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\title{Interest Rate Modeling}
\author{Jaime Frade}
\date{\today}

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\frame{\titlepage}


\section[Outline]{}
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\section{Objective}
\frame
{
The main focus of this project is to create a model for 3 month LIBOR interest rates.  This will be modeled three different ways
  \begin{itemize}
  \item<1-> Regression Time Series
  \item<2-> ARIMA
  \item<3-> Cox, Ingersoll, Ross (CIR) stochastic model.      
  \end{itemize}

}
\section{Purpose}
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{
	The three models will be accessed and a short rate will be forecasted with some error.    The forecasted rate will be used produce value of a European call option, with $S$, spot rate and ,$K$, strike price.   The payoff for this call option is 
	\[
	\max[(S-K], 0) 
\]
	
	
}
\section{Background}
\frame
{
	The process for the short rate models a rate in a risk-neutral world.  Interest rate behavior is similar to the behavior of stocks; however, rates appear to be pulled back to some long-run average level over time.  This mean reversion is taken into consideration in only the CIR model.
}




\section{Data}
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{
Interest rates were collected on a daily basis for the past three years.  Data was obtained from the Federal Reserve website. 
}
\subsection{3 month LIBOR}
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{
\pdfimage width 8cm {libor.png}
}
\subsection{3 month T-rate}
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{
\pdfimage width 8cm {3Trate.png}
}
\subsection{30 year T-rate}
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{
\pdfimage width 8cm {30Trate.png}
}

\section{Regression Time series}
\frame
{To avoid multicollinearity, I regressed several models to predict either the current 3 month LIBOR rate or changes LIBOR rate.  I decided to use a 1 step lag in the 3 month T-rate to predict Libor.  

\\
\textbf{Model:}
	\[
	Y = \beta_0 + \beta_1*(X_{t-1})
\]
}

\subsection{Forcasted value}
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{
Using a value of $1.5\%$, I predicted the 3 month LIBOR rate to be $4.55\%$, with a $95\%$ confidence level $(4.518720\%, 4.585424\%)$
\hspace{.25cm}
\\
}


\section{ARIMA}
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{
ARIMA processes are just integrated ARMA processes. In other words, a process is ARIMA of order d if its d-th derivative is ARMA. The model can be written
	\[
	\phi(B)\,(1-B)^d X_t = \theta(B) Z_t
\]
}
\subsection{Forcasted value}
}
\section{CIR}
\frame
{
	A risk neutral process for $r$ which contains a mean reverting drift as well as way to model non-negative rates.  The change in the short rate in a short period of time is proportional to $\sqrt{r}$.
	\[
	dr = a(b-r)dt + \sigma \, \sqrt{r} \, dz
\]
}
\subsection{Forcasted value}
\frame
{
Using Monte Carlo simulations and other assumptions about the parameters of the model, the following value for the LIBOR rate was forcasted. $4.5831179\%$
}

\section{European Call option}
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{
RTS model: Cost of 0.045600363
\\
CIR model: Cost of option is 0.045892182
}


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