\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 74, pp. 1--26.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/74\hfil A comparison principle]
{A comparison principle for an American option on several assets:
Index and spread options}
\author[Peter Laurence \& Edward Stredulinsky\hfil EJDE--2003/74\hfilneg]
{Peter Laurence \& Edward Stredulinsky } % in alphabetical order
\address{Peter Laurence \newline
Department of Mathematics, Universit\`a di Roma 1, Rome, Italy}
\email{laurence@mat.uniroma1.it}
\address{Edward Stredulinsky \newline
Department of Mathematics, University of Wisconsin, Richland,
WI 53581, USA}
\email{estredul@uwc.edu}
\date{}
\thanks{Submitted March 10, 2003. Published July 7, 2003.}
\subjclass[2000]{35K85, 35Q99}
\keywords{American options, variational inequalities, free boundary,
\hfill\break\indent
parabolic equations, finance, symmetrization, optimal stopping,
rearrangements}
\begin{abstract}
Using the method of symmetrization, we compare the price
of the American option on an index or spread to that of the
solution of a parabolic variational inequality in one spatial
variable. This comparison principle is established for a broad
class of diffusion operators with time and state dependent
coefficients. The purpose is to take a first step towards deriving
symmmetrized problems whose solutions bound solutions of
multidimensional American option problems with variable
coefficients when the computation of the latter lies beyond the
scope of the most powerful numerical methods.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corolary}
\section{Introduction}
An American option gives its holder the right to buy a
stock or basket of stocks at a given price called the strike
prior to but not later than a given time $T$, from the time of
inception of the contract. What distinguishes an American
option from a European option is the possibility of
early exercise. In this paper we focus on American options on two or
more assets. A standard example is an index option that is based on the
geometric or arithmetic means of several assets. The S\&P 100 index option,
traded on the Chicago Board of Options Exchange is an
American option on a value weighted index of 100 stocks.
Two other examples are American options on the maximum or on the spread
of two stocks.
It is well known since 1973 \cite{Me} that an American call option on a
single stock which pays no dividends and which follows a geometric Brownian
motion will not be exercised prior to expiration and
therefore is, for valuation purposes, equivalent to a European
option. This is not the case for American put options. Moreover,
in most cases of practical interest, the stocks underlying the call
options pay dividends and early exercise is then often not
optimal.
No closed form solutions are known for American options,
even in the case of one asset, except for the so-called
perpetual option, which is of limited practical interest.
In the case of a single underlying asset, a long tradition exists
in the finance literature of seeking analytical solutions which
yield good approximations to the value of the
option and to the value $S_f(t)$, $0 \leq t \leq T$, at which it is optimal
to exercise the option. Some good references in this direction
are Barone-Adesi and Whaley \cite{Ba-Wh} and the recent paper by
Ju and Zhong \cite{Ju-Zh} which contains an extensive bibliography of
previous work.
The literature on options with several underlying assets is less
extensive. A pioneering paper is that of Broadie and Detemple
\cite{Br-De} who use probabilistic techniques to describe the shape of
the free boundary for some of the most important contracts.
Villeneuve \cite{Vi} further enriched and strengthened the
mathematical underpinnings of Broadie and de Temple's work. It
should be emphasized that the qualitative results obtained in
these papers are set in the standard Black-Scholes framework, in
which all parameters, such as the volatility, are constant, and to
our knowledge, little is known about the free boundary or about
the options value, when we leave this setting. Thus it is
desirable to identify {\it features of the underlying volatility,
dividend and interest processes} from which bounds can be
derived, that depend on only {\it partial information} concerning
these processes. Note that even in one dimension, the effect of a
state dependent or of a stochastically driven volatility, has
significant influence on American option's value, as discussed in
the recent papers by Broadie, Detemple, Ghysels and Torres
\cite{Br-De-Gh-To,Br-De-Gh-To2}) and the assumption that American index option follows a
geometric Brownian motion is weakly founded. Moreover in the case
of American options on several underlying assets even the most
recent numerical methods based on Monte-Carlo algorithms, require
extensive computational power, especially in the case of a
continuous exercise envisaged here
\footnote{In
practice American options permit at best daily exercise. Using
even the best Monte Carlo methods, which involves one hundred
regressions at each exercise date, an option on 3 underlying
assets, with 100 days to expiration and 5000 paths per stock
generated, yields a conservative estimate of $100\times 5000\times
3 = 150 000$ bits to be stored}
of multiple underlying assets and
state and time dependent parameters. Thus it is desirable to
develop analytic methods which yield useful and stable bounds,
that can be used as benchmarks by the investor. These bounds will
not be optimal in general, ie. if all the parameters, such as
the volatility, are known with precision, one can in principle
obtain sharper bounds or better comparison equations. By
``stable'' we mean that the bounds change little if the partial
information we have about the parameters is altered a little.
In this paper, we address the following question. Given an
American option on several assets we seek
to obtain upper bounds for its value which
rely only on partial information about the volatility, interest rate
and dividend processes governing the stocks evolution under the
risk-neutral measure. Such partial information
is the best that one can expect in most cases.
Indeed the process of calibration
or ``backing out'' a reasonable estimate of the stocks
volatility from market data is one of the most active areas
of research in modern finance, see for instance Bakshi, Cao, and Zhiwu
\cite{Ba-Ca-Zh}.
Of the many approaches that have been taken, perhaps the ones most similar,
in the financial context,
to the one we will take here, is that of Avellaneda, Levy and Paras
\cite{Av-Le-Pa}, and that of Lyons \cite{Lyons},
which was applied in the American option setting by Buff \cite{Bu}.
These are similar not in the techniques used, but rather
in the types of assumptions that are made about the volatility
process. For instance in Buff's development of Avellaneda et al.'s
uncertain volatility approach, its assumed that the
volatility lies in a ``band'' $[\sigma_{min}, \sigma_{max}]$.
Many pricing methods for index options assume that
the latter follows a geometric Brownian motion.
Recent work has found this assumption to be
weakly founded in some cases. See Broadie, Detemple,
Ghysels and Torres \cite{Br-De-Gh-To}.
