\PassOptionsToPackage{unicode}{hyperref}
\PassOptionsToPackage{hyphens}{url}
%
\documentclass[reqno,final]{amsart}
\usepackage{tcolorbox}
%\usepackage{authblk}
\newtcolorbox{mybox}{boxrule=0pt,toprule=1.2pt,bottomrule=1.2pt,width=\textwidth,arc=0mm,colback=gray!50}
\newtcolorbox{mybox1}{boxrule=0pt,width=\textwidth,arc=0mm}
\newtcolorbox{mybox2}{boxrule=0pt,bottomrule=1.2pt,width=\textwidth,arc=0mm}
\usepackage{fancyhdr}
\usepackage{amsmath,amssymb}
\usepackage{pstricks}
\usepackage{lmodern}
\usepackage{multirow}
\usepackage{lipsum}
\usepackage{caption}
\usepackage{subcaption}




\newcommand\blfootnote[1]{%
  \begingroup
  \renewcommand\thefootnote{}\footnote{#1}%
  \addtocounter{footnote}{-1}%
  \endgroup
}
\setcounter{page}{50}
\usepackage{longtable}
\usepackage{iftex}
\ifPDFTeX
  \usepackage[T1]{fontenc}
  \usepackage[utf8]{inputenc}
  % \usepackage{ebgaramond}
  \usepackage{textcomp} % provide euro and other symbols
\else % if luatex or xetex
  \usepackage{unicode-math}
  \defaultfontfeatures{Scale=MatchLowercase}
  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
\fi
% Use upquote, if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
\IfFileExists{microtype.sty}{% use microtype if available
  \usepackage[]{microtype}
  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
\makeatletter
\@ifundefined{KOMAClassName}{% if non-KOMA class
  \IfFileExists{parskip.sty}{%
    \usepackage{parskip}
  }{% else
    \setlength{\parindent}{0pt}
    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
}{% if KOMA class
  \KOMAoptions{parskip=half}}
\makeatother
\usepackage{xcolor}
\usepackage{longtable,booktabs,array}
\usepackage{calc} % for calculating minipage widths
% Correct order of tables after \paragraph or \subparagraph
\usepackage{etoolbox}
\makeatletter
\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
\makeatother
% Allow footnotes in longtable head/foot
\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
\makesavenoteenv{longtable}
\usepackage{graphicx}
\usepackage[top=2.5cm,right=2.3cm,bottom=2.5cm,left=2.3cm,head=.5cm,foot=.5cm]{geometry}
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
\makeatother
% Scale images if necessary, so that they will not overflow the page
% margins by default, and it is still possible to overwrite the defaults
% using explicit options in \includegraphics[width, height, ...]{}
\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
% Set default figure placement to htbp
\makeatletter
\def\fps@figure{htbp}
\makeatother
\usepackage[normalem]{ulem}
\setlength{\emergencystretch}{3em} % prevent overfull lines
\providecommand{\tightlist}{%
  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
\ifLuaTeX
  \usepackage{selnolig}  % disable illegal ligatures
\fi
\ifPDFTeX
  \TeXXeTstate=1
  \newcommand{\RL}[1]{\beginR #1\endR}
  \newcommand{\LR}[1]{\beginL #1\endL}
  \newenvironment{RTL}{\beginR}{\endR}
  \newenvironment{LTR}{\beginL}{\endL}
\fi
\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
\urlstyle{same} % disable monospaced font for URLs
\hypersetup{
  hidelinks,
  pdfcreator={LaTeX via pandoc}}

\author{}
\date{}

\begin{document}
\renewcommand{\headheight}{11pt}
\pagestyle{fancy} \fancyhf{} \fancyhead[RO,LE]{\small\thepage}
\fancyhead[RE]{\small\shortauthors} \fancyhead[LO]{\small\shorttitle}

\pagestyle{fancy}

\begin{mybox}
  \centering
  \LARGE{\fontsize{16}{24}\selectfont\textbf{Optimality}}
  \\[3mm]
  \href{}{\fontsize{12}{14}\selectfont{www.opt.reapress.com}}
  \\[1mm]
  \vspace*{-1.12cm}
  \hspace*{-3.75cm}
  \normalsize{\includegraphics[width=1.20711in,height=0.91358in]{image1.png}}
  \textbf{\fontsize{10}{12}\selectfont{Opt.Vol. 1, No. 1 (2024) 50–58.}}
  \vspace*{-.15cm}
\end{mybox}
\vspace*{-5cm}
\vspace*{5cm}

\textcolor{brown}{\selectfont{Paper Type: Original Article}}

% Title of paper
% first part for the header part
% second part for the title of the paper in the first page
\title[\textbf{LOEX Option: A Combination of Exchange Option and Lookback Option}
]
{
\textbf{LOEX Option: A Combination of Exchange Option and Lookback Option}
}   

