Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Math 595, Section FM: Financial
Mathematics
Spring 2008 (1st-half)
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Class: |
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Instructor: |
Rick
Gorvett |
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154
Henry Admin. Bldg. |
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374
Altgeld Hall |
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Class
Dates: |
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Phone:
244-1739 |
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January
15 through |
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Office
Hours: |
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or by appointment |
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E-mail: gorvett@uiuc.edu |
Course website: http://www.math.uiuc.edu/~gorvett/m595s08/home.html
(See below for class summaries, links to readings, and
suggested research topics)
Course Summary
This graduate mini-course will survey the field of
financial mathematics. After a brief review of relevant probability theory and
an introduction to several specific types of financial instruments (primarily
financial derivatives, including forwards, options, and swaps), we will begin
by examining financial modeling in a discrete-time framework. This framework
primarily involves the use of binomial trees. Emphasis will be given to the
understanding and implications of the concepts of probability measures,
no-arbitrage, and risk-neutrality.
This discrete-time perspective then leads, motivationally
and mathematically, to the continuous-time framework. We will examine the
application of stochastic calculus and Brownian motion to the modeling of
financial processes, and to the evaluation of financial derivatives, including
the Black-Scholes model. Also examined will be
mathematical approaches to the modeling of the term structure of interest
rates.
Textbooks
No specific text will be used or required. However, possible useful references might include any of the following:
Prerequisites
There are no formal prerequisites, other than graduate standing in mathematics or approval of the instructor. Some exposure to probability theory is assumed.
Grading
The course grade will be based upon a final project involving a topic in financial mathematics. Possible topics, and specific requirements associated with this research project, will be identified and discussed during the course.
Accessibility Statement
To insure that disability-related concerns are properly addressed throughout the semester, students with disabilities who require reasonable accommodations to participate in this class are asked to contact me within the first week of class.
Cumulative List of Some Suggested Research Topics
Here are some of many possible research topics for class papers:
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Topic |
Brief Description |
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Correlations among economic and financial variables |
Examine historical evidence of relationships between various economic and financial random variables, and explore ways, and the impact, of modeling those correlations to estimate future values of those variables. |
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Implications of a behavioral finance framework |
Explore the implications of findings in behavioral finance on the mathematical modeling of financial and economic variables. |
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Non-normality of financial and economic processes |
Analyze historical data for potential non-normality of changes in process values, and explore the implications of non-normality on the modeling of financial and economic variables. |
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Actuarial Research Clearing House |
Papers and abstracts (particularly under subject heading Finance) from the 2007 Actuarial Research Conference. Might provide some food for thought. |
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Research Experience for Undergraduates |
Sponsored by the National Science Foundation, this REU was given at UIUC in Summer 2007. Here is a summary paper, with several of the student projects dealing with financial mathematics. Might provide some idea seeds. |
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Sensitivity testing of binomial pricing model parameters |
Build a binomial option pricing model (in Excel or another software). Use it to test the sensitivity of option prices to changes in parameters (for example, the volatility of stock price or interest rate movements). |
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Sensitivity testing of binomial tree construction techniques |
Build a binomial option pricing model (in Excel or another software). Use it to test the sensitivity of option prices and underlying asset price evolution to changes in construction frameworks (for example, binomial model versus Cox-Ross-Rubenstein approaches). |
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Binomial trees evaluated under different behavioral paradigms |
The evaluation of American options or callable bonds in a binomial pricing framework is traditionally performed under simple monetary more-is-better behavioral assumptions (e.g., the owner of the American option will always exercise at an intermediate node if the exercise-now payoff is greater than the expected discounted future payoff value). This project would consider the impact on option pricing of possible different behavioral paradigms that might alter that decision (e.g., laziness, fuzziness, risk-taking). |
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Analysis and evaluation of various measures of risk |
Compare, contrast, and evaluate different measures of risk. Perhaps propose your own measure, along with a mathematical evaluation of the coherence (see articles here and here) of your proposed measure. |
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Evaluate value-at-risk (VaR) and consider its sensitivity to underlying assumptions |
Evaluate the adequacy of VaR (wikipedia article here), and consider how changes in assumptions (e.g., the stochastic process followed by an asset) will mathematically impact the VaR measure. |
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Computational modeling of interest rates and other economic variables |
Use a financial model (link here, including report sections, and the model itself at the Appendix D link) to explore the sensitivity of financial scenario modeling to different assumptions and parameters. |
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Develop a risk measure via selected axioms |
Choose axioms which you think should apply to a risk measure (which might be different that the coherence properties). Examine the resulting risk measure, its structure and its properties. |
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Regime switching approach to financial or economic time series |
Using the Hardy regime switching paper as a basis, reproduce part of the paper using a different financial or economic time series, testing the appropriateness of a regime switching model to describe that series. |
Class Summaries and Links to
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Date |
Class Topic |
Specific Concepts Covered |
Suggested Links and |
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Jan 15 |
Introduction and background |
Introduction to class. Probability and statistics background (probability space, statistical distributions). Financial background (guiding principles). |
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Jan 17 |
Financial background. Introduction to binomial option pricing. |
Derivatives and derivative markets. Option types and characteristics. Binomial trees for stock options. |
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Jan 22 |
Single period binomial option pricing model |
Replicating portfolios. Risk neutral framework. Risk neutral probabilities. Mathematical relationships of up and down stock movements. |
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Jan 24 |
Multiple period binomial pricing model |
Construction of binomial trees. European options on stocks. American options on stocks. |
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Jan 29 |
Multiple period binomial pricing model (cont.) |
Interest rate trees. Calibration. Callable bonds. |
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Jan 31 |
Continuous-time financial models |
Wiener process. Markov process. Martingales. Arithmetic and geometric Brownian motion. |
Convergence of Binomial to Black-Scholes (Don Chance, LSU) |
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Feb 5 |
Continuous-time financial models
(cont.) |
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Feb 7 |
Continuous-time financial models (cont.) |
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Feb 12 |
Interest rates |
Spot versus forward rates. Term structures and yield curves. |
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Feb 14 |
Interest rate modeling |
Historical behavior of interest rates. |
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Feb 19 |
Interest rate modeling (cont.) |
Models of interest rates, including Vasicek, CIR, HJM. |
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Feb 21 |
Risk measures |
List of possible measures. Payoff versus loss random variables. Coherence properties. |
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Feb 26 |
Risk measures (cont.) |
Mathematical descriptions. Value at Risk subadditivity problem. VaR techniques. |
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Feb 28 |
Modeling stock returns |
GBM/Lognormal. AR, ARCH, and GARCH. Regime-switching. |