In considering the use of more complicated models,
an important consideration is their {\it tractability}.
Monte Carlo methods for options on multiple assets
that take into account the daily exercise feature and
the multifactor structure are expensive, especially
when calibrating market data to a rich structure
of input parameters and allowing these to have a
non trivial functional form. The method proposed here
explores a direction which trades off precision for tractability.
At present it
produces upper bounds only and further numerical
work is required to assess how sharp these are.
It's tractability derived from the fact that
the upper bounds produced by solving a one dimensional
parabolic variational inequality require
only a few seconds on a Pentium 2 PC.
However preliminary numerical results indicate that the
comparison principle derived in this paper does not
produce bounds that are not sharp enough to be of interest
in practice. Thus the present paper should be seen
as a first attempt to adapt the method of symmetrization
to the American option problem and the method will need
to be refined in the future (in progress).
To obtain our bounds we will use the method of symmetrization
to estimate the solutions of the options on multiple
assets in terms of the solution $V_k(r, \tau) : B_k\times [0, T]$
of a spatially one dimensional parabolic variational inequality
\begin{multline}
(V_k)_{\tau} - \lambda_{co}^2(\tau) (V_k)_{rr} - \frac{(n -1) \lambda_{co}^2(\tau) }{r} (V_k)_{r} + D(\tau) (V_k)_r + C(\tau) V_k
- F(r, \tau))\\
\begin{aligned}
& = 0 \quad \mbox{on } \{ V_k > 0\} \\
& \geq 0 \quad\mbox{on } B_k\,, \end{aligned}
\label{inequ}
\end{multline}
where the coefficients $\lambda_{co}(\tau) , D(\tau), C(\tau)$
and source term $F$ are determined by the original volatilities,
interest rate and dividend rates by a recipe that we
will describe in section 3, and where $k$ corresponds to
the standard cut-off wherein the problem is localized to
a ball of radius $k$.
This equation has time dependent coefficients, with the exception
of the term $1/r$ preceding the first order derivative $(V_k)_r$,
familiar in physics in deriving the Laplacian in {\it spherical}
coordinates. Because of its appearance in a variety
of physical contexts, the case of {\it equations} of
the form (\ref{inequ}) has been studied numerically
by mathematical physicists and efficient algorithms
can be adapted to deal with the variational inequality.
The method of symmetrization was introduced by Schwarz cite{Sc},
Steiner \cite{St}, and Hardy-Littlewood-Polya \cite{Ha-Li-Po}.
It's close connection with isoperimetric inequalities was realized
by Polya and Szego , and summarized in their book.
Bandle \cite{Ba1} and Talenti \cite{Ta1} pioneered the introduction of
symmetrization and rearrangement techniques in the area of partial differential equations.
Kawohl \cite{Ka} and Mossino \cite{Mo} described the state of the
art and found many refinements in their books.
The technique has since been substantially developed
in work by Alvino, Lions and Trombetti \cite{Al-Li-Tr1},
Gustafsson and Mossino \cite{Gu-Mo}, Diaz and Mossino \cite{Di-Mo},
Ferone and Volpicelli \cite{Fe-Vo}, and Kesavan \cite{Ke},
to mention only a few. In recent work by Kinateder and Mac Donald \cite{Ki-Mc}
the closely connected problem of the
distribution of first exit times in its dependence on the domain is considered.
The symmetrization method will be applied to the parabolic variational
inequalities modeling the multidimensional American option problem.
The theoretical framework for this
was provided by Jaillet, Lamberton and Lapeyre {Ja-La-La}
who showed how to adapt the theory described in Bensoussan-Lions {Be-Li}
theory to the present setting.
Our results also apply in the case of an American option on a single asset,
when the parameters are state and time dependent, and bound the price
of the option by the solution of a variational
inequality with purely time dependent coefficients.
In light of the recent work of Broadie, Detemple, Ghysels and Torres
\cite{Br-De-Gh-To}
on the profound effect that stochastic volatility and
dividends can have on the price of the American option
on one dividend paying option, it is possible that
our comparison principle might be of interest also in this setting.
To help illustrate what follows in a simple case, in
{\bf Figure 1}, we illustrate the value of a put on one dividend paying asset,
in the original variables, before the logarithmic transformation
is introduced, at a given time $t$ prior
to expiration. The payoff function is $ (K - S)^+$.
The value function is tangent to the payoff at one and only one
point which corresponds to the free boundary $S(t)$.
In {\bf Figure 2}, we show the same Figure but after the logarithmic change
of the independent variable and the change of dependent
variable $u \to u/K$. In the new variables the payoff
function is $( 1 - e^x)^+$, so this change of variables also has
the effect of making the transition between `in the money'
and out of the money, occur the value at $x=0$.
In {\bf Figure 3} we computed the difference between the
option value and the payoff in the new variables.
Note that in these new variables, $v$ is nearly {\it symmetric} around
the $y$ axis. It will {\it not} in general be exactly
symmetric, but note that the inverse image of any value
$y = c , 0 < c < \max v$ consists exactly of two points.
It turns out that this implies that the bounds obtained
are sharper if $r \geq d$, then when $r < d$ (and the opposite is true
for calls). The reason
for this is that $v$ satisfies in the new variables an
equation of the form
\begin{equation}
v_{\tau} - \sigma^2 v_{xx} - (r - d - \sigma^2 ) v_x + r v =
\delta_{\{ x =0\}} + (d e^{x} - r) {\bf 1}_{ x < 0}
\label{onedimensionalcase}
\end{equation}
Note that right hand side of (\ref{onedimensionalcase}) is the sum of
a delta function at the value $x=0$ and a function which is monotonically
increasing {\it if and only if} $r \geq d$.
This paper is organized as follows:
Section 2 -- Formulation of the variational inequalities, background
material on symmetrization and isoperimetric inequalities.
Section 3 -- Parameters of symmetrized comparison problem.
Section 4 -- Statement of Main Results.
Section 5 -- Proof of Main Results.