% first author information
%\author[Ghasem Pour et al. \textbar Opt. 1(1) (2024) 50-58]{ 
\author[Ghasem Pour]{ 
\textbf{Rajabali Ghasem Pour }$^{1}$\includegraphics[width=0.12333in,height=0.12333in]{image2.png}}\blfootnote{\includegraphics[width=0.14333in,height=0.14333in]{image6.png}~Corresponding Author:S. Pourmohammad Azizi}

% Second author information

\author[Azizi]{
\textbf{S. Pourmohammad Azizi } $^{2,*}$}

\author[Waloo\textbar \ Opt.(1) (2024) 50-58]{
\textbf{Sajad Ahmad Waloo }$^{3}$}

% Authors corresponding affiliations 
\address{\scriptsize\textsuperscript{1}Department of Mathematics and Physics, University of Campania Luigi Vanvitelli, Caserta, Italy; rajabali.ghasempour@studenti.unicampania.it; \\
 \scriptsize\textsuperscript{2}Department of Mathematics and Computer Sciences, Allameh Tabataba’i University, Tehran, Iran;  21053004@mail.ntou.edu.tw; \\  \scriptsize\textsuperscript{3}Department of Electrical Engineering, National Taiwan Ocean University, Keelung, Taiwan; 11053075@mail.ntou.edu.tw;
}

\maketitle

% Paper citation information



\textbf{Citation:}
% \begin{table}[!ht]
% \begin{tabular}{l|l|l|}
% \cline{2-3}
%  $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$& {\fontsize{6.5}{24} \begin{tabular}[c]{@{}l@{}}\emph{Received: 14 March 2023}   \\ \emph{Revised: 21 May 2023} \\ \emph{Accepted:02/11/2023}\end{tabular}} & {\fontsize{6.5}{24}\begin{tabular}[c]{@{}l@{}}\emph{Ghasem Pour, R., \ Azizi, S., \& Waloo, S. (2024).\ LOEX option: a combination of} \\\emph{exchange option and lookback option.} \em{Optimality,1}\\ \emph{(1), 50-58}\end{tabular}} \\ \cline{2-3} 
% \end{tabular}
% \end{table}

\begin{table}[!ht]
\begin{tabular}{l|l|l|}
\cline{2-3}
 $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$& \begin{tabular}[c]{@{}l@{}}Received: 14 March 2023   \\ Revised: 21 May 2023 \\ Accepted:02/11/2023\end{tabular} & \begin{tabular}[c]{@{}l@{}}  Ghasem Pour, R., \ Azizi, S., \& Waloo, S. (2024). LOEX option: \\ a combination of exchange option and lookback option. 
 \\
 Optimality,1(1), 50-58 \end{tabular} \\ \cline{2-3} 
\end{tabular}
\end{table}

% \begin{table}[h!]
% \begin{tabular}{|p{6cm} |p{8cm}|}
% \hline
% \\
% \begin{minipage}[b]{\linewidth}
% \begin{quote}
% \emph{\scriptsize Received: 14 March 2023}
% \\
% \emph{\scriptsize Revised: 21 May 2023}
% \\
% \emph{\scriptsize Accepted:02/11/2023}
% \end{quote}
% \end{minipage}& \vspace*{-1.1cm}
%  \begin{minipage}[b]{\linewidth}
% \scriptsize
% Ghasem Pour, R., \ Azizi, S.,\& Waloo, S. (2024).\ LOEX option: a combination of exchange option and lookback option. \emph{Optimality,1}(1), 50-58.
% \end{minipage} \\
% \hline
% \end{tabular}
% \end{table}

\thispagestyle{empty}
\textbf{Abstract}

\begin{mybox2}
In this article, we consider modeling and pricing a combination of two options (Exchange option and Lookback option) that we call the LOEX option. It is a type of exotic option, or clearer, path-dependent option because its price depends on the maximum and minimum prices of the assets to be considered. In reality, we introduce a conditional claim on two assets: the holder can change the first asset with the highest price to the second with the lowest price or cancel the transaction at the strike time. In this paper, after describing and modeling this option, we use numerical methods to estimate the pricing using the MATLAB software and present the results.

\vspace*{.5cm}
\textbf{Keywords:} Exchange option, Lookback option, Black–Scholes Option pricing, Mesh-free method. 
\end{mybox2} 

\blfootnote{\includegraphics[width=0.14333in,height=0.14333in]{image4.png}}
\blfootnote{\includegraphics[width=0.28333in,height=0.28333in]{image5.png}~Licensee System Analytics. This  article is an open-access article distributed under the terms and conditions of the Creative}
\blfootnote{\quad\quad\ \  Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0).}
%\hypertarget{section}{%
%\section{}\label{section}}
%
%\hypertarget{section-1}{%
%\section{}\label{section-1}}
%
%\hypertarget{section-2}{%
%\section{}\label{section-2}}
%
%\hypertarget{section-3}{%
%\section{}\label{section-3}}

{\fontsize{16}{24}\selectfont \hypertarget{introduction}{%
\section{\texorpdfstring{1\textbar Introduction
}{1\textbar Introduction }}\label{introduction}}}


Finance is a huge world. One part of this world is financial instruments, and there are a lot of papers about modeling the pricing of these instruments. In this paper, we consider introducing the” LOEX option.” The LOEX option comes with a combination of two options. We designed this security by combining the Exchange option and the Lookback option. In other words, this option comes from applying the Lookback option on the Exchange option. The LOEX option is a kind of conditional claim focusing on two assets that consider one with the highest price and the other with the lowest. 


