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Monte Carlo Methods in Insurance and Finance
Department of Mathematics, University of Copenhagen, Block 1, 2010

Course Details:

Lecturer: Jeffrey Collamore; ph.: 3532 0782; e-mail: collamore-at-math.ku.dk
Lectures: Wednesday 10-12 in Aud. 5; Wednesday 13-15 in Aud. 6.
Evaluation: There will be a 30-minute oral exam and homework exercises. The oral exam will count for the primary part of your final grade, although the homework exercises will also count for a smaller part of the grade. In particular, all homework problems must be attempted and the homework must be "passed" in order to participate in the oral exam.

Prerequisites: Many results will be established from basic principles, but a measure-theoretic course on probability theory will generally be assumed.

Course material: P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer-Verlag, Berlin, 2004 (Ch. 1-6).
With a few exceptions, the reading assignments will all be taken from this text. The text may be purchased from various sources, such as Amazon.co.uk or the university bookstore.
Another useful and highly recommended reference (not required) is:
S. Asmussen and P. Glynn, Stochastic Simulation, Springer-Verlag, Berlin, 2007.

Course description: This will be an introductory course on Monte Carlo simulation techniques. Topics will include: basic principles and sampling methods; variance reduction; quasi-Monte Carlo; discretization methods for stochastic differential equations; applications. Monte Carlo methods are of applied relevance because real-life problems in insurance, finance, and other applied areas are often too complicated to be solved using explicit analytical methods. When simulation is done naively, various problems can arise (e.g., the variance of the estimate may be large compared with the estimate). For continuous-time processes satisfying a stochastic differential equation, one also needs to compare the continuous process to the discrete-time approximation and determine the accuracy of various possible approximations. There are also methodological issues (e.g., effective means for generating random samples). Throughout the course, examples will be drawn from both insurance mathematics and finance. In general, the topics of the course will be of broad relevance in insurance, finance, applied probability, and statistics.

Schedule for the lectures:

08.09.10: Glasserman, Ch.1-2; specifically: pp. 1-19 (lecture 1); pp. 39-44 and pp. 53-58 (lecture 2)*.
15.09.10: Glasserman, Ch. 2; specifically, pp. 58-77.
22.09.10: Glasserman, Ch. 3; specifically, pp. 79-114 and handout on Levy processes.
29.09.10: Glasserman, Ch. 4; specifically, pp. 185-220 and pp. 236-243.
06.10.10: Glasserman, Ch. 4; specifically, pp. 255-279.
13.10.10: Glasserman, Ch. 4, pp. 255-279 (cont.) and handout (lecture 1); Ch. 6 (lecture 2).
20.10.10: Glasserman, Ch. 6 (cont., lecture 1); Ch. 5 (Sec. 5.1-5.2, lecture 2).

*A description of the classical ruin problem (for those who haven't seen it before) can be found in any book on non-life insurance or ruin theory (e.g., the books by S. Asmussen or T. Mikosch) or the following:
P. Embrechts, C Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance or Finance, Springer-Verlag, Berlin, 1997, Section 1.1 and (for futher details) Section 1.2.

Homework exercises: Click here. (Updated October 7; final version.)

Comments on the homework:
You are allowed to work in groups of at most three people.
All problems must be handed in either electronically, to me personally, or to my mailbox on the first floor by Tuesday, October 26 (and you need to do this to participate in the oral exam).
Please give a complete description of all steps in your arguments; i.e., you should explain theoretically what you have simulated--it is not sufficient to just present the numerical results.

Exam dates: November 2-3.
Reexam date and time: March 4, 10:00 in A105.

Exam topics:
1. Simulating random variables. (Roughly, Glasserman Ch. 2.)
2. Simulating Brownian motion. (Roughly, Glasserman Ch. 3 plus supplement on Levy processes.)
3. Variance reduction, I. (Glasserman Ch. 4 but excluding importance sampling.)
4. Variance reduction, II. (Glasserman Ch. 4 plus supplement.)
5. Numerical SDEs. (Glasserman Ch. 6.)
6. Quasi-Monte Carlo/random number generation*. (Glasserman Ch. 5, 2.)

The above topics should be regarded as guides: you should try to subdivide the lectures into six categories based on these topics. Remember that this is an exam; thus, you should aim to be reasonably detailed in your presentation.

*Quasi-Monte Carlo is a shorter topic, which you should also try to relate to those topics described in 1.-5. above; in particular, you should begin with a discussion of deterministic v. random methods and the problem of generating uniform random variables with the M.C. method. Moreover, you are only responsible for those sections in Ch. 5 which were discussed in lecture.