Section 6 -- Explicit form for regularized inhomogeneous term.
\section{Formulation of the Problem}
The american option problem is an optimal stopping problem
for a vector of stocks $S_t = (S_t^1,\cdots S_t^n)$
which follow a diffusion process
\begin{equation}
dS_t^i = ( r(S_t, t) - d_i(S, t)) S_t^i dt
+ \sum_{ij} S_t^i \tilde{\sigma}_{ij}(S, t) dZ_t^j
\label{process1}
\end{equation}
where $r$ and $d_i$ are respectively the short rate and the
continuously compounded dividend rate of the $i$-th stock,
$\tilde{\sigma}_{ij}(S, t), i=1,\cdots, n, j=1,\cdots, n$ is the
$n\times n$ dimensional volatility matrix, and
$(Z_t)_{t\geq 0}$ is a standard $\mathbb{R}^n$ valued Brownian motion on a
probability space $(\Omega, \mathcal{F}, P)$ with respect to the measure
$P$, where $P$ is the so-called risk neutral measure.
Throughout this paper we will frequently use the Einstein
summation convention wherein the summation
sign is omitted when summing over repeated indices,
The value process $u$ of the american option problem time is a
solution of the following problem
\begin{equation}
\tilde{u}(S, t)=
\sup_{\tau\in \mathcal{T}} E[ e^{-\int_t^{\tau} r(S_u, u)du}
\psi(S_{\tau}) : S_t= S] \,
\end{equation}
where the stopping time $\tau$
varies over all $\mathcal{F}_t$ adapted random variables and
$\psi(S)$ is the option payoff. Here ($\mathcal{F}_t)_{t\geq 0}$
denotes the $ P$ completion of the natural filtration associated
to $(Z_t)_{t\geq 0}$. Intuitively the optimal stopping problem
consists in finding the stopping strategy $\tau$ that maximizes
the expected gain to the holder of the option. Changing
variables as follows
\begin{equation} \label{changeofvar}
\begin{gathered}
x_i = ln(S_i/K), \quad i=1, \cdots n \\
u= \tilde{u}/K \\
\tau = T - t\,,
\end{gathered}
\end{equation}
where $K$ is the strike of the option and let
$\mathcal{C}_t$, $\mathcal{E}_t$, denote respectively the continuation
and exercise region for the option, with $\mathcal{C}_t \cup
\mathcal{E}_t= \mathbb{R}^n$. It can then be shown that $u$ is a weak
solution (in a sense made precise below) of
{\bf Problem $(\gamma_1)$}:
\begin{gather}
u_{\tau} - \sigma_{i j}(x, \tau) \frac{\partial^2 u}{\partial
x_i
\partial x_j} - (r - d_i - \sigma_{ii}) \frac{\partial u}{\partial
x_i} + r u = 0\, \label{equationinRn} \\
x\in \mathcal{C}_{\tau}, \quad \tau \in[0, T] \nonumber \\
u = \psi \quad \mbox{on } \mathcal{E}_{\tau} \label{coincidencecondition}\\
u = \psi \quad \hbox{on } \partial\mathcal{C}_{\tau} \nonumber \\
\frac{\partial u}{\partial \nu} = \frac{\partial \psi}{\partial\nu}\
quad\hbox{on }\partial\mathcal{C}_t \label{freebdryconditions}
\end{gather}
The free boundary condition $\frac{\partial u}{\partial \nu} =
\frac{\partial \psi}{\partial \nu}\quad\hbox{on}\quad
\partial\mathcal{C}_t$ has a meaning only at regular points of the
free boundary and, to our knowledge, no complete analysis of the
regularity of the free boundary is presently available, especially
in the case $n\geq 2$.
The spatial part of the operator in (\ref{equationinRn}) is denoted
$\mathcal{L}_{ S}$
\begin{equation}
\mathcal{L}_S = - \sigma_{i j}(x,\tau) \frac{\partial^2 }{\partial x_i \partial x_j}
- (r - d_i -\sigma_{ii}) \frac{\partial }{\partial x_i} + r , \label{spatial}
\end{equation}
where
\begin{equation}
\sigma_{ij}=\frac{1}{2 K^2} \sum_k
\tilde{\sigma}_{ik}\tilde{\sigma}_{jk}\,. \label{defnsigma}
\end{equation}
The rigorous weak formulation of the problem is
formulated in terms of variational inequalities and was detailed
by Jaillet Lamberton and Lapeyre \cite{Ja-La-La} based on earlier work by
Bensoussan and Lions and is subject to the following hypotheses:
\begin{itemize}
\item[(H1)] $r(x, \tau)$ and $d_j(x,\tau),
j=1,.. n$ are bounded $C^1$ functions from
$ \mathbb{R}^n\times [ 0, T]$ to $\mathbb{R}$, with bounded derivatives and
$r\geq 0, d_j \geq 0$.
\item[(H2)] The entries $\tilde{\sigma}_{ij}, i, j=1, \cdots n$, in the matrix
$\underline{\tilde{\sigma}}(x, \tau)$, are bounded $C^1$ functions
from $\mathbb{R}^n\times [0, T]$ to $\mathbb{R}$.
$\underline{\tilde{\sigma}}$ admits continuous second partial derivatives
with respect to $x$ satisfying a Holder condition in $x$ uniformly
with respect to $(x, \tau)$ in $\mathbb{R}^n\times [ 0, T]$.
\item[(H3)] The matrix $\underline{\sigma}= \frac{1}{2}
\underline{\tilde{\sigma}}\cdot\underline{\tilde{\sigma}}^{t}$
satisfies the following property:
There exists $\eta > 0$ such for all $(x, t)\in [0, T] \times\mathbb{R}^n$
and all $\xi\in \mathbb{R}^n$,
\begin{equation}
\sum_{1\leq i , j \leq n} \sigma_{i, j} (x, t) \xi_i
\xi_j \geq \eta \sum_{i=1}^n \xi_i^2
\end{equation}
\item[(H4)] The option payoffs $\psi$ considered depend only on $x$ and in
addition satisfy the condition:
There exists $M > 0$ such for all $x\in \mathbb{R}^n$
\begin{equation}
|\psi(x)| + \sum_{j=1}^n |\frac{\partial
\psi}{\partial x_j}(x)| \leq M e^{M |x|} \,.