The research background in this area is generally divided into two parts because there is no research directly related to this option: 
{\fontsize{14}{24}\section*{Exchange Option}}
The Exchange Option is one of the interesting instruments provided by Margrabe (1978)\cite{marg}. Several studies have been carried out on the modeling of this security, which can be cited by El Karoui et al. (1995), Antonelli et al. (2010), Cheang and Chiarella (2011), Kim and Koo (2016), and Wang(2016)\cite{el, anto, chea, kim1, fagh, wang} . El Karoui et al. obtained a close formula for the Exchange Option using the change of numeraire, and Kim and Koo 
provided a close formula for the state of the two assets being the same drift using a Mellin transform. Antonelli et al. worked on this option considering the other structure of Volatility. Cheang and Chiarella, as well as Wang, modeled the Jump diffusion for this structure. 




{\fontsize{14}{24}\section*{Lookback option}}


The first studies in the context of the Lookback option were Goldman et al. (1979) and Conze and Viswanathan (1991) \cite{gold, conz}research, which provided the exact solution for the floating and fix strike model. In this regard, Dai et al. (2004) \cite{dai} provided a close formula for quanto Lookback options. He et al. (1998) \cite{he} have priced it using numerical methods and Monte Carlo by obtaining a joint density function. Jeon and Yoon (2016) \cite{jeon} also got a closed formula for put-back options using the Mellin transform. 
  

As we said, in this paper, we are combining the Exchange option and the Lookback option to provide a new option called the LOEX option, and for the first time in this study, modeling and pricing it. In addition to being a risk hedging instrument, this financial derivative can have another function; due to its structure, we can use it as a market stimulus. For example, this option can be used to move the ask queues of an asset. 
  

To achieve the intended purpose, we will introduce the Exchange and Lookback options in the next section and provide the price equation for each. In Section 3, we will use the mathematical concepts of LOEX option modeling, develop a PDE for its price model, and provide a way to solve the PDE. Finally, in Section 4, we will run the model with MATLAB software, compare it with the Exchange option price, and show the results. 



{\fontsize{16}{24}\hypertarget{introduction}{%
\section{\texorpdfstring{2\textbar Preliminary Concepts
}{2\textbar Preliminary Concepts }}\label{Preliminary Concepts}}}


In this section, we consider to provide the instruments that made this idea come to our mind. To start, we suppose that $(\Omega, F, Q)$ is the probability measure space where $\Omega$ is a sample space, $F$ is an event space and $Q$ is a probability function and we work in this space. In the following, we will define the Exchange and Lookback options and show the price equation. To this end, we consider the following two risky assets: 

\begin{equation}
\begin{array}{l}
dS = rSdt + {\sigma _1}Sdw_t^1,\\
d{S^*} = r{S^*}dt + {\sigma _2}{S^*}dw_t^2,
\end{array}
\end{equation}
where $r$ is the interest rate that is drift of these equations and ${\sigma _i},i \in \{ 1,2\} $ are the volatility of the assets. 





{\fontsize{14}{24}\hypertarget{Exchange Option}{%
\section{2.1\textbar Exchange Option}\label{Exchange Option}}}

An exchange option is a conditional claim that the holder can convert assets to another asset at strike time or NOT. Assume that $C(t, S, S^* )$ is the Exchange option price at time $t$ with the price of assets $S$ and $S^*$ and the payoff function is as follows: 
\begin{equation}
C(T,S(T),{S^*}(T)) = {(S(T) - {S^*}(T))^ + }.
\end{equation}
Now, using the Feynman–Kac formula: 
\begin{equation}
\frac{{\partial C}}{{\partial t}} + rS\frac{{\partial C}}{{\partial S}} + r{S^*}\frac{{\partial C}}{{\partial {S^*}}} + \frac{1}{2}\sigma _1^2S_{}^2\frac{{{\partial ^2}C}}{{\partial S_{}^2}} + \rho {\sigma _1}{\sigma _2}S{S^*}\frac{{{\partial ^2}C}}{{\partial S\partial {S^*}}} + \frac{1}{2}\sigma _2^2{S^{{*^2}}}\frac{{{\partial ^2}C}}{{\partial {S^{{*^2}}}}} - rC = 0
\end{equation}
which $\rho dt = dw_t^1dw_t^2$ and its terminal condition is as follows: 
\begin{equation}
C\left( {T,S,{S^*}} \right) = {({S^*} - S)^ + }.
\end{equation}
The closed formula for the Exchange option according to PDE (3) is as follows \cite{kim1} 
\begin{equation}
C(t,{S_1},{S_2}) = {S_1}\Phi (d) - {S_2}\Phi (d - \sigma \sqrt {T - t} ),
\end{equation}
where
\begin{equation}
\begin{array}{l}
d = \frac{{ln\left( {\frac{{{S_1}}}{{{S_2}}}} \right)}}{{\sigma \sqrt {T - t} }} + \frac{{\sigma \sqrt {T - t} }}{2},\\
\sigma  = \sqrt {\sigma _1^2 + \sigma _2^2 - 2\rho {\sigma _1}{\sigma _2}} ,\\
\Phi (x) = \int\limits_{ - \infty }^x {{e^{ - 0.5{x^2}}}dx} .
\end{array}
\end{equation}


