\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 153, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2007/153\hfil Construction of Green's functions]
{Construction of Green's functions for the Black-Scholes equation}
\author[M. Y. Melnikov, Y. A. Melnikov \hfil EJDE-2007/153\hfilneg]
{Max Y. Melnikov, Yuri A. Melnikov}
\address{Max Y. Melnikov \newline
Labry School of Business and Economics,
Cumberland University,
Lebanon, TN 37087, USA}
\email{mmelnikov@cumberland.edu, Phone 615-547-1260}
\address{Yuri A. Melnikov \newline
Department of Mathematical Sciences,
Middle Tennessee State University,
Murfreesboro, TN 37132, USA}
\email{ymelniko@mtsu.edu, Phone 615-898-2844, Fax 615-898-5422}
\thanks{Submitted August 28, 2007. Published November 14, 2007.}
\subjclass[2000]{35K20, 58J35}
\keywords{Black-Scholes equation; Green's function}
\begin{abstract}
A technique is proposed for the construction of Green's functions for
terminal-boundary value problems of the Black-Scholes equation. The
technique permits an application to a variety of problems that vary by
boundary conditions imposed. This is possible by extension of an approach
that was earlier developed for partial differential equations in applied
mechanics. The technique is based on the method of integral Laplace
transform and the method of variation of parameters. It provides closed form
analytic representations for the constructed Green's functions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\section{Introduction}
The well-known function, in financial mathematics \cite{n1,s1,w1},
\begin{equation}
G(S,t;\widetilde{S})=\frac{\exp (-r(T-t))}{\widetilde{S}[2\pi \sigma
^{2}(T-t)]^{1/2}}\exp \Big(-\frac{[\ln (S/\widetilde{S})+(r-\sigma
^{2}/2)(T-t)]^{2}}{2\sigma ^{2}(T-t)}\Big) \label{e1}
\end{equation}%
is referred to as the Green's function of the backward in time parabolic
partial differential equation
\begin{equation}
\frac{\partial v(S,t)}{\partial t}+\frac{\sigma ^{2}S^{2}}{2}\frac{\partial
^{2}v(S,t)}{\partial S^{2}}+rS\frac{\partial v(S,t)}{\partial S}-rv(S,t)=0
\label{e2}
\end{equation}
which is called the Black-Scholes equation \cite{b1}. To be more specific
mathematically, we note that \eqref{e1} represents the Green's function for
the homogeneous terminal-boundary value problem corresponding to
\begin{gather}
v(S,T)=f(S) \label{e3} \\
|v(0,t)|<\infty \quad \text{and}\quad |v(\infty ,t)|<\infty . \label{e4}
\end{gather}
This problem was posed for the Black-Scholes equation in the quarter-plane
$\Omega =(0~~t>-\infty )$ of the $S,t$-plane.
In the above setting, $v=v(S,t)$ is the price of the derivative product,
$f(S)$ is the pay-off function of a given derivative problem at the
expiration time $T$, with $S$ and $t$ being the share price of the
underlying asset and time, respectively. The parameters $\sigma $ and $r>0$
represent the volatility of the underlying asset and the risk-free interest
rate, respectively. The variable $\widetilde{S}\in (0,\infty )$ in \eqref{e1}
plays the role of a \textit{source point}.
A special comment is required as to the symbolism used in specifying the
\textit{boundary} conditions in \eqref{e4}. Both the end-points of the
domain for the independent variable $S$ represent the so-called
\textit{singular points} \cite{s2} to the Black-Scholes equation, in which case the
corresponding boundary conditions cannot formally assign certain values to
the solution of the governing differential equation. Instead, the conditions
in \eqref{e4} imply that the solution that we are looking for has to be
bounded as the variable $S$ approaches both zero and infinity.
