Information Theory and Applications to Finance, Fall 1998
The lectures will cover the theory of
relative entropy in statistics and information theory
and its applications to theoretical and applied Finance.
The course is divided into 3 parts:
information theory with emphasis on statistical estimation (3 lectures)
investing (maximizing growth of wealth) and recent generalizations that
take into account risk-aversion (Bielicki and Pliska) (2 lectures)
for calibrating asset-pricing models using stochastic control and optimization (8 lectures).
The course is intended for graduate students in mathematics, computer
science, economics or finance, and for Wall Street quantitative analysts who
want to explore the use of entropy, stochastic control and optimization techniques
for calibrating asset models (for options, interest rates, credit?).
A strong level of mathematical proficiency is assumed, as well as knowledge of asset pricing theory
(e.g. Mathematics of Finance I or II).
Grades will be assigned by completing three take-home assignments, involving
theoretical questions and computer programming.
The material will be drawn from Cover and Thomas: Elements of Information
Theory and from journal papers.
Names before the title of a lecture indicate the primary bibliographical source.
When there are no names, the material will be drawn from my personal notes
and I will give additional references in class.
1. Cover and Thomas
Elements of Information Theory
Chapter on Entropy and Relative Entropy
2. Cover and Thomas
Chapter on Large Deviations and Entropy
3. Cover and Thomas, Buchen and Kelly
Statistical Estimation and Entropy,
Estimation of pricing measure from option prices in a single period model
4. Cover and Thomas, E. Thorp Kelly investing
- 5. Bielicki and Pliska
- Risk-Averse growth-rate optimization.
- 6. Elements of stochastic control theory and zero-viscosity limit of
quasi-linear Hamilton-Jacobi-Bellman PDEs
- 7. Avellaneda
- Entropy algorithms for calibrating asset-pricing models.
- 8. Application to modeling forward rates. Least-squares vs. constrained
minimization, higher-order regularizations, cubic splines, etc.
- 9. Numerical methods and stability analysis for HJB equations and
- 10. Rubinstein, Dupire, Shimko, Derman, Kani
- Estimation of volatility surfaces using
the ``implied tree'' approach.
- 11. Avellaneda, Friedman, Holmes and Samperi
- Calibration of volatility
surfaces via relative-entropy minimization.
- 12. Term-structure models entropy
algorithm for simultaneous estimation
of volatility and drift surfaces. Interest-rate options.
- 13. Exploring entropy
algorithms for jump-diffusion processes & conclusion.