A fundamental result in financial mathematics states that in a financial market there exist no arbitrage opportunities (i.e. possibilities for making a risk-free profit) if and only if there exists a so-called equivalent martingale measure. This is an important result since it links the economic notion of no-arbitrage to the mathematical notion of a martingale. It explains the central role of martingale theory in financial mathematics.
In the first part of this course we explain this so-called "Fundamental Theorem of Asset Pricing". We will provide full proofs in a relatively simple discrete-time setting and outline the much more general theorem of Delbaen and Schachermayer. In particular, we will explain that it is natural to model assets prices in an arbitrage-free economy as semimartingales and to assume the existence of equivalent martingale measures.
In the second part of the course we study these general semimartingale models for arbitrage-free markets in some detail. We derive general results on the pricing and hedging of derivatives, completeness, etc. Subsequent more special topics that might be addressed include American options, term structure models, stochastic volatility models, utility-based pricing, etc.
literature:
We mainly use the following lecture notes:
exam: The EXAM is in take-home form. You have to do the following exercises:
required knowledge: Measure theoretic probability, stochastic integration
What we have done so far:
| Date: | Material from the lecture notes: | Corresponding exercises: |
| 5/9 | A: Chapter 1, Theorem A.1.1 | 1.1, A.1 |
| 12/9 | A: Sections 2.1 - 2.3 | 2.1 - 2.4 |
| 19/9 | A: Theorem A.1.2, Section 2.4 | 2.5 - 2.10 |
| 26/9 | A: rest of Chapter 2 | 2.11, 2.12 |
| 3/10 | A: Chapter 3 | 3.1, 3.2 |
| 10/10 | C: Sections 5.1 - 5.4 | 5.4, 5.16, 5.18, 5.38 |
| 17/10 | C: Sections 5.6 - 5.8 | 5.53, 5.61, 5.62 |
| 31/10 | A: Chapter 4 (no proofs) | 4.1, 4.2 |
| 7/11 | B: Sections 4.1, 4.2 | 4.2, 4.3, 4.12 |
| 14/11 | B: Section 4.3, Section 4.4 very sketchy. |