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Mathematical Finance 1

A.Y. 2017/2018

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:**

**Assessment result:**

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

**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 second fundamental theorem of asset pricing: complete markets. 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 in finite and infinite dimensional spaces. The set K and the cone C of super replicable claims. The NA, NFL and NFLVR conditions and the weak closure of C. The I fundamental theorem of asset pricing: equivalence between NFL and M^e not empty. 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. 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" of a contingent claim. 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. Two characterizations of optimal stopping time. Tao min e tao max. P*-a.s. hedging if tao is 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 point of the arbitrage free interval. Attainable American claims, characterization of attainable claims.

VII Superhedging American contingent claims

P-Supermertingale, uniform Doob decomposition.

Superhedging of American contingent claims. Upper Snell envelope with properties.

Superhedging strategies, Minimal amount needed to superhedge.

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 second fundamental theorem of asset pricing: complete markets. 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 in finite and infinite dimensional spaces. The set K and the cone C of super replicable claims. The NA, NFL and NFLVR conditions and the weak closure of C. The I fundamental theorem of asset pricing: equivalence between NFL and M^e not empty. 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. 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" of a contingent claim. 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. Two characterizations of optimal stopping time. Tao min e tao max. P*-a.s. hedging if tao is 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 point of the arbitrage free interval. Attainable American claims, characterization of attainable claims.

VII Superhedging American contingent claims

P-Supermertingale, uniform Doob decomposition.

Superhedging of American contingent claims. Upper Snell envelope with properties.

Superhedging strategies, Minimal amount needed to superhedge.

SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES - University credits: 9

Practicals: 20 hours

Lessons: 49 hours

Lessons: 49 hours

Professor:
Frittelli Marco

Professor(s)

Reception:

on appointment

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