

Scientific papers proposed by the teacher.

H. Follmer, A. Schied: "Stochastic Finance", 3rd Edition, de Gruyter, 2010.

C. Aliprantis, K. Border: "Infinite Dimensional Analysis", 3rd Edition, Springer 2006.

Scientific papers proposed by the teacher.

**Assessment methods and Criteria**

The final examination consists of an oral exam (in the way described in the emergency phase paragraph).

Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

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

Lessons: 42 hours

Professor:
Frittelli Marco

Educational website(s)

Professor(s)

Reception:

on appointment

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