The function in \eqref{e1} represents the only Green's function for %
\eqref{e2} that is available in financial mathematics for decades. This
study proposes a new approach that enables one to construct Green's
functions to the Black-Scholes equation not only for the boundary conditions
in \eqref{e4} but also for a variety of others. The approach flows out from
a technique proposed earlier \cite{m1} for boundary value problems in
applied mechanics. It is not based on the classical formalism for the
diffusion equation as in \cite{n1,s1}. Instead, the emphasis is made on the
parabolic single-parameter partial differential equation forward in time
\begin{equation} \label{e5}
\frac{\partial u(x,\tau )}{\partial \tau }=\frac{\partial ^{2}u(x,\tau ) }
{\partial x^{2}}+(c-1)\frac{\partial u(x,\tau )}{\partial x} -cu(x,\tau )
\end{equation}
which is traditionally obtained \cite{n1,w1} from \eqref{e2} by introducing
new independent variables
\begin{equation} \label{e6}
x=\ln S\quad\text{and}\quad \tau =\frac{\sigma ^{2}}{2}(T-t)
\end{equation}
and setting $u(x,\tau )=v(S,t)$.
The parameter $c$ in \eqref{e5} is defined in terms of $r$ and $\sigma ^{2}$
of the Black-Scholes equation as $c=2r/\sigma ^{2}$.
To illustrate the effectiveness of our approach, a validation example is
considered in the next section where we derive the Green's function of
\eqref{e1}. After the approach is validated, it is used, in the following
sections, to tackle some other terminal-boundary value problems for the
Black-Scholes equation. New Green's functions are obtained none of which
have earlier been presented in literature.
\section{A validation example}
By introducing new variables $x$ and $\tau $ in compliance with the
relations in \eqref{e6}, the terminal-boundary value problem of
\eqref{e2}-\eqref{e4} transforms to the following initial-boundary value problem
\begin{gather}
u(x,0)=f(\exp x) \label{e8} \\
|u(-\infty ,\tau )|<\infty ,\quad |u(\infty ,\tau )|<\infty \label{e9}
\end{gather}
for \eqref{e5} on the half-plane $(-\infty t>-\infty )$ of the $S,t$ -plane. Let the terminal condition be given by
\eqref{e3}, while the Dirichlet boundary conditions
\begin{equation} \label{e19}
v(S_{1},t)=0, \quad v(S_{2},t)=0
\end{equation}
are imposed on the edges $S=S_{1}$ and $S=S_{2}$ of $\Omega $.
Note that the above setting for the Black-Scholes equation sounds quite
practical for the financial engineering, whereas its Green's function is not
yet available in literature.
By the transformations of \eqref{e6}, the setting in \eqref{e2}, \eqref{e3}
and \eqref{e19} converts to the following initial-boundary value problem
\begin{gather}
u(x,0)=f(\exp x), \label{e21} \\
u(a,\tau )=0,\quad u(b,\tau )=0 \label{e22}
\end{gather}
for \eqref{e5} on the semi-infinite strip $(at>-\infty )$. Indeed, if we manage to find the
solution to the problem in \eqref{e2}, \eqref{e3} and \eqref{e29} in an
\textit{integral form} like that in (3.10), then the kernel of the integral
represents the Green's function that we are looking for.
The second condition in \eqref{e29} is referred to, in mathematical physics,
as either \textit{mixed} or \textit{Robin }type. To our best knowledge,
mixed boundary conditions have never been considered yet in association with
the Black-Scholes equation. It is even unclear if such problem settings are
timely for financial engineering. But from mathematics stand-point, they do
not look unfeasible and could possibly find realistic applications in the
field of finance in years to come.
Upon introducing new variables $x$ and $\tau $ as suggested in \eqref{e6},
one converts the setting in \eqref{e2}, \eqref{e3} and \eqref{e29} to the
initial-boundary value problem
\begin{gather}
u(x,0)=f(\exp x) \label{e31} \\
|u(-\infty ,\tau )|<\infty ,\quad \frac{\partial u(b,\tau )}{\partial x}+
\overline{\varrho }u(b,\tau )=0
\end{gather}%
for \eqref{e5} on the quarter-plane $(-\infty ~~