\end{equation}
All option payoffs of interest in practice satisfy this growth condition.
\item[(H5)] The interest rate $r(x, \tau)$ is bounded below by some
positive constant $r_0$.
\end{itemize}
The variational inequalities considered are formulated in
certain function spaces which we now describe.
Let $m$ be a non negative integer and suppose that
$1 \leq p \leq \infty$ and $0 < \mu < +\infty$.
$W^{m, p, \mu}$ denotes the space of all functions $u$
whose distributional derivatives of order less than or equal to $m$ lie
in $L^p(\mathbb{R}^n, e^{- \mu |x|}dx)$. For brevity we will
use the notation $H_{\mu}$ to denote the space $W^{0, 2, \mu}(\mathbb{R}^n)$ and
$V_{\mu}$ to denote the space $W^{1, p, \mu}$. The inner
product on $H_{\mu}$ is denoted $(\cdot, \cdot)_{\mu}$. Define a a bilinear
form on $V_{\mu}$ for each $t\in [ 0, T]$ as follows:
For all $u , v \in V_{\mu}$,
\begin{equation}
\begin{aligned}
a^{\mu}(\tau, u, v) =& \sum_{i, j =1}^n
\int_{\mathbb{R}^n} \sigma_{i, j}( x, \tau) \frac{\partial u}{\partial x_i}
\frac{\partial v}{\partial x_j} e^{ - \mu |x|} dx \\
&- \sum_{i}^n \int_{\mathbb{R}^n} \left( r_i - d_i - \sigma_{ii}
- \sum_{j}^n (\sigma_{i j})_{x_j}\right)\frac{\partial u}{\partial x_i}
v e^{ - \mu |x|} dx \\
&- \sum_{i, j=1}^n \int_{\mathbb{R}^n}\left(\sigma_{i j}\frac{x_j}{|x|}
\frac{\partial u}{\partial x_i}\right) v e^{ - \mu |x|} dx
+ \int_{\mathbb{R}^n} r( x, \tau) \, u v e^{ - \mu |x|}dx
\end{aligned}
\label{weightedVI1formulation}
\end{equation}
\subsection*{Coerciveness}
There exist constants $\alpha > 0 $ and $\rho >0 $
such that for all $\tau\in [0, T]$ and for all $u\in V_{\mu}$
\begin{equation} \label{e2.10}
a^{\mu}(\tau, u, u) + \rho |u|_{\mu}^2 \geq \alpha \|u\|_{\mu}^2\,,
\end{equation}
where we use single bars $|\cdot |$ to denote the norm in $H_{\mu}$ and
double bars $\|\cdot\|$ to denote the norm in $V_{\mu}$.
The coerciveness is ensured by hypotheses (H3).
Under the regularity and non-degeneracy assumptions on the coefficients,
i.e. conditions (H1)--(H5) we have the following result
\begin{theorem} \label{thm1}
If $\psi\in V_{\mu}$, there
exists a unique solution to the following parabolic variational
inequalities defined on $[0, T]\times\mathbb{R}^n$
\begin{equation}
\begin{gathered}
u \in L^2([ 0, T]; V_{\mu}) ,\quad \frac{\partial u}{\partial \tau}
\in L^2( [ 0, T]; H_{\mu}) \\
u \geq \psi \quad \hbox{a.e in } \mathbb{R}^n\times [0, T], \quad u(0)= \psi \\
\forall v \in V_{\mu}\quad v \geq \psi \Rightarrow (
\frac{\partial u}{\partial \tau}, v - u)_{\mu} + a^{\mu}(\tau , u
, v -u) \geq 0
\end{gathered}\label{weightedVIregularity}
\end{equation}
where the bilinear form was defined in (\ref{weightedVI1formulation}).
\end{theorem}
\noindent{Remarks } 1) The terms $\sum_{i=1}^n
\frac{\partial \sigma_{i j}}{\partial x_i}(x, \tau) $ and
$\sum_{j=1}^n \sum_{i=1}^n \sigma_{i j}( x, \tau)
\frac{x_i}{|x|} \frac{\partial u}{\partial x_j}$ in
(\ref{weightedVI1formulation}), arise from
an integration by parts, by bringing the derivative respectively
on the volatility coefficient
$\sigma_{i j}$ and on the exponential damping factor $e^{ - \mu |x|}$.\\
2) Unlike Jaillet-Lamberton-Lapeyre, we will work
in the backward variable $\tau = T - t$.
The proof of Theorem \ref{thm1}
is outlined in Jaillet, Lamberton and Lapeyre. It is based on the
treatment in Bensoussan -Lions \cite[Chapter 3, Section 4]{Be-Li}.
\subsection*{Class of Payoffs considered}
The payoffs of practical interest for American options on several
underlying assets, fall into two principal categories.
\noindent {\bf Payoffs Class A:}
These payoffs are illustrated by payoffs
on a basket or on a spread. After normalizing the
strike to be equal to one with the change of variables
(\ref{changeofvar}) these payoffs may be written
\begin{gather}
\eta(x) = (x)^+ \\
\psi_C = \eta( \Phi_C) ,\quad \Phi_C = \sum_{i=1}^n (w_i e^{x_i} - 1)
\\
\psi_P = \eta(\Phi_P), \quad \Phi_P = \sum_{i=1}^n (1 - w_i e^{x_i})\,
\end{gather}
where for an index option all constants $w_i$'s are positive
and for a spread some of the $w_i$ are positive and others negative.
The simplest example of an index option is an option on the
average of two assets where $\psi_C= ( \frac{1}{2}(e^{x_1} + e^{x_2}) - 1)^+$
for a spread on two assets $\psi_C = (e^{x_1} - e^{x_2} - 1)^+$. \smallskip
\noindent {\bf Payoffs Class B:}
Consider $\eta( \max_{i=1}^n( \Phi_i))$ where $\Phi_i$ is a smooth function.
