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## IAM743 - Special Topics: Malliavin Calculus and its Applications

**Credit: **
3(3-0); **ECTS: **
8.0

**Instructor(s): **
Consent of IAM

**Prerequisites: **
Consent of Instructor(s)

#### Course Catalogue Description

Important markets such as commodities or credit derivatives are essentially incomplete. The recent financial crisis has increased even more the importance of pricing and hedging in incomplete markets. Therefore these lectures concentrate on advanced methods of stochastic finance required in the context of incomplete markets. We will consider both, process in discrete and continuous time.

The content of the course covers in particular the following topics: market efficiency, market incompleteness; perfect hedges; equivalent martingale measures; attainable payoffs; asset management; contingent claims; replicating portfolio; dynamical arbitrage theory; arbitrage-free pricing; geometric characterization of arbitrage; von Neumann representation; robust Savage representation; expected utility; fair value; certainty equivalent; risk premium; risk aversion; equilibrium pricing; relative entropy; convex risk measures; robust representation; coherent risk measures; VAR; average VAR; upper/lower hedging prices; superhedging duality; risk indifference pricing; HJB equations; dynamical programming.

#### Course Objectives

The aim of this course is to introduce the fundamental ideas of Malliavin calculus and discuss some of the applications used in financial mathematics. The emphasis will be on understanding derivative and divergence operator, duality formula and Clark-Ocone formula. It is a high level calculus course which will help students to enhance their way of thinking in analysis and financial mathematics.#### Course Learning Outcomes

Student, who passed the course satisfactorily will be able to:- use Clark-Ocone formula in the integral representation of stochastic calculus
- identify Malliavin derivative operator, Skorohod integral operations in solving financial/mathematical problems
- formulate integral theorems in solving financial/mathematical problems

#### Tentative (Weekly) Outline

Important markets such as commodities or credit derivatives are essentially incomplete. The recent financial crisis has increased even more the importance of pricing and hedging in incomplete markets. Therefore these lectures concentrate on advanced methods of stochastic finance required in the context of incomplete markets. We will consider both, process in discrete and continuous time. The content of the course covers in particular the following topics: market efficiency, market incompleteness; perfect hedges; equivalent martingale measures; attainable payoffs; asset management; contingent claims; replicating portfolio; dynamical arbitrage theory; arbitrage-free pricing; geometric characterization of arbitrage; von Neumann representation; robust Savage representation; expected utility; fair value; certainty equivalent; risk premium; risk aversion; equilibrium pricing; relative entropy; convex risk measures; robust representation; coherent risk measures; VAR; average VAR; upper/lower hedging prices; superhedging duality; risk indifference pricing; HJB equations; dynamical programming.More Info on METU Catalogue

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