{\fontsize{14}{24}\hypertarget{Lookback Option}{%
\section{2.2\textbar Lookback Option}\label{Lookback Option}}}


  
The Lookback option is an exotic option that depends on the asset price path. In general, the Lookback option is divided into four categories. In this section, we summarize everything and present the price equation. Consider the following symbols: 
\begin{equation}
\begin{array}{l}
M(S(t),t) = \mathop {\max }\limits_{0 \le \omega  \le t} S(\omega ),\\
m(S(t),t) = \mathop {\min }\limits_{0 \le \omega  \le t} S(\omega ).
\end{array}
\end{equation}
Now we have Lookback options and their price equations \cite{umeo} 
\subsubsection*{Floating strike Lookback call option}
\begin{equation}
\begin{array}{l}
payoff \to S(T) - m(S(T),T)\\
Eq. \to S\Phi \left( {\frac{{\ln \left( {\frac{S}{{m(S,t)}}} \right) + \left( {r + \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right) - m(S,t){e^{ - r(T - t)}}\Phi \left( {\frac{{\ln \left( {\frac{S}{{m(S,t)}}} \right) + \left( {r - \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right)\\
 - \frac{{S\sigma _1^2}}{{2r}}\Phi \left( {\frac{{\ln \left( {\frac{{m(S,t)}}{S}} \right) - \left( {r + \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right) + {e^{ - r(T - t)}}\frac{{S\sigma _1^2}}{{2r}}\Phi \left( {\frac{{\ln \left( {\frac{{m(S,t)}}{S}} \right) + \left( {r - \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right).
\end{array}
\end{equation}
\subsubsection*{Floating strike Lookback put option}
\begin{equation}
\begin{array}{l}
payoff \to M(S(T),T) - S(T)\\
Eq. \to  - S\Phi \left( { - \frac{{\ln \left( {\frac{S}{{M(S,t)}}} \right) + \left( {r + \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right) + M(S,t){e^{ - r(T - t)}}\Phi \left( { - \frac{{\ln \left( {\frac{S}{{M(S,t)}}} \right) + \left( {r - \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right)\\
 + \frac{{S\sigma _1^2}}{{2r}}\Phi \left( {\frac{{\ln \left( {\frac{S}{{M(S,t)}}} \right) + \left( {r + \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right) - {e^{ - r(T - t)}}\frac{{S\sigma _1^2}}{{2r}}{\left( {\frac{M}{S}} \right)^{\frac{{2r}}{{\sigma _1^2}}}}\Phi \left( {\frac{{\ln \left( {\frac{S}{{M(S,t)}}} \right) - \left( {r - \frac{{\sigma _1^2}}{2}} \right)(T - t)}}{{\sigma \sqrt {T - t} }}} \right).
\end{array}
\end{equation}
\subsubsection*{Fixed strike Lookback call option}
\[payoff \to MAX(M(S(T),T) - K,0)\]
At the time t, the price would be ${e^{ - r(T - t)}}E[\max (\max (M_0^t(S(t),t),M_t^T(S(T),T)) - K,0)],$
\begin{equation}
\begin{array}{l}
Eq. \to \left\{ {\begin{array}{*{20}{c}}
{H(t,S;K)}&{M_0^t(S(t),t) \le K}\\
{{e^{ - r(T - t)}}(M_0^t(S(t),t) - K) + H(t,S;M_0^t(S(t),t))}&{M_0^t(S(t),t) \ge K}
\end{array}} \right.,\\
H(t,S;N) = S\Phi ({d_1}) - N{e^{ - r(T - t)}}\Phi ({d_2}) + {e^{ - r(T - t)}}\frac{{S\sigma _1^2}}{{2r}}\left( {{e^{r(T - t)}}\Phi ({d_1}) - {{\left( {\frac{S}{N}} \right)}^{\frac{{ - 2r}}{{\sigma _1^2}}}}\Phi \left( {{d_1} - \frac{{2r\sqrt {T - t} }}{\sigma }} \right)} \right).
\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}
{d_1} = \frac{{\ln (\frac{S}{N}) + (r + \frac{{\sigma _1^2}}{2})}}{{\sigma \sqrt {T - t} }},\\
{d_2} = \frac{{\ln (\frac{S}{N}) + (r - \frac{{\sigma _1^2}}{2})}}{{\sigma \sqrt {T - t} }}.
\end{array}
\end{equation}

