Contents |
Course Title
Special Topics: Financial Mathematics of Incomplete Markets
Course Code
IAM 743
Credit
(3-0)3
Prerequisites
Consent of the instructor.
Content
Motivation: 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, processes 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.
Aims
The primary aim is to familiarize with advanced mathematical techniques of stochastic finance which are applicable in the context of incomplete markets.
Learning Outcomes
Upon successful completion of the course, it is expected that the student will be familiar with some of the advanced techniques of stochastic finance required in order to develop a consistent framework and methods for pricing and hedging in incomplete markets. The inclined research student will be prepared to start mathematical research on the topics related to incomplete markets.
Suggested Textbooks
Lecture notes will be prepared and further references to research papers will be given during the course. Some of the core content of this course is covered in:
- H. Föllmer, A. Schied: Stochastic Finance, de Gruyter, 2002
- I. Karatzas, S. E. Shreve, Methods of Mathematical Finance, Springer, 1998
Outline
The tentative program of this course is as follows:
- 1.week: 1-period model, arbitrage theory, martingales, market completeness
- 2.week: geometric characterization of arbitrage-free models
- 3.week: expected utility, risk aversion; risk premium
- 4.week: relative entropy
- 5.week: equilibrium pricing
- 6.week: convex risk measures
- 7.week: average value at risk
- 8.week: contingent claims and hedging
- 9.week: sub/super hedging
- 10.week: super hedging duality
- 11.week: optimal martingale measures
- 12. week: utility indifference pricing
- 13.week: risk-indifference pricing
- 14.week: HJB-equation, dynamical programming
Resources
- LaTEX