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\AtBeginDocument{{\noindent\small
Ninth MSU-UAB Conference on Differential Equations and Computational
Simulations.
\emph{Electronic Journal of Differential Equations},
Conference 20 (2013), pp. 79--91.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}\setcounter{page}{79}
\title[\hfilneg EJDE-2013/Conf/20/ \hfil Stabilized Adams type method]
{Stabilized Adams type method with a block extension for the
valuation of options}
\author[S. N. Jator, D. Y. Nyonna, A. D. Kerr \hfil EJDE-2013/conf/20 \hfilneg]
{Samuel N. Jator, Dong Y. Nyonna, Andrew D. Kerr} % in alphabetical order
\address{Samuel N. Jator \newline
Department of Mathematics and Statistics,
Austin Peay State University,
Clarksville, TN 37044, USA}
\email{Jators@apsu.edu}
\address{Dong Y. Nyonna \newline
Department of Accounting, Finance, and Economics,
Austin Peay State University, Clarksville,
TN 37044, USA}
\email{NyonnaD@apsu.edu}
\address{Andrew D. Kerr \newline
Department of Physics and Astronomy,
Austin Peay State University, Clarksville,
Clarksville, TN 37044}
\email{akerr@my.apsu.edu}
\thanks{Published October 31, 2013.}
\subjclass[2000]{65L05, 65L06}
\keywords{Stabilized Adams method; extended block; options;
\hfill\break\indent Black-Scholes partial differential equation}
\begin{abstract}
We construct a continuous stabilized Adams type method (CSAM) that
is defined for all values of the independent variable on the range
of interest. This continuous scheme has the ability to provide a
continuous solution between all the grid points with a uniform
accuracy comparable to that obtained at the grid points. Hence,
discrete schemes which are recovered from the CSAM as by-products
are combined to form a stabilized block Adams type method (SBAM).
The SBAM is then extended on the entire interval and applied as a
single block matrix equation for the valuation of options on a
non-dividend-paying stock by solving a system resulting from the
semi-discretization of the Black-Scholes model. The stability of the
SBAM is discussed and the convergence of the block extension of the
SBAM is given. A numerical example is given to show the accuracy of
the method.
\end{abstract}
\maketitle
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
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\section{Introduction}
The Black-Scholes option pricing model is one of the most celebrated
achievements in financial economics in the previous four decades. The
model gives the theoretical value of European style options on a
non-dividend-paying stock given the stock price, the strike price,
the volatility of the stock, the time to maturity, and the risk-free
rate of interest. However, since it is optimal to exercise early an
American put option on a non-dividend paying stock, the
Black-Scholes formula cannot be used.
Hull \cite{Hull}
argues that it is never optimal for an American call option on a
non-dividend-paying stock to be exercised early. Therefore the
Black-Scholes formula can be used to value American Style call
options on non-dividend-paying stocks.
In fact, no exact analytic
formula for valuing American put options on non-dividend paying
stocks exists. As a result, numerous numerical procedures are
utilized. A discussion of some of these numerical techniques is
found in Hull \cite{Hull}. In addition to that, several other
numerical procedures for solving the Black-Scholes model abound in
the literature (see Chawla et al. \cite{CAE} and Khaliq et al.
\cite{KVK}). Since there is the possibility of an early exercise,
Khaliq et al. \cite{KVK}) consider the pricing of an American put
option as a free boundary problem. In effect, the early exercise
feature of the American put option transforms the Black-Scholes
linear differential equation into a non-linear type. In order to do
away with the free and moving boundary, Khaliq et al. \cite{KVK} add
a small continuous penalty term to the Black-Scholes equation and
treat the nonlinear penalty term explicitly. They conclude that
their method maintains superior accuracy and stability properties
when compared to standard methods that are based on the Newton-type
iteration procedure in valuing American options.
