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\begin{document}
Name:  \underline{~~~Jaime Frade~~~}
\begin{center}
Financial Homework 6.3  (Already turned in 6.1-6.2 in mailbox)
\\
\end{center}
\begin{enumerate}
\item{6.3}  Verify the following:

\begin{equation}\label{eq:ques}
d_tP(t,T) = P(t,T)[\Sigma(t,T)d_t\widetilde{W_t}+ r_t\,d_t] 
\end{equation}

Given the following, will use to verify (\ref{eq:ques}):
\begin{eqnarray}
\widetilde{W_t} & = & W_t + \int^T_t \gamma_sds \label{eq:w}
\\
\Sigma(t,T) & = & -\int^T_t\sigma(t,u)du \label{eq:sum}
\\
Z(t,T) & = & \beta^{-1}_tP(t,T) \label{eq:SDE}
\\
 &  & Z(t,T) \textrm{is the SDE of the discounted bond, where}  \nonumber
\\
 &  &  \beta_t \Rightarrow \textrm{cash bond and} \nonumber
\\
 &  &  P(t,T) \Rightarrow \textrm{price of the T-maturity bond.}\nonumber
\\
d_tZ(t,T) & = & Z(t,T)\Sigma(t,T)d\widetilde{W_t} \label{eq:dbond}
\\
 &  & \textrm{for discounted bond under under} \hspace{.15cm} \mathbb{Q}
\end{eqnarray}

Since $\beta_t$ is a zero-vaildity process satisfying SDE, 
\begin{equation}
d\beta_t = r_t\beta_tdt, \label{eq:SDE2}
\end{equation}

Using above, will verify (\ref{eq:ques}) by using the product rule and simplify using above equations.

\begin{eqnarray}
d_t(P(t,T)) & = & d_t(\beta_t\,Z(t,T)) \label{eq:1-1}
\\
&  & \textrm{applying the product rule obtains}
\\
& = & \beta_t\,\underbrace{d_tZ(t,T)}_{(\ref{eq:dbond})} + Z(t,T)\underbrace{d_t\beta_t}_{(\ref{eq:SDE2})}
\\
& = & \beta_t\,Z(t,T)\Sigma(t,T)d\widetilde{W_t} + Z(t,T)r_t\beta_tdt
\\
& = & \beta_tZ(t,T)(\Sigma(t,T)d\widetilde{W_t} + r_tdt)
\\
& = & P(t,T)(\Sigma(t,T)d\widetilde{W_t} + r_tdt) 
\end{eqnarray}
$\therefore$
\end{enumerate}
\end{document}