## Theses

#### Abstract

Since the time of the Black-Scholes model published in 1973, the research about mathematical finance models has never stopped. The original Black-Scholes model is for stock and stock derivatives pricing. However, stock derivatives is not the only kind of financial instrument in the market. Fix income derivatives also plays a very important role in the financial market, appealing to many researchers to explore more about their pricing model.

The fundamental theory of Black-Scholes is still employed in the pricing model for fix income derivatives, but there is something else making the research even more complicate: the definition function for the risk neutral interest rate. Like the stock price, part of the risk neutral interest rate also follows Brownian Motion, but still keeps certain term structure as the basic property of interest rate. There are many famous models in history to determine the risk neutral interest rate, but they have some disadvantages in estimating the spot interest rate. In this paper, we will use the historical data to build a spot neutral interest rate estimation model that can give us more accurate information about the imbalance of the fix income derivative prices.

In this research, we use the yield to maturity of the Treasury bonds as our target, and collect the 10 years data of all kinds of Treasury bonds from Jan 3rd, 1994 to Dec 31st. Then we take part of the data which comes from a period when the economy was relatively stable to conduct the data analysis. Then we notice that the change of the interest rate has the shape of its graph asthe intersection of two parabolas with opposite directions. Based on this discovery, we build our model and test it with the other part of data from our collection, and our model turns to work well.

To verify the accuracy of the model, we use the built-in model in MATLAB which is based on the similar theory of ours to do a model comparison. The result of the comparison shows that our model works better than the model in MATLAB.

The spot interest rate estimation model in this research gives a new way to describe the properties of interest rate, and also give a more accurate estimation about the future interest rate. The bond, or fix income derivative, pricing model based on this interest rate model should be able to help investors to make better decisions from a new point of view.

#### Library of Congress Subject Headings

Interest rates--Mathematical models; Finance--Mathematical models

8-13-2014

Thesis

#### Degree Name

Applied and Computational Mathematics (MS)

#### Department, Program, or Center

School of Mathematical Sciences (COS)

Bernard Brooks

Nathan Cahill

Raluca Felea

Physical copy available from RIT's Wallace Library at HG6043 .L8 2014

#### Campus

RIT – Main Campus

ACMTH-MS

COinS