\subsubsection*{Fixed strike Lookback put option}
\[payoff \to MAX(K - m(S(T),T),0)\]
At the time $t$, the price would be ${e^{ - r(T - t)}}E[\max (K - \min (m_0^t(S(t),t),m_t^T(S(T),T)),0)],$
\begin{equation}
\begin{array}{l}
For\begin{array}{*{20}{c}}
{}
\end{array}K \le m_0^t(S(t),t)\\
Eq. \to K{e^{ - r(T - t)}}\Phi ( - {d_2}) - S\Phi ( - {d_1}) + {e^{ - r(T - t)}}\frac{{S\sigma _1^2}}{{2r}}\left( { - {e^{r(T - t)}}\Phi ( - {d_1}) - {{\left( {\frac{S}{K}} \right)}^{\frac{{ - 2r}}{{\sigma _1^2}}}}\Phi \left( { - {d_1} + \frac{{2r\sqrt {T - t} }}{\sigma }} \right)} \right),\\
For\begin{array}{*{20}{c}}
{}
\end{array}K \ge m_0^t(S(t),t)\\
Eq. \to {e^{ - r(T - t)}}(K + m_0^t(S(t),t)(\Phi ({d_2}) - 1)) - S\Phi ( - {d_1}),\\
\begin{array}{*{20}{c}}
{}
\end{array}\begin{array}{*{20}{c}}
{}
\end{array}\begin{array}{*{20}{c}}
{}
\end{array}\begin{array}{*{20}{c}}
{}
\end{array} + {e^{ - r(T - t)}}\frac{{S\sigma _1^2}}{{2r}}\left( {{e^{r(T - t)}}\Phi ( - {d_1}) + {{\left( {\frac{S}{{m_0^t(S(t),t)}}} \right)}^{\frac{{ - 2r}}{{\sigma _1^2}}}}\Phi \left( { - {d_1} + \frac{{2r\sqrt {T - t} }}{{{\sigma _1}}}} \right)} \right).
\end{array}
\end{equation}
$d_1$ and $d_2$ are the same as (11) formula. 

{\fontsize{16}{24}\hypertarget{LOEX Option}{%
\section{\texorpdfstring{3\textbar LOEX Option
}{3\textbar LOEX Option }}\label{LOEX Option}}}


In this section, we model and price the LOEX option using the concepts described in the previous section. As stated, this option is made by implementing the Lookback option on the Exchange option. The holder can replace an asset with the highest price from the start of the contract to maturity time with the lowest price of another asset in that period. Since the price of this option depends on the asset price path, this option is an exotic option. 

To begin, we consider formula (1) and the following symbols: 
\begin{equation}
\begin{array}{l}
M(S(t),t) = \mathop {\max }\limits_{0 \le \omega  \le t} S(\omega ),\\
m({S^*}(t),t) = \mathop {\min }\limits_{0 \le \omega  \le t} {S^*}(\omega ).
\end{array}
\end{equation}

Now assume $G(t, S, S^* )$ is the price of a LOEX option at time $t$ and the price of assets $S$ and $S ^*$, the payoff corresponding to this option is as follow: 
\begin{equation}
G(T,S(T),{S^*}(T)) = {(M(S(T),T) - m({S^*}(T),T))^ + }.
\end{equation}
Also, the price of this option at $t$, due to the $Q$ martingale measure is as follow: 
\begin{equation}
\begin{array}{l}
G(t,S,{S^*}) = {e^{ - r(T - t)}}{E^Q}[G(T,S(T),{S^*}(T))|M(S(t),t),m({S^*}(t),t)] = \\
{e^{ - r(T - t)}}E[{(M(S(T),T) - m(S(T),T))^ + }|M(S(t),t),m({S^*}(t),t)].
\end{array}
\end{equation}
The LOEX option price differential with the Ito lemma will follow the following SDE: 
\begin{equation}
\begin{array}{l}
dG = \left( {{G_t} + {\alpha _1}S{G_S} + {\alpha _2}{S^*}{G_{{S^*}}} + \sigma _1^2{S^2}{G_{SS}} + \sigma _2^2{S^{{*^2}}}{G_{{S^*}{S^*}}} + \rho {\sigma _1}{\sigma _2}S{S^*}{G_{S{S^*}}}} \right)dt\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {{\sigma _1}S{G_S} + {\sigma _2}{S^*}{G_{{S^*}}}} \right)dW.
\end{array}
\end{equation}
Now we use the Feynman–Kac formula: 
\begin{equation}
\frac{{\partial G}}{{\partial t}} + rS\frac{{\partial G}}{{\partial S}} + r{S^*}\frac{{\partial G}}{{\partial {S^*}}} + \frac{1}{2}\sigma _1^2S_{}^2\frac{{{\partial ^2}G}}{{\partial S_{}^2}} + \rho {\sigma _1}{\sigma _2}S{S^*}\frac{{{\partial ^2}G}}{{\partial S\partial {S^*}}} + \frac{1}{2}\sigma _2^2{S^{{*^2}}}\frac{{{\partial ^2}G}}{{\partial {S^{{*^2}}}}} - rG = 0,
\end{equation}
that its terminal condition is as follows: 
\begin{equation}
G(T,S(T),{S^*}(T)) = {(M(S(T),T) - m({S^*}(T),T))^ + }.
\end{equation}