In the present paper {\it we will limit the discussion to payoffs of class A}.
Option payoffs of class B
require an additional regularization in our treatment
due to the presence of both positive part and max in their definition.
This will likely decrease the tightness of the upper bounds
and is best addressed in the context of a different kind
of symmetrization.
Payoff functions of Class A are only Lipschitz and so, for technical reasons, we will need to regularize them. This is achieved by
approximating the payoff $\psi$ by a function $\psi_{\epsilon}$ which lies in $W^{2, p, \mu}(\mathbb{R}^n)$ and such that $\psi_{e} \to \psi$ uniformly in $\mathbb{R}^n$.
The explicit form of this regularization will not play a role
until \S 6, see (\ref{regularizedinhomo}).
Denoting for brevity, the solution of obstacle
problem for given $\psi$ by $u[\psi]$,
our strategy below will be to obtain estimates for $u[\psi_{\epsilon}]$
and to then carry over these estimates to $u(\psi)$ using the
following result: Under the same conditions as those in Theorem \ref{thm1},
we have that
\begin{equation}
\| u[\psi] - u[\psi_{\epsilon}]\|_{L^{\infty}([0, T]\times\mathbb{R}^n])}
\leq \|\psi - \psi_{\epsilon}\|_{L^{\infty}(\mathbb{R}^n)}
\label{contraction}
\end{equation}
for the proof of which we refer to Bensoussan-Lions and and
Jaillet-Lamberton-Lapeyre.
In the results below, we will frequently work with $u=u[\psi_{\epsilon}]$.
We use the following result, which shows that the problem on all of
$\mathbb{R}^n$ can be approximated by a sequence of problems
on balls $B_k = \{ x : |x| < k\}$ with $k\to\infty$.
Let
\begin{gather*}
B_k = \{x \in \mathbb{R}^n : |x| < k\}, \quad
\partial B_k = \{ x\in \mathbb{R}^n: |x| = k\}, \\
H_k = L^2(B_k), \quad
V_k = \{ f \in H_k, \nabla f \in H_k\}
\end{gather*}
and define a bilinear form on $V_{k}$ for each $t\in [0, T]$
as follows: For $u, v\in V_{k}$,
\begin{equation}
\begin{aligned}
a_{k}(t; u, v) =& \sum_{i, j=1}^n \int_{B_k}
\sigma_{i j}(x, t) \frac{\partial u}{\partial x_i}
\frac{\partial v}{\partial x_j} dx \\
&- \sum_{i=1}^n \int_{B_k} \left(\sum_{j=1}^n ( r - d_i -\sigma_{ii}
- \sum_{j=1}^n \frac{\partial \sigma_{ij}}{\partial x_j}
\right) \frac{\partial u}{\partial x_i} v \,dx
+ \int_{B_k} r u v\, dx
\end{aligned} \label{defnofa_k}
\end{equation}
\begin{theorem}\label{thm2}
Under assumptions (H1)--(H5), there exists a unique solution $u_k$ of the
variational inequality
\begin{equation}
\begin{gathered}
u_k\in L^2( [ 0, T]; V_k), \quad
\frac{\partial u_k}{\partial t} \in L^2([0, T]; H_k) \\
u_k\geq \psi\quad \hbox{a.e. in }\quad [ 0, T] \times B_k
\\
\forall v\in V_k \quad\hbox{if}\quad v\geq \psi
\quad \hbox{then}\quad (\frac{\partial u_k}{\partial t},
v - u_k)_k + a_k(t, u, v -u)\geq 0 \\
u_k = \psi \quad\hbox{if}\quad x\in \partial B_k \\
u_k(0) = \psi \nonumber
\end{gathered} \label{problemonB_k}
\end{equation}
\end{theorem}
Moreover the approximate solutions $u_k$ have the property that, for $t\in
[0, T]$ they converge uniformly to $u$ as $k\to\infty$ on a ball
of radius $k/2$. This result is implicitly contained in the
research report by Jaillet, Lamberton and Lapeyre [1990] which is
a long version of their 1988 article. Since this report is not
easily available, for the reader's convenience, the salient points
of the arguments are described in Appendix 1. The choice of a ball
of radius $k/2$ is arbitrary. Any radius $c(k)$ such that
$\lim_{k\to \infty} \,(k - c(k) )\, =\, +\infty$ is possible, as
is clear from (\ref{escapefromball}).
\subsection*{Transformation to a inhomogeneous equation in the continuation
region}
We transform the solution of the problem guaranteed by Theorem \ref{thm2} to an
equivalent problem with {\it zero payoff} and a {\it non-zero source term}
in the continuation region, by making the transformation
$u_k \to v_k= u_k - \psi$.
For brevity let
\begin{equation}
L^2_k = L^2(B_k)\,,\quad
V_0^k = W^{1, 2}_0(B_k)\,.
\label{defnV_0^k}
\end{equation}
We then obtain the following problem
\begin{equation}
\begin{gathered}
v_k\in L^2( [ 0, T]; V_0^k )\,, \quad \frac{\partial v_k}{\partial t}
\in L^2([0, T]; L_2^k) \\
\quad v_k\geq 0 \quad \hbox{a.e. in } [ 0, T] \times B_k \\
\quad \forall\, w_k\in V_0^k \quad \mbox{with}\quad w_k \geq 0 \\
(\frac{\partial v_k}{\partial t},
w_k - v_k) + a_k(t, v_k, w_k -v_k) \geq -a_k(t, \psi_{\epsilon}, w_k -v_k)\\
v_k(0) = 0
\end{gathered}\label{smoothinhomog}
\end{equation}
In later sections the explicit form of the inhomogeneous term
$-\mathcal{L}_S \psi$, (see (\ref{spatial})), where $\psi$ is the payoff
function of a basket option, will be useful, and is given below. The
calculation is most easily carried out in the original variables
$S$ and then transfered to the new variables. This form for the
right hand side can be derived in two ways. One is to regularize
the payoff function $\psi$ from above and pass to the limit. The
other is to apply Federer's coarea formula directly to the
Lipschitz function $\psi$. In the case of a call on an index or a
call on a spread we get
\noindent\textbf{Calls:}
\begin{equation}
\begin{aligned}
- \mathcal{L}_S (\psi_C)
&= \sigma_{i j} w_i w_j e^{x_i + x_j}
\frac{1}{|\nabla \Phi_C|_{\Phi_C = 0^+}} \delta_{\{\Phi_C = 0 \}}\\
&\quad + ( r - d_i ) w_i e^{x_i} {\bf 1}_{\{\sum w_i e^{x_i} > 1\}} (\sum
w_i e^{x_i} - 1)^+ \\
&=\sigma_{i j} w_i w_j e^{x_i + x_j} \frac{1}{|\nabla \Phi_C|_{\Phi_C= 0^+}}
\delta_{\{\Phi_C=0\} } + (r - w_i d_i e^{x_i}) {\bf 1}_{\{\sum w_i e^{x_i} > 1
\}}\,.
