- Summer 2014 Projects
**Estimation of Option Prices with Stochastic Volatility Models**

Dr. Hui Gong

prerequisites: a course in statistics; experience with MATLAB and R is highly expected.An option gives you a right to buy or sell an asset in a future time at a contract price. However, you need to purchase this right. So how much should you pay to buy this option in that you still can profit in the future? For any asset, its prices are time series data and can be modeled by different time series models. Recently there has been a growing interest in using stochastic volatility models in pricing options. This project is to study several stochastic volatility models. The research will include: deriving option pricing formulas for these stochastic volatility models; estimating the parameters in these models with the assistance of software MATLAB and/or R; and comparing the pricing outcomes. The project will also use simulation-based technique to investigate the correlations between the option pricing and the parameters of the stochastic volatility models.

[1]. Black, F., and Scholes, M., (1973): The pricing of options and corporate liabilities. Journal of Political Economy, 81(3),637-654.

**The Influence of Risk-taking Behavior on the Evolution of Infectious Diseases**

Dr. Daniel MaxinPrerequisite: differential equations

We study several epidemic models where the infection transmission term includes a risk-taking behavior component. By this component we mean the totality of measures that an individual will undertake in order to either avoid or reduce his/her chances of infection if the disease is perceived to be dangerous or to increase the exposure risk if the disease is perceived as mild. The main objective is to study the influence of risk-taking behavior in a variety of models (both one-sex or gender-structured). Of particular importance is investigating whether this behavioral component may change the course of the epidemic from disease clearance to persistence. A more detailed description of this project can be accessed

.*here***Pattern Avoidance in Double Lists**

Dr. Lara PudwellPrerequisites: at least one of the following: discrete mathematics, combinatorics, programming experience

A permutation of length n is an arrangement of the numbers {1,...,n}. There are n! permutations of length n (for example, the 3! permutations of {1,2,3} are 123, 132, 213, 231, 312, and 321), but it is more interesting to answer questions of the form "How many permutations have (a given property)?"

Pattern-avoiding permutations have been a popular and increasingly useful area of study over the past few decades. Permutation p (of length n) contains permutation q (of length m) if there exist m digits of p that are in the same relative order as the digits of q. For example, p=15243 contains two copies of q=312 because of the digits of 524 (resp. 523), like 312, are in the order large, small, medium. If p doesn't contain q than p is said to avoid q.

The number of permutations of length n avoiding 12 (resp. 21) is 1. The number of permutations of length n avoiding 123 (or any other pattern of length 3) is given by binomial(2n,n)/(n+1). Many other variations and generalizations have been studied that produce nice enumerative (counting) results.

In this project we will look at special lists of the numbers {1,...,n} that contain 2 copies of each digit, and determine the number of such lists that avoid a given pattern.