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\begin{document}
%slide one: Lookback put option Problem
\textbf{Recall:}  \underline{Definition 1.3}:  European floating strike lookback put option

\begin{equation}
Payoff = \max(M-S_T, 0) 
\end{equation}

where $S_T$ is the final stock price and $M = \max_{0 \leq \tau \leq\,T}S_{\tau}$.

Let $P=P(S,M,t)$ stand for the option price, which satisfies 
\begin{equation}\label{eq:model}
 \left\{ \begin{array}{l l}
\dfrac{\partial P}{\partial t} +  \dfrac{\sigma^2}{2}S^2\dfrac{\partial^2 P}{\partial S^2}  + (r_d - r_f)S\dfrac{\partial P}{\partial S} - r_dP = 0,  &  \textrm{$0 \leq S \leq M$ \hspace{.05cm} and \hspace{.05cm}$0\leq t <T$},
\\
\;
\\
P(S,M,T)= M-S 
\\
\;
\\
\dfrac{\partial P}{\partial M}(M,M,t) = 0 
\end{array} \right.
\end{equation}

where $r_d>0$ and $r_f>0$ represent the risk-free domestic rate and foreign rate, respectively.


With the transformation:
\begin{equation}
x= \ln \dfrac{M}{S} \hspace{.5in} V(x,t)= \dfrac{P(S,M,t)}{S},
\end{equation}

Then (\ref{eq:model}), can be reduced to 
\begin{equation}\label{eq:one}
 \left\{ \begin{array}{l l}
\dfrac{\partial V}{\partial t} +  \dfrac{\sigma^2}{2}\dfrac{\partial^2 V}{\partial x^2}  + \left(r_d - r_f-\dfrac{\sigma^2}{2}\right)\dfrac{\partial V}{\partial x} - r_fV = 0,  &  \textrm{$x > 0$ \hspace{.05cm} and \hspace{.05cm}$0\leq t <T$},
\\
\;
\\
V(x,T)= e^x-1 
\\
\;
\\
\dfrac{\partial V}{\partial x}(0,t) = 0 
\end{array} \right.
\end{equation}




\newpage

%slide 2: binomial tree model

\begin{equation}
u = e^{\sigma\sqrt{\Delta t}} \hspace{.5cm} d = \dfrac{1}{u} \hspace{.5cm}  K_0 = \dfrac{M}{S} = 1
\end{equation}

\begin{equation}
u^j = \dfrac{M_n}{S_n}  \hspace{.5cm}  V^N_j = u^j-1
\end{equation}

\begin{equation}
p=\frac{\rho e^{-r_f \Delta t}-d}{u-d}
\end{equation}


\begin{equation}
K_2= 1 = u^0 \hspace{.5cm} K_2 = u^1 \hspace{.5cm} K_2 = \dfrac{S}{d^2S} = u^2  \hspace{.5cm} u^3  \hspace{.5cm} u^4
\end{equation}



$p$ $1-p$ $$p=\frac{\rho e^{-r_f \Delta t}-d}{u-d}$$ $\rho=e^{r_d \Delta t}$ 
$V_0^n=\frac{1}{\rho} \left( puV_0^{n+1} + (1-p)dV_1^{n+1}\right)$ $for \ j=0$
$V_j^n=\frac{1}{\rho} \left( puV_{j-1}^{n+1} + (1-p)dV_{j+1}^{n+1}\right)$ for $j>0$

\begin{equation}
\rho=e^{r_d \Delta t}
\end{equation}



\end{document}