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Mathematical Finance 1

A.Y. 2021/2022

Learning objectives

Introduction to Mathematical Finance: option pricing in complete and incomplete markets: the fundamental theorems of asset pricing. Pricing of American contingent claims. Applications.

Expected learning outcomes

Knowledge of contingent claim financial markets and the methodology of option pricing. Hedging of American and European contingent claims.

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation

**Course syllabus**

I Introduction to financial markets and options

Simple and compound interest. Preference relations and their numerical representation via utility functions. Risk aversion and certainty equivalent.

One period markets: binomial and trinomial models. Definitions and properties of options. The No Arbitrage principle. Replicable contingent claims. Risk neutral measures. Complete and incomplete markets. Option pricing and hedging. Put-Call parity.

II Brief introduction to stochastic processes

Probability spaces and L^p spaces. Conditional expected value and its properties. Stochastic processes, adapted and predictable processes. Natural filtration. Discrete time martingales. Supermartingales and Doob decomposition. Submartingale property of the square of a martingale. Equivalent martingale measures. Martingale property of the elementary stochastic integral.

III Discrete time models

Multi-period markets. Hedging strategies and replicable contingent claims. The value process. Equivalent martingale measures and no arbitrage pricing. The first fundamental theorem of asset pricing: NA is equivalent to the existence of an equivalent martingale measure. The pricing formula of replicable claims.

Pricing and hedging in the binomial model. Pricing formulae and hedging strategies by backward induction. Examples and pricing of particular contingent claims: Lookback, Asian, Chooser, Compound options.

IV Complete and Incomplete Markets

Brief introduction to convex analysis. Dual spaces and weak topologies. Polar and bipolar of a convex cone.

Admissible trading strategies. The set K and the cone C of super replicable claims. The proof of the I FTAP in the one period conditional case. The proof in the multiperiod case. The NA, NFL and NFLVR conditions. The weak closure of C. Equivalence between NFL and M^e not empty. The second fundamental theorem of asset pricing: complete markets. Characterization of the set M of martingale measures, as the (normalized) polar set of C. Density of the set of equivalent martingale measures. Characterization of the cone C of bounded super replicable claims as the polar of M^a. Characterization of replicable claims. Proof of the II fundamental theorem of asset pricing.

Incomplete markets: the pricing problem. The super replication cost and the no arbitrage interval. Duality theorem for the super replication cost under the assumption that C is weakly closed. Maximization of expected utility. Definition of the "Fair Price". The associated equivalent martingale measure. The case of exponential utility. Duality between utility maximization and relative entropy minimization.

V American contingent claim in complete markets

American claims, stopping times and exercise strategies, Doob decomposition. Stopped process, Doob's stopping theorem for martingales and supermartingales.

Seller approach, the Snell envelope and properties.

Buyer approach, optimal stopping problem, the Snell envelope is the solution of the optimal stopping problem. Two characterizations of optimal stopping times. Stopping times min e max. P*-a.s. hedging with a P* optimal stopping time.

Comparison between European and American call and put: theory and interpretation.

VI American contingent claim in incomplete markets

American claims, arbitrage free prices, upper and lower points of the arbitrage free interval. Attainable american claims, characterization of attainable claims.

VII Superhedging american contingent claims

P-Supermertingale, uniform Doob decomposition. Upper Snell envelope with properties. Equivalent formulation of the upper Snell envelope. Superhedging of american contingent claims.

Superhedging strategies, minimal amount needed to superhedge.

VIII On alternative pricing methods

Quantile hedging. Shortfall risk minimization. Super replicazion and risk measures. Local risk minimization.

Simple and compound interest. Preference relations and their numerical representation via utility functions. Risk aversion and certainty equivalent.

One period markets: binomial and trinomial models. Definitions and properties of options. The No Arbitrage principle. Replicable contingent claims. Risk neutral measures. Complete and incomplete markets. Option pricing and hedging. Put-Call parity.