{\fontsize{14}{24}\subsection{}}

{\fontsize{16}{24}\hypertarget{PDE solution}{%
\section{\texorpdfstring{ PDE solution
}{ PDE solution }}\label{LOEX Option}}}


Sometimes, an issue is a problem that we cannot access the close form of problems, so the numerical methods help us change this circumstance. There are a lot of numerical methods that are used for these problems \cite{safa} \cite{aziz1}.  In this solution section, we consider solving PDE (17) by numerical methods. The solution method used for the PDE (17) is the mesh-free numerical approximation method \cite{fass, fagh2}. Therefore, by property of exchange and lookback options, we consider boundary and initial conditions as follows: 
\begin{equation}
\begin{array}{l}
G(0,S,{S^*}) = {(S - {S^*})^ + },\\
G(\tau ,0,{S^*}) = 0,\\
G(\tau ,S,0) = S,\\
\mathop {\lim }\limits_{S,{S^*} \to \infty } G(\tau ,S,{S^*}) = \mathop {\lim }\limits_{S,{S^*} \to \infty } \frac{S}{{{\psi ^{{S^*}}}}},
\end{array}
\end{equation}
where $ \psi $ is a large number. 

For applying the meshfree numerical approximation method, suppose the change of assets $S$ and $S*$ is in ithe interval $[0, S_{max}]$ and $[0, S^*_{max}]$ respectively, and also discrete the first interval in $M$ points and the second $N$ points so we have the following radial basis function: 

\begin{equation}
G(t,S,{S^*}) = \sum\limits_{i = 1}^{MN} {{\lambda _i}(t)\varphi (S,{S^*})} ,
\end{equation}
where
\begin{equation}
\varphi (x,y) = \sqrt {{x^2} + {y^2} + {\varepsilon ^2}} ,
\end{equation}
and $\lambda$ is a function that depends on time. 
            
We rewrite the PDE’s (17) derivatives as follow: 
\begin{equation}
\begin{array}{l}
{G_{{S^{(*)}}}}(t,S,{S^*}) = \sum\limits_{i = 1}^{MN} {{\lambda _i}(t){\varphi _{{S^{(*)}}}}(S,{S^*})}  = :{\varphi _{{S^{(*)}}}}\lambda ,\\
{G_{{S^{(*)}}{S^{(*)}}}}(t,S,{S^*}) = \sum\limits_{i = 1}^{MN} {{\lambda _i}(t){\varphi _{{S^{(*)}}{S^{(*)}}}}(S,{S^*})}  = :{\varphi _{{S^{(*)}}{S^{(*)}}}}\lambda ,\\
{G_t}(t,S,{S^*}) = \sum\limits_{i = 1}^{MN} {{{\dot \lambda }_i}(t)\varphi (S,{S^*})}  = :\varphi \dot \lambda ,
\end{array}
\end{equation}
where $\varphi ,{\varphi _{{S^{(*)}}}},{\varphi _{{S^{(*)}}{S^{(*)}}}}$supposed that are matrix as follow:
\begin{equation}
\begin{array}{l}
{\varphi _{i,j}} = \varphi ({S_i},S_j^*),\\
{\varphi _{{S^{(*)}}}}_{_{i,j}} = {\varphi _{{S^{(*)}}}}({S_i},S_j^*),\\
{\varphi _{{S^{(*)}}{S^{(*)}}}}_{_{i,j}} = {\varphi _{{S^{(*)}}{S^{(*)}}}}({S_i},S_j^*).
\end{array}
\end{equation}
Then, we could have the PDE (17) as below: 
\begin{equation}
\varphi \dot \lambda  + L\lambda  = 0,
\end{equation}
where
\begin{equation}
L = rS{\varphi _S} + r{S^*}{\varphi _{{S^*}}} + \frac{1}{2}\sigma _1^2S_{}^2{\varphi _{SS}} + \rho {\sigma _1}{\sigma _2}S{S^*}{\varphi _{S{S^*}}} + \frac{1}{2}\sigma _2^2{S^{{*^2}}}{\varphi _{{S^*}{S^*}}} - r\varphi .
\end{equation}
Using the concept of derivative, suppose the interval of time is $[0, T] $ whereas mentioned, $T$ is strike time and consider $K$ as the number of equal discrete points of this interval with $dt$ difference so 
\begin{equation}
\dot \lambda  = \frac{{\lambda ({t_{k + 1}}) - \lambda ({t_k})}}{{dt}} = :\frac{{{\lambda ^{k + 1}} - {\lambda ^k}}}{{dt}}
\end{equation}
and using the numerical $\theta$-method, we can rewrite the ODE (24) as follows: 
\begin{equation}
A{\lambda ^{k + 1}} = B{\lambda ^k},
\end{equation}
where
\begin{equation}
\begin{array}{l}
A = \varphi  + \theta dtL,\\
B = \varphi  - (1 - \theta )dtL,\\
1 \le k \le K - 1.
\end{array}
\end{equation}
{\fontsize{14}{24}\section*{Algorithm}}