\end{aligned}
\end{equation}
For puts we have:
\noindent\textbf{Puts}
\begin{equation}
- \mathcal{L}_S (\psi_P) = \sigma_{i j} w_i w_j e^{x_i + x_j}
\frac{1}{|\nabla \Phi_P|_{\Phi_P = 0^+}} \delta_{\{\Phi_P = 0\}} +
(w_i d_i e^{x_i} - r ){\bf 1}_{\{ 1 > \sum w_i e^{x_i}\}}
\end{equation}
where $\delta_{\{\Phi = 0\}}$ is shorthand for the
$n -1$ dimensional Hausdorff measure restricted to the $n-1$
dimensional surface $\{\Phi = 0\}$, ie. $\mathcal{H}_{n-1} \lfloor
\{\Phi = 0\}$. Note that the well known term $rK {\bf 1}_{\sum w_i
e^{x_i} > 1 }$ that usually appears in the right hand side in our
case has become $r {\bf 1}_{\sum w_i e^{x_i} > 1}$, since we work
in the normalized variables.
\subsection*{Semi-discretization of the regularized problem}
The estimates obtained below, using symmetrization techniques on the regularized, localized problem, require the technique of semi-discretization,
also known as Rothe's method. On the domain $B_k$
consider the following sequence of approximating problems.
Let $0\leq \tau_1\leq\tau_2 \cdots, \tau_n=T$ be the partition associated to the
$n$-th approximating problem where
\begin{equation}
{\Delta\tau}_n = \frac{T}{n}
\end{equation}
Define an approximating elliptic
variational problem by letting
\begin{equation} (V_0^k)^+ = \{ u \in
H_0^1(B_k) : u \geq 0\}
\end{equation}
(recall that $V_0^k$ was defined
earlier in (\ref{defnV_0^k})). and
\begin{equation}
a_k^{m, n} (u, v) =
\frac{1}{{\Delta \tau}_n} \int_{m {\Delta\tau}_n}^{ (m+1){\Delta
\tau}_n} a_k(s, u, v) ds, \quad\ u, v \in (V_0^k)^+,
\end{equation}
where $a_k$ was defined in (\ref{defnofa_k}). Let
\begin{equation}
G_{\epsilon}^D= - \mathcal{L}_S \psi_{\epsilon}^D, \quad \quad D = C
\quad\mbox{or}\quad P
\end{equation}
Recall that $\psi_{\epsilon} $ is a regularization of $\psi$ and then define
\begin{equation}
(G_{\epsilon}^D)^{m, n} = \frac{1}{{\Delta \tau}_n}
\int_{m {\Delta\tau}_n}^{ (m+1){\Delta
\tau}_n} G_{\epsilon}^D(s) ds \,.
\end{equation}
When we average the bilinear form
$a_k(\cdot , u, v)$ over the time interval $ [m{\Delta \tau}_n, (m
+1) {\Delta \tau}_n]$ , we average coefficients of the operator ,
eg. $r(x, \cdot), \sigma_{i j}(x, \cdot), d(x, \cdot)$.
Now for fixed $n$, let $v_k^{m, n} \in (V_0^k)^+, m = 1,.., n$ be the solution to the following elliptic
variational problem, defined recursively by $v_k^{0, n} = 0$ and
for all $w\in (V_0^k)^+$,
\begin{equation}
a_k^m( v_k^{m,n} , w - v_k^{m, n}) - ( (G_{\epsilon}^D)^{m,n}, w - v_k^{m,n})
\geq \big(\frac{v_k^{m-1,n} - v_k^{m, n}}{{\Delta\tau}_n} , w - v_k^{m, n}
\big)\,.
\end{equation}
Rewrite this expression as
\begin{equation}
\begin{aligned}
&a_k^m( v_k^{m, n} , w - v_k^{m, n}) +
\frac{1}{{\Delta\tau}_n} (v_k^{m, n}, w - v_k^{m, n}) \\
&\geq \big(\frac{1}{{\Delta\tau}_n} v_k^{m -1, n} + ((G_{\epsilon}^D)^{m,n},
w -v_k^{m, n})\big),
\quad \forall w \in (V_0^k)^+
\end{aligned} \label{approximatingvariationalproblem}
\end{equation}
Writing the problem in this form, makes clear that the coefficient
of the zero-th order term is increased by a factor
proportional to $\frac{1}{(\Delta \tau_n)}$ and so by well known
results our elliptic variational problem is solvable when
$(\Delta \tau_n)$ is small enough, for arbitrary inhomogeneity $ \frac{1}{ (\Delta \tau_n)} v_k^n + (G_{\epsilon}^D)^{m,n}$.
Moreover, one has the convergence result in Bensoussan-Lions \cite{Be-Li}
which shows that the solution of this problem converges weakly
in $L^2([0, T]: H_0^1(B_k))$ and weak * in $L^{\infty}([0, T]: L^2(B_k))$
to a solution of problem (\ref{smoothinhomog}).
This completes our discussion of introduction to the
optimal stopping problem and its rigorous formulation.