II Brief introduction to stochastic processes

Probability spaces and L^p spaces. Conditional expected value and its properties. Stochastic processes, adapted and predictable processes. Natural filtration. Discrete time martingales. Supermartingales and Doob decomposition. Submartingale property of the square of a martingale. Equivalent martingale measures. Martingale property of the elementary stochastic integral.

III Discrete time models

Multi-period markets. Hedging strategies and replicable contingent claims. The value process. Equivalent martingale measures and no arbitrage pricing. The first fundamental theorem of asset pricing: NA is equivalent to the existence of an equivalent martingale measure. The pricing formula of replicable claims.

Pricing and hedging in the binomial model. Pricing formulae and hedging strategies by backward induction. Examples and pricing of particular contingent claims: Lookback, Asian, Chooser, Compound options.

IV Complete and Incomplete Markets

Brief introduction to convex analysis. Dual spaces and weak topologies. Polar and bipolar of a convex cone.

Admissible trading strategies. The set K and the cone C of super replicable claims. The proof of the I FTAP in the one period conditional case. The proof in the multiperiod case. The NA, NFL and NFLVR conditions. The weak closure of C. Equivalence between NFL and M^e not empty. The second fundamental theorem of asset pricing: complete markets. Characterization of the set M of martingale measures, as the (normalized) polar set of C. Density of the set of equivalent martingale measures. Characterization of the cone C of bounded super replicable claims as the polar of M^a. Characterization of replicable claims. Proof of the II fundamental theorem of asset pricing.

Incomplete markets: the pricing problem. The super replication cost and the no arbitrage interval. Duality theorem for the super replication cost under the assumption that C is weakly closed. Maximization of expected utility. Definition of the "Fair Price". The associated equivalent martingale measure. The case of exponential utility. Duality between utility maximization and relative entropy minimization.

V American contingent claim in complete markets

American claims, stopping times and exercise strategies, Doob decomposition. Stopped process, Doob's stopping theorem for martingales and supermartingales.

Seller approach, the Snell envelope and properties.

Buyer approach, optimal stopping problem, the Snell envelope is the solution of the optimal stopping problem. Two characterizations of optimal stopping times. Stopping times min e max. P*-a.s. hedging with a P* optimal stopping time.

Comparison between European and American call and put: theory and interpretation.

VI American contingent claim in incomplete markets

American claims, arbitrage free prices, upper and lower points of the arbitrage free interval. Attainable american claims, characterization of attainable claims.

VII Superhedging american contingent claims

P-Supermertingale, uniform Doob decomposition. Upper Snell envelope with properties. Equivalent formulation of the upper Snell envelope. Superhedging of american contingent claims.

Superhedging strategies, minimal amount needed to superhedge.

VIII On alternative pricing methods

Quantile hedging. Shortfall risk minimization. Super replicazion and risk measures. Local risk minimization.

**Prerequisites for admission**

Probability theory

**Teaching methods**

Lectures at the blackboard, software presentation (see the statements in the paragraph on the emergency phase).

**Teaching Resources**

S. Pliska: "Introduction to Mathematical Finance", Blackwell, 1997.

H. Follmer, A. Schied: "Stochastic Finance", 4° Edition, de Gruyter, 2016.

S. Shreve: "Stochastic Calculus for Finance I and II", Springer, 2004.

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

D. Williams: "Probability with martingales", Cambridge University Press, 1991.

Scientific papers delivered by the teacher.

H. Follmer, A. Schied: "Stochastic Finance", 4° Edition, de Gruyter, 2016.

S. Shreve: "Stochastic Calculus for Finance I and II", Springer, 2004.

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

D. Williams: "Probability with martingales", Cambridge University Press, 1991.

Scientific papers delivered by the teacher.

**Assessment methods and Criteria**

The final examination consists an oral exam.

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: 9

Lessons: 63 hours

Professor:
Frittelli Marco

Educational website(s)

Professor(s)

Reception:

on appointment

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