\begin{itemize}
\item[1.]Using initial condition, obtain ${\lambda ^k},k = 1$; 
\item[2.]Compute $A{\lambda ^{k + 1}} = B{\lambda ^k}$; 
\item[3.]$G = \varphi {\lambda ^k}$;
\item[4.]Apply boundary condition on $G$; 
\item[5.]${\lambda ^{k + 1}} = {\varphi ^{ - 1}}G,k = k + 1$, and if $k < K$ go step 2.
\end{itemize}


{\fontsize{16}{24}\hypertarget{Numerical Result}{%
\section{\texorpdfstring{4\textbar Numerical Result
}{4\textbar Numerical Result}}\label{Numerical Result}}}


In this section, we use the solving method in section (3.1) and the Kim and Koo (2016) data to implement the LOEX option that ${\sigma _1},\,{\sigma _2} = 0.125,\,\rho  =  - 0.35,\,r = {\rm{ }}0.05$  and finally, we compare our model with their price model.

\begin{table}[ht]
\caption{Variables}
\centering
\begin{tabular}{|cc|}
\hline
Variable             & Value \\
\hline
$S_{max}, S^*_{max}$ & 100   \\
T                    & 1     \\
M, N                 & 20    \\
$\psi$               & 1.5   \\
dt                   & 0.01 \\ \hline
\end{tabular}
\end{table}

Using the MATLAB software, the LOEX option price curve will be as figure (1.A), and with the same data and the Kim and Koo (2016) price formula, we get the Exchange option price curve as figure (1. B). 

\begin{figure}[ht]
\begin{subfigure}{.4\textwidth}
  \centering
  % include the first image
  \includegraphics[width=\linewidth]{1.png}  
  \caption{LOEX option price curve }
  \label{f111}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
  \centering
  % includes the second image
  \includegraphics[width=.8\linewidth]{2.png}  
  \caption{Exchange option price curve}
  \label{f122}
\end{subfigure}
\caption{Option price curves}
\label{f10}
\end{figure}

In Figure (2), we consider presenting the Discrepancy between LOEX option and the Exchange option as follows: 

\begin{figure}[ht]
\centering
  \includegraphics[width=0.6\linewidth]{3.png}
  \caption{Difference between options}
  \label{f5}
\end{figure}

In the figure (2), there is a huge difference between the LOEX option and the Exchange option. The price changes of the LOEX option are more than the price changes of the Exchange option. According to this option's structure, this difference's existence was not unpredictable. In the following, we can observe the Discrepancy of sensitivity of two options according to asset price differential in figure (3). 


\begin{figure}[ht]
\begin{subfigure}{.5\textwidth}
  \centering
  % include the first image
  \includegraphics[width=\linewidth]{4.png}  
  \caption{$\frac{{\partial G}}{{\partial S}} - \frac{{\partial C}}{{\partial S}}$}
  \label{f11}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
  \centering
  % includes the second image
  \includegraphics[width=\linewidth]{5.png}  
  \caption{$\frac{{\partial G}}{{\partial {S^*}}} - \frac{{\partial C}}{{\partial {S^*}}}$}
  \label{f12}
\end{subfigure}
\caption{Discrepancy of sensitivity of two options}
\label{f1}
\end{figure}



{\fontsize{16}{24}\hypertarget{Conclusion}{%
\section{\texorpdfstring{5\textbar Conclusion
}{5\textbar Conclusion}}\label{Conclusion}}}




In this article, we have presented a new option called the LOEX option. In fact, in this study, we introduced the new option by combining the Exchange and Lookback options. In this way, we have modeled this option, solved it by numerical methods, and provided it with the MATLAB software. Finally, after obtaining the price of the Exchange option, we have compared it with the LOEX option curve. The price difference figure shows a significant difference between the two assets. Due to the LOEX option's dependency on the path, this option is much more expensive than the Exchange option, and this difference increases with the increase in the price of the two assets. 

 Furthermore, we can develop the model and make it more realistic for the market. For example, we can add components like credit risk and improve the performance of the price model. 



{\fontsize{16}{24}\section{Acknowledgments}}
The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.
 
{\fontsize{16}{24}\section{Author Contribution}}
             R. Ghasem Pour: methodology, software, and editing. S.P. Azizi: conceptualization. S.A.Waloo: writing        
             and editing. All authors have read and agreed to the published version of the manuscript.
{\fontsize{16}{24}\section{Funding}}
The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions.
Data Availability
All data supporting the reported findings in this research paper are provided within the manuscript.
{\fontsize{16}{24}\section{Conflicts of Interest}}
The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study's design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.