In the next section we give some background material on
symmetrization so that we may then introduce the
radially symmetric (in spatial variables) comparison problem.
\subsection{Background material on Symmetrization \label{symm}}
We recall some background material on rearrangements
and symmetrization. If $\phi\in L^1(\Omega)$ we let
\begin{equation}
\mu_{\phi}(t) = | \{ x\in\Omega: \phi(x) > t\}|, \quad t\in \mathbb{R},
\end{equation}
where, if $A$ is a Lebesgue measurable set, $| A |$ denotes the $n$ dimensional Lebesgue measure of $A$. $\mu_{\phi}(t)$ is called the distribution function of $\phi$. Also we define the monotone decreasing rearrangement of $\phi$ by
\begin{equation}
\phi^{*}(s) = \sup\{t : \mu_{\phi}(t) > s : s\in[0, |\Omega|]\}
\end{equation}
The increasing rearrangement of $\phi$ is defined by
\begin{equation}
\phi_*(s) = \phi^{*}(|\Omega| - s)\quad s \in [0, |\Omega|]
\end{equation}
We also let $\Omega^{*}$ be the solid
ball with the same volume as $\Omega$ and
define the Schwartz symmetrization of $\phi$ by
\begin{equation}
\phi^{\sharp}(x) = \phi^{*}(\omega_n |x|^n) \quad x\in \Omega^{*},
\end{equation}
where $\omega_n$ is the volume of he unit sphere in $\mathbb{R}^n$.
If $\mu_{\phi}$ is strictly decreasing and continuous
$\phi^*$ is the smallest decreasing function from
$[0, \Omega]$ such that $\phi^*(\mu_{\phi}(t))\geq t $
for every $t\in \mathbb{R}$. A basic property of $\phi^{*}$ , $\phi^{\sharp}$
is that $\phi$ and $\phi^{*}$ and $\phi^{\sharp}$ have the same distribution
function. This implies that for any Borel function $F$ we have
\begin{equation}
\int_{\Omega} F(\phi) = \int_{-\infty}^{+\infty} F(t) d\mu_{\phi}(t)
= \int_{\Omega^{\sharp}} F(\phi^{\sharp}) = \int_0^{|\Omega|} F(\phi^{*})
\end{equation}
In \S \ref{mainresults} we will use the notation $\phi^{*, k}$ and
$\phi^{\sharp, k}$ to emphasize that we are considering symmetrizations
relative to the domain $B_k$.
\subsection*{Properties of rearrangements}
The following properties of rearrangements will be
frequently used in the sequel.
\begin{itemize}
\item[(i)] (a) For a constant $c$,
$(cf)^{*}= c f^{*}$ and \\
(b) $ (f + c)^{*} = f^{*} + c$.
\item[(ii)]
If $h$ is a monotone increasing function then
$(h(f))^{*} = h(f^{*})$
\item[(iii)]
If $\hbox{\bf 1}_{A}$ is the characteristic function of a
set $A$ then
\begin{equation}
\left(\hbox{\bf 1}_{A}\right)^{*} = \hbox{\bf 1}_{A^{*}}
\end{equation}
\item[(iv)]
The Hardy-Littlewood inequalities
\begin{gather}
\int f g \leq \int f^{*} g^{*} \label{bothsame}\\
\int f g \geq \int f^{*} g_* \label{opposite}
\end{gather}
\item[(v)]
\begin{equation*}
\int_0^{\mu} (f + g)^{*} \leq \int_0^{\mu} f^{*} + \int_0^{\mu} g^{*}
\end{equation*}
\item[(vi)]
$$\int_{v > t} f \leq \int_{v^{*}> t } f^{*} $$
\item[(vii)]
If $h\geq 0$ is a non-increasing function,
\begin{equation*}
\int_0^{\mu} ( f + g)^* h(s) ds \leq \int_0^{\mu} ( f^* + g^*) h(s) ds
\end{equation*}
\item[(viii)] For non negative $f$ and $g$, Chong and Rice \cite{Ch-Ri},
\begin{equation}
\int_0^{\mu} (fg)^* \leq \int_0^r f^* g^*\,.
\end{equation}
\end{itemize}
\subsection*{Background material on Minkowski-Buseman
isoperimetric inequality}
Let $D$ be a closed set (not necessarily convex) with boundary satisfying certain
Lipschitz conditions . Given a symmetric non-negative quadratic form
$Q(x, x) = \sum Q_{i j} x_i x_j$, consider the action of $Q$ on the unit vector
$\mathbf{n}$ to the surface $\partial D$ and integrate it's square root over the surface
\begin{equation}
\bar{\Lambda} = \int_{\partial D} \sqrt{Q(\mathbf{n}, \mathbf{n})} \,dS
\end{equation}
Since $Q$ is a convex function (see for instance Bonnessen and Fenchel
\cite{Bo-Fe})
the weight $\zeta := Q^{1/2}(x, x)$, which is homogeneous of degree one,
can be used to determine the boundary $\partial C^Q$ of a convex set $C^Q$
as follows
\begin{equation}
\partial C^Q = \{\zeta^{-1}(u) u : u \in S^{n-1} \}
\end{equation}
Note that $C^Q$ is precisely that convex set determined by the
condition $y : \zeta( y) \leq 1 $. Indeed, for $u\in S^{n-1}$ $
\zeta( \frac{u}{\zeta(u)}) = 1$, since $\zeta$ is homogeneous of
degree one.Then we consider $(C^Q)^0$ the convex set polar to
$C^Q$. This is defined in terms of the support function
\begin{equation}
S(u)= \max \{(u, x) : x\in C^Q\} = \max_{\xi\in \mathbb{R}^n\neq 0}
\frac{(u, \xi)}{\zeta(\xi)},
\end{equation}
of the convex set as follows
\begin{equation}
(C^Q)^{0} = \{ y \in \mathbb{R}^n : S(u) \leq 1\}
\label{defnpolarset}
\end{equation}
We are now in a position to state a
special case of the Minkowski-Buseman inequality \cite{Bus}.
\begin{equation}
\bar{\Lambda}\geq n | D|^{\frac{n -1}{n}} |
(C^Q)^0|^{1/n} \label{Buseman}
\end{equation}
In the case of the convex sets considered
in the present paper, this inequality can be obtained from a
scaling argument using the principle axes of the quadratic form
$Q$.
