# CSI 779 / STAT 789

## Statistical Modeling of Financial Data

#### 207 Innovation Hall

Instructor: James Gentle; email: jgentle@gmu.edu

This course will cover a variety of methods of computational statistics in the analysis of financial data. The emphasis will be on the mathematical models, the statistical methods, and the computations, rather than on topics in the domain of finance.

Many of the standard results in finance rely on simplifying assumptions about the distribution of random components. These results can be examined by Monte Carlo methods, and can be modified by bootstrapping.

#### Prerequisites

While some background in finance would be useful, it will not be necessary. Some knowledge of statistical theory and methods (roughly equivalent to STAT 554 and STAT 652) is a prerequisite. Knowledge of advanced calculus and differential equations is also required.

#### Software

No particular software package will be required. The main software I use is R/S-Plus, but Matlab and other packages can also be used for the assignments and the project. However, students are encouraged to obtain and use R so that the exercises and discussions in class will be easier to follow.

There are a number of useful books on R. A list of books is available at the "Books" link on the main webpage for the R Project.

Frank Harrell has a very useful website on S/R resources. One of the links at that site is to a very useful introductory manual on S (and R).

#### Data

Data, of course, are the raw materials of any statistical analyses. There is a wealth of easily accessible financial data. Traders need timely data. Persons studying fine structure of price movements require intraday data or even ticker data. For this course we need neither timely data nor intraday day. We will generally be interested in daily, weekly, or monthly closing prices and the volume corresponding to that period. An easy source of the kind of data we need is Yahoo.

Price and volume data can be obtained by entering the symbol. Symbols for indices begin with ^; for example, the symbol for the Dow Jones Industrial Average is ^DJI; for the S&P 500, it is ^SPX; for the Nasdaq Composite, it is ^IXIC; and for the CBOE Volatility Index, it is ^VIX.

Prices and open interest in exchange-traded options can also be obtained at this Yahoo site.

### Topics

• Mathematical preparations
• Basic probability theory
• Stochastic processes
• Stochastic calculus
• Jump processes
• Pricing
• Arbitrage
• Present value
• Derivatives
• Pricing models
• Effect of jumps
• Path dependency
• Scenario generation
• Monte Carlo methods

Performance in the class will be evaluated based on

• an in-class midterm (25%)
• a final exam consisting of a take-home portion and an in-class portion (35%)
• a project to evaluate/compare derivative pricing models (30%)
• a number of smaller assignments (10%)

Students may discuss and otherwise collaborate on the project and the homework, but what is submitted for grading must be written by the individual students.

Each student will prepare a web page for presentation of the project and for some of the smaller assignments.

### Texts and References

There are a number of useful books on various topics that together comprise "computational finance", or "financial engineering".

#### Main text

The text is Quantitative Methods in Derivatives Pricing: An Introduction to Computational Finance, by Domingo Tavella (2002).
The general flow of the course follows this text.

#### Models of derivative pricing

• Neil A. Chriss (1997), Black-Scholes and Beyond: Option Pricing Models, Irwin Professional Publishing, Chicago.
• John C. Hull (2005), Options, Futures, & Other Derivatives, sixth edition, Prentice Hall, Upper Saddle River, NJ.
• P. J. Hunt and J. E. Kennedy (2000), Financial Derivatives in Theory and Practice, John Wiley & Sons, New York.
• Yue-Kuen Kwok (1998), Mathematical Models of Financial Derivatives, Springer-Verlag, Singapore.
• Domingo Tavella and Curt Randall (2000), Pricing Financial Instruments; The Finite Difference Method, John Wiley & Sons, New York.
• Paul Wilmott (1998), Derivatives, John Wiley & Sons, Chichester, UK.
• Paul Wilmott, Sam Howison, and Jeff Dewynne (1995), The Mathematics of Financial Derivatives; A Student Introduction, Cambridge University Press, Cambridge, UK.

• Jean-Pierre Fouque, George Papanicolaou, and K Ronnie Sircar (2000), Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK.
• Gatheral, Jim (2006), The Stochastic Surface; A Practitioner's Guide, John Wiley & Sons, New York.
• Javaheri, Alireza (2005), Inside Volatility Arbitrage; The Secrets of Skewness, John Wiley & Sons, New York.

#### Summary of derivative pricing formulas

• Espen Gaarder Haug (1998), The Complete Guide to Option Pricing Formulas, McGraw-Hill, New York.

#### Probability theory, with an emphasis on stochastic processes

• Leo Breiman (1992), Probability, classic edition, SIAM, Philadelphia

#### Stochastic calculus

• Bernt Øksendal (1998), Stochastic Differential Equations; An Introduction with Applications, Springer-Verlag, Berlin.
• Steven E. Shreve (2005), Stochastic Calculus for Finance I;The Binomial Asset Pricing Model, Springer-Verlag, New York.
• Zeev Schuss (1980), Theory and Applications of Stochastic Differential Equations, John Wiley & Sons, New York.
• J. Michael Steele (2001), Stochastic Calculus and Financial Applications, Springer-Verlag, New York.

#### Computational methods

• Paul Glasserman (2004), Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York.
• Peter Jäckel (2002), Monte Carlo Methods in Finance, John Wiley & Sons, Chichester, UK.
• Peter E. Kloeden, Eckhard Platen, and Henri Schurz (1997), Numerical Solution of SDE Through Computer Experiments, Springer-Verlag, Berlin.
• Svetlozar T. Rachev, Editor, (2004) Handbook of Computational and Numerical Methods in Finance, Birkhäuser, Boston.

#### General reference on financial assets

• William F. Sharpe, Gordon J. Alexander, and Jeffery V. Bailey (1997), Investments, sixth edition, Prentice Hall, Upper Saddle River, NJ.