{\fontsize{16}{24}
\begin{thebibliography}{20}
\bibitem{marg}
Margrabe, W. (1978).
\newblock The value of an option to exchange one asset for another.
\newblock {\em The Journal of Finance}, 33(1), 177--186. https://doi.org/10.2307/2326358






\bibitem{el}
El Karoui, N., Geman, H., \& Rochet, J. C. (1995.)
\newblock Changes of numeraire, changes of probability measure and option pricing.
\newblock {\em Journal of Applied Probability}, 32(2), 443--458. https://doi.org/10.2307/3215299






\bibitem{anto}
Antonelli, F., Ramponi, A., \& Scarlatti, S. (2010).
\newblock Exchange option pricing under stochastic volatility: a correlation expansion.
\newblock {\em Review of Derivatives Research}, 13(1), 45--73. https://doi.org/10.1007/s11147-009-9043-4






\bibitem{chea}
Cheang, G. H. L., and Chiarella, C. (2011).
\newblock Exchange options under jump-diffusion dynamics.
\newblock {\em Applied Mathematical Finance}, 18(3), 245--276. http://dx.doi.org/10.2139/ssrn.1352126





\bibitem{kim1}
Kim, G., \& Koo, E. (2016).
\newblock Closed-form pricing formula for exchange option with credit risk.
\newblock {\em Chaos, Solitons \& Fractals}, 91, 221--227. https://doi.org/10.1016/j.chaos.2016.06.005



\bibitem{fagh}
Azizi, S. P., Huang, C. Y., Chen, T. A., Chen, S. C., \& Nafei, A. (2023).
\newblock Bitcoin volatility forecasting: An artificial differential equation neural network.
\newblock {\em AIMS Mathematics}, 8(6), 13907--13922. https://doi.org/10.3934/math.2023712




\bibitem{wang}
Wang, X. (2016).
\newblock Pricing power exchange options with correlated jump risk.
\newblock {\em Finance Research Letters}, 19, 90--97. https://doi.org/10.1016/j.frl.2016.06.009




\bibitem{gold}
Goldman, M. B., Sosin, H. B., \& Gatto, M. A. (1979).
\newblock Path dependent options: "Buy at the low, sell at the high".
\newblock {\em The Journal of Finance}, 34(5), 1111--1127. https://doi.org/10.2307/2327238








\bibitem{conz}
Conze, A. (1991).
\newblock Path dependent options: The case of lookback options.
\newblock {\em The Journal of Finance}, 46(5), 1893--1907. https://doi.org/10.2307/2328577








\bibitem{dai}
Dai, M., Wong, H. Y., \& Kwok, Y. K. (2004).
\newblock Quanto lookback options.
\newblock {\em Mathematical finance: an international journal of mathematics, statistics and financial economics}, 14(3), 445--467. https://doi.org/10.1111/j.0960-1627.2004.00199.x


\bibitem{he}
He, H., Keirstead, W. P., \& Rebholz, J. (1998).
\newblock Double lookbacks.
\newblock {\em Mathematical Finance}, 8(3), 201--228. https://doi.org/10.1111/1467-9965.00053




\bibitem{jeon}
Jeon, J., \& Yoon, J.-H. (2016).
\newblock A closed-form solution for lookback options using Mellin transform approach.
\newblock {\em East Asian Mathematical Journal}, 32(3), 301--310. koreascience.or.kr/article/JAKO201617559403564.page


\bibitem{umeo}
Umeorah, N. O. (2017).
\newblock Pricing barrier \& lookback options using finite difference numerical methods.
\newblock PhD thesis, North-West University (South Africa), Potchefstroom Campus.


\bibitem{safa}
Safaei, M., Neisy, A., \& Nematollahi, N. (2018).
\newblock New splitting scheme for pricing American options under the Heston model.
\newblock {\em Computational Economics}, 52(2), 405--420. https://doi.org/10.1007/s10614-017-9686-4



\bibitem{aziz1}
Pourrafiee, M., Nafei, A. H., Banihashemi, S., \& 
 Azizi, S. P. (2020).
\newblock Comparing entropies in portfolio diversification with fuzzy value at risk and higher-order moment.
\newblock {\em Fuzzy Information and Engineering}, 12(1), 123--138. https://doi.org/10.1080/16168658.2020.1811481





\bibitem{fass}
Fasshauer, G. E., Khaliq, A. Q. M., \& Voss, D. A. (2004).
\newblock Using meshfree approximation for multi-asset American options.
\newblock {\em Journal of the Chinese Institute of Engineers}, 27(4), 563--571. https://doi.org/10.1080/02533839.2004.9670904






\bibitem{fagh2}
Azizi, S. P., Jafari, H., Faghan, Y., \& Neisy, A. (2022).
\newblock Inverse Problem Approach to Machine Learning with Application in the Option Price Correction.
\newblock {\em Optical Memory and Neural Networks}, 31(1), 46--58. https://doi.org/10.3103/S1060992X22010088







\end{thebibliography}
}


\end{document}
