# Background

Financial engineering is a multidisciplinary field that involves the use of mathematical techniques and analyses to solve financial problems. It uses a variety of tools such as statistics, concepts of economics, computer science, and applied mathematics to help address the financial market, present and future. Financial engineering is primarily used as an analysis technique for financial corporations such as corporate banks, hedge funds, and investment banks.

Risk is an important concept in Financial Engineering. It can be defined as the amount of exposure to loss. Investments are generally made in anticipation of a positive return but the value of this return is uncertain, unpredictable and potentially unfavorable. The practice of financial engineering uses tools to determine a set of decisions that minimize future risk.

# Problem

Normality is assumed in many problems initially as the distribution is understood comprehensively through many prior studies and the foundational central limit theorem infers that for a sufficiently large data-set the mean is distributed normally. This enables us to draw effective conclusions from a data-set and explain the observations using well-defined principles.

The normal distribution assumption is so prevalent in many domains including financial asset pricing models of S&P 500. The normal distribution is often utilized because of its universal occurrence in many phenomena as well as its relative simplicity. A key feature of its simplicity is the three-sigma rule specifying that virtually all of the values are constrained within three standard deviations of the distribution’s mean.

S&P 500 data is assumed to be normal because analysis and plotting of the sufficiently large data set actually shows that most of the price movements are described adequately by the normal distribution (bell-shaped curve). However, the key problem with applying this assumption with S&P 500 data is that the exceptional price movements in the distribution are not as extremely infrequent as assumed with a data-set explained by normal distribution.

# Objective

The quest for reliable financial modeling techniques has increased in response to the highly volatile and seemingly unpredictable nature of the financial markets. Large losses and returns occur more frequently than predicted under the assumption of normality.

The aim of this project is to develop a fat-tailed distribution model that fits the S&P500’s daily percent returns to be applied in standard financial calculations.

# Sponsor: Dr. Kuo Chu Chang

Dr. Chang’s technical interests are mostly in the areas of Bayesian inference and decision theory, multi-target tracking, multi-source data fusion, situation assessment, and uncertainty in artificial intelligence. For more than thirty years, Dr. Chang has conducted research on wide range of distributed data fusion and probabilistic inference under uncertainty.

Dr. Chang received his M.S. and Ph.D. degrees in Electrical Engineering from the University of Connecticut in 1983 and 1986 respectively. From 1983 to 1992, he was a senior research scientist in Advanced Decision Systems (ADS) division, Booz-Allen & Hamilton. In 1992, he joined the Systems Engineering and Operations Research department, George Mason University, where he is currently a professor. He is also the Director for the Sensor Fusion Lab at George Mason University.