Furthermore, Chawla et al. \cite{CAE} employ a technique based on
the Generalized Trapezoidal Formulas (GTF) and compare the
computational performance of the scheme obtained with the
Crank-Nicolson scheme for the case of European option pricing. They
note that their GTF (1/3) scheme is superior to the Crank-Nicholson
scheme. While all these techniques try to accomplish the same goal
by solving the Black-Scholes differential equation for a particular
derivative security, they are applied only after transforming the
model to be forward in time. In this paper, we propose SBAM that is
$A$-stable and applied to solve the model in its original form
without transforming it into a forward parabolic equation. Thus,
consider the Black-Scholes model
\begin{equation} \label{e1}
\frac{\partial V}{\partial t}
+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2} V}{\partial S^{2}}
+rS\frac{\partial V}{\partial S}-rV=0,
\end{equation}
subject to the initial/boundary conditions
\begin{gather*}
V (0, t) = X, \\
V (S, t) \to 0\quad\text{as }S \to \infty,\\
V (S, T) = \max (X-S, 0),
\end{gather*}
where $V (S, t)$ denotes the value of the option, $\sigma$
the volatility of the underlying asset, $X$, the exercise price, $T$
the expiry, and $r$ the interest rate.
The method considered in this article involves the method of line
approach to solve \eqref{e1} in which we discretize the space derivatives
in such as way that the resulting system of ordinary differential
equations is stable (see Lambert \cite{Lam1}, Ramos and Vigo-Aguiar
\cite{RA}, and Cash \cite{CA}). We then discretize time by using the
SBAM. In particular, we seek a solution in the strip (rectangle)
$[a, b]\times[c, d]$ by first discretizing the variable $S$ with
mesh spacings $\Delta S=1/M$,
\[
S_{m}=m\Delta S, \quad m=0, 1, \dots, M.
\]
We then define $v_{m}(t)\approx V(S_{m}, t)$,
$\mathbf{v}(t)=[v_0(t), v_{1}(t), \dots, v_{M-1}(t)]^{T}$, and
replace the partial derivatives
$\frac{\partial^{2} V(S,t)}{\partial S^{2}}$ and
$\frac{\partial V(S, t)}{\partial S}$
occurring in \eqref{e1} by central difference approximations to obtain
\begin{gather*}
\frac{\partial^{2} V(S_{m}, t)}{\partial S^{2}}
=[v(S_{m+1},t)-2v(S_{m},t)+v(S_{m-1},t)]/(\Delta S)^{2};
\\
\frac{\partial V(S_{m}, ~t)}{\partial S}
=[v(S_{m+1},t)-v(S_{m-1},t)]/(\Delta S),\quad m=0, 1, \dots, M-1.
\end{gather*}
Problem \eqref{e1} then leads to the resulting semi-discrete
problem
\begin{align*}
\frac{d v_{i}(t)}{dt}
&=-\frac{1}{2}\sigma^{2}S_{i}^{2}[v_{i+1}(t)-2v_{i}(t)+v_{i-1}(t)]/(\Delta
S)^{2}\\
&\quad -rS_{i}[v_{i+1}(t)-v_{i-1}(t)]/(\Delta S)+r v_{i}(t)=0,
\end{align*}
which can be written in the form
\begin{equation} \label{e2}
\frac{d \mathbf{v}(t)}{d t} =\mathbf{f}(t,\mathbf{v}),
\end{equation}
where
$\mathbf{f}(t,\mathbf{v})=\textbf{A v}+ \mathbf{Q}$ and
$\mathbf{A}$ is an
$M-1 \times M-1$ matrix arising from the central difference
approximations to the derivatives of $S$ and $\mathbf{Q}$ is a vector
of given constants. Problem \eqref{e2} is now a system of ordinary
differential equations which can be solved by the SBAM.
We will assume the scalar form of \eqref{e2} for notational simplification
and will return to the system at the implementation stage in Section
5. We note that the Adams Moulton is one of the most popular methods
available for solving \eqref{e2}. The $k$-Adams Method is given by
\begin{equation} \label{e3}
v_{n+k}-v_{n+k-1}=h\sum _{j=0}^{k}\beta_jf_{n+j}
\end{equation}
where $\beta_{j}$ are constants. We note that $v_{n+j}$ is the numerical
approximation to the analytical solution
$v(t_{n+j})$, $f_{n+j}=f(t_{n+j},v(t_{n+j}))$, $j=0,\dots ,k$.
For non-stiff problems, \eqref{e3} performs excellently, while for stiff
for problems, \eqref{e3} is restricted by the step-size. For instance, for
$k=4$, \eqref{e3} gives the standard $4-$step Adams-Moulton Method which is
of order 5 and has a stability interval of $[-1.84, 0]$. The method
\eqref{e3} is implemented in a step-by-step fashion in which on the
partition $\pi _N$, an approximation is obtained at $t_n$ only
after an approximation at $t_{n-1}$ has been computed, where
\[
\pi _N:a=t_0