## Manipulation of the Greeks Within Black-Scholes

### Faculty mentor: Dr. Julius Esunge

In this research, we will look at derivatives as a function of accurately predicting risks and pricing. Both of which have the intention of either creating wealth monetarily, managing risks within respective industries, or in some fashions, a combination of both. Specifically, we will analyze the Greeks in the Black-Scholes equation and how they change the outcome of a call within the American markets. The project will involve analytical methods to derive and explain the usefulness of each of the Greeks. Also, statistical analysis will be performed on recent options data to show the practical aspects of each Greek. Mathematics, probability, and statistics are pivotal to accurately predicting both risks and pricing in these ubiquitous applications.

## Deterministic and Stochastic Models for HIV-1 Dynamics

### Faculty mentor: Dr. Leo Lee

In this research project, I investigated deterministic and stochastic versions of a model for Human Immunodeficiency Virus Type 1 (HIV-1) dynamics. First, the deterministic model is introduced, and numerical techniques are used to obtain an approximate solution to the system. Then, a stochastic model is developed from the deterministic system. Patient data is introduced, and the Monte Carlo Method is used to find an approximate solution to the stochastic system. The results of this project demonstrate the behavior of HIV-1 in an infected patient under the effects of reverse transcriptase and protease inhibitors, and the introduction of randomness to the system of equations allows us to account for the randomness that occurs biologically within the model, thereby making our results more biologically sound.

## The Geometry of Surfaces and its Applications Using Mathematica

### Faculty mentor: Dr. Yuan-Jen Chiang

We first introduce the concepts of surface theory including the coordinate patch, coordinate transformation, normal vector, tangent plane, etc. We next compute the first fundamental form of a surface: the matrix (gij) of metric coefficients, gij = xi x xj , the inner product with respect to the basis {x1, x2} of the tangent space of the surface. We then discuss the second fundamental form, Weingarten map (i.e., shape operator), Christoffel symbols, the geodesic, geodesic curvature, principal curvature, Gauss curvature, mean curvature, normal curvature, parallelism, etc. We will apply the proceeding terms to a few concrete examples by different calculations. We will utilize the software Mathematica to sketch various surfaces and their geometric properties. 