| lecturers | A.W. van der Vaart and S. Gugushvili (both at VU) |
| credits | 6 |
| period | Fall 2010 |
| schedule | Mondays 10.00-12.45 36-42 10:00 12:45 HG-08A04 from September 6 to December 13. No lectures on October 25 (=week 43), November 8 and November 15. |
| location | VU; weeks 36 to 42: HG-08A04 (HG=main building), weeks 44 to 50: WN-M639 (WN=science building), Science Building Vrije Universiteit |
| exam | December 24, 8:45 - 11:30. Registration (via the VU-website) is mandatory. For room consult the VU webpages for definitive information. Books, notes or calculators not permitted. |
| Retake: February 14, 18:30 - 21:15. | |
| registration | Registration for the course is unnecessary, but registration as a VU-student is mandatory (at the VU-student administration in the main building). You need a VU student number to take the exam and receive credits. |
| contents | Financial institutions trade in
risk, and therefore must measure and control such
risks. Financial instuments such as options, swaps, forwards, caps and
floors, etc. play an important role in risk management, and to handle
them one needs to be able to price them. This course gives an
introduction to the mathematical tools and theory behind risk
management.
A "stochastic process" is a collection of random variables, indexed by a set T. In financial applications the elements of T model time, and T Is the set of natural numbers (discrete time), or an interval in the positive real line. "Martingales" are processes whose increments over an interval in the future have zero expectation given knowledge of the past history of the process. They play an important role in financial calculus, because the price of an option (on a stock or an interest rate) can be expressed as an expectation under a so-called martingale measure. In this course we develop this theory in discrete and continuous time, with an emphasis on the second. Most models for financial processes in continuous time are based on a special Gaussian process, called Brownian motion. We discuss some properties of this process and introduce "stochastic integrals" with Brownian motion as the integrator. Financial processes can next be modelled as solutions to "stochastic differential equations". After developing these mathematical tools we turn to finance by applying the concepts and results to the pricing of derivative instruments, by studying models for the "term structure of interest rates", and to risk measurement and management. Foremost, we develop the theory of no-arbitrage pricing of derivatives, which are basic tools for risk management. |
| literature | We shall mainly use the Lecture Notes. For more background, see for instance |
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| grade | The grade will be based on the grades for the 4 computer assignments (4 times 10 %) and the grade (60 %) for the final, written exam. If all assignments and the final exam are graded at least 4, the final grade is the weighted average of these; otherwise it is the minimum of all grades. |
| required knowledge | Basic probability theory, linear algebra, calculus, working knowledge of the spreadsheet program EXCEL. This course is meant for students in Business Mathematics, Econometrics, Mathematics or Quantitative Finance, who have not and do not plan to take measure theory, but are interested in financial mathematics. Students who like mathematical rigour are recommended to take courses in measure-theoretic probability, stochastic integration and finally a course such as financial stochastics. |
| assignments | The assignments should be emailed to Shota Gugushvili (shota (at) few.vu.nl). The answers usually consist of modifications to an Excel file and answers to specific questions. Do not put the text in the Excel file, but send a separate document with your answers (preferably a single pdf). Give the files a (unique) name of the form "lastname-assignmentnumber.???". Also mention your last name, student number and the assignment number in the subject line of the message. |
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| Week: | Material from the lecture notes: |
| 1 | Chapter 1 and Sections 2.1 and 2.2 of Chapter 2 |
| 2 | Rest of Chapter 2 and Sections 3.1, 3.2, 3.3, 3.4 of Chapter 3 |
| 3 | Rest of Chapters 3 and Chapter 4 |
| 4 | Sections 5.1 to 5.6 of Chapter 5 |
| 5 | Chapter 5 up to Example 5.13 |
| 6 | Rest of Chapter 5 and beginning of Chapter 6 (up to equation (6.2)). |
| 7 | Rest of Chapter 6 except Theorem 6.8 and Sections 6.5 and 6.7 |
| 8 | Rest of Chapter 6 and Sections 7.1 and 7.2 |
| 9 | Section 7.4 and beginning of Chapter 8 |
| 10 | Chapter 8 until Theorem 8.15 |
| 11 | Rest of Chapter 8 |
| 12 | Chapter 9: Sections 9.1, 9.2, 9.3, 9.4 |
| 13 | Exercise session |