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:

1. General information theory with emphasis on statistical estimation (3 lectures)
2. Kelly investing (maximizing growth of wealth) and recent generalizations that take into account risk-aversion (Bielicki and Pliska) (2 lectures)
3. Algorithms 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
ibid 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 the linearizations.
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.