This inequality holds for any
convex function $\zeta$ (ie. not necessarily of the form
$Q^{1/2}$) which is homogeneous of degree one. Note that this
inequality is always stronger than that obtained by applying the
standard isoperimetric inequality in conjunction with the lower
bound on the quadratic form $Q$.
\begin{equation}
\zeta(x) = Q(x, x)^{1/2} \geq \alpha^{1/2} |x|\,, \label{lowerboundquadraticform}
\end{equation}
where $\alpha$ is the smallest eigenvalue of the positive quadratic form
$Q$, and where the classical isoperimetric inequality reads
\begin{equation}
\mbox{surface area}\,(\partial D) \geq n \omega_n^{1/n}
|D|^{\frac{n - 1}{n}} \label{classicisoperimetric}
\end{equation}
Indeed, merely combining (\ref{lowerboundquadraticform})
and (\ref{classicisoperimetric}),
would yield
\begin{equation*}
\bar{\Lambda} \geq \alpha^{1/2} n \omega_n^{1/n} |D|^{\frac{n - 1}{n}},
\end{equation*}
and the inequality (\ref{Buseman}) is stronger because
we always have
\begin{equation*}
|(C^Q)^0 |^{ 1/n}\geq \alpha^{1/2} \omega_n^{1/n}
\end{equation*}
To see this, it suffices to note that the dual convex set $(C^Q)^{0}$
always contains the ball of radius $\alpha^{1/2}$.
Indeed for the support function $S(u)$ of $C^Q$ we have
\begin{equation*}
S(u) = \sup_{\xi \in S^{n-1}} \frac{ __}{ \zeta(\xi)} \leq
\frac{1}{\alpha^{1/2}} |u|,
\end{equation*}
since for $\xi\in S^{n-1}$ $Q(\xi, \xi) \geq \alpha$.
Therefore if $|u|\leq \alpha^{1/2}$ then $S(u) \leq 1$, and so $u\in
(C^Q)^0$ as required.
The parameters of the symmetrized problem are defined in the next section.
\section{Parameters of symmetrized one dimensional-in-space parabolic
variational inequality}\label{parameters}
\subsection*{Diffusion coefficient}
Let $\Lambda(\tau)$ be the largest purely time
dependent, symmetric,
positive definite matrix smaller than $\underline{\sigma}(x, \tau)$,
ie.
\[
\Lambda(\tau) \leq \underline{\sigma}(x, \tau) \quad \forall x\in \mathbb{R}^n
\]
and if $\tilde{\Lambda}(\tau)$ is any other purely
time dependent matrix that is smaller (in the sense of matrices, ie. the difference is a positive matrix) than
$\underline{\sigma}(x, \tau)$ (which exists since $\underline{\sigma}(x, t)$
is a smooth symmetric matrix) then
\[
\tilde{\Lambda}(\tau) < \Lambda(\tau)
\]
Let $Q_{\Lambda}(\xi, \xi) $ be the associated quadratic form.
Let
\begin{equation}
\lambda_{co}(\tau) =
\big\{\frac{|\{y \in \mathbb{R}^n : (Q_{\Lambda} y, y) \leq 1)\}^o|}{\omega_n}\big\}^{1/n},
\label{definitionoflambda}
\end{equation}
where $\omega_n$ is the
volume of the unit ball in $\mathbb{R}^n$. The definition of the polar
convex set was given in (\ref{defnpolarset}). In the present
case it coincides with the set $\{z\in \mathbb{R}^n : (\Lambda^{-1}(\tau)
z , z) \leq 1 \}$, where $\Lambda^{-1}(\tau)$ is the inverse of
$\Lambda(\tau)$. As an example, when $n=2$ and $\lambda_{11} = a^2
$, $\lambda_{22} =b^2$ and $\lambda_{12} = \lambda_{21} = 0$ the
convex set $C$ is the ellipse $a^2 x^2 + b^2 y^2 \leq 1$ and the
polar set $C^0$ is the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2}
\leq 1$, i.e., the eccentricities are switched. The volume of the
polar convex set is $\pi a b$, so the volume divided by the volume
of the unit sphere is $ab$ and $\sqrt{ab}$ is larger than the
square root of the minimum eigenvalue, $\min(\sqrt{a^2} ,
\sqrt{b^2})$, of the matrix $\Lambda$. The consideration of the
polar convex set is quite natural in cases where the volatility
matrix has eigenvalues that are {\it substantially different} in
magnitude.
In the context of partial
differential equations it's usefulness was pointed out and
illustrated in Alvino, Ferone, Trombetti and Lions \cite{Al-Fe-Tr-Li}. In
such cases it gives a considerably sharper estimate than the one
that would be obtained by using simply the ellipticity constant of
the matrix, which corresponds to the minimum of the eigenvalues.
\subsection*{Drift term} First define
\begin{equation}
\mathbf{\bar{D}}_{ij}(x, \tau) = r(x,t) - d_i(x, t) -
\sigma_{ii} (x, t) + \sum_{j=1}^n (\sigma_{i j})_{x_j} (x, t) \label{effectivedrift}
\end{equation}
{\bf Remark}
It is important to note that the definition of $D(x, \tau)$ involves
the partial derivative of the volatilities with respect to the logarithm of the stocks. This means that if we attempt to use the model to obtain
upper bounds for options in a given market, we build into our implementation of the model a guess at the size and (certainly) of the {\it sign}
of these partial derivatives ( a generalization to the multi-factor
model of incorporating information about the so-called ``smile'') and input this information
into our ``effective parameters'', i.e. the parameters of our
parabolic variational inequality with one spatial variable.
Denote by $\hat{\lambda}_{i j}(\tau), i, j=1,\cdots n$, the entries
of the matrix $\underline{\Lambda}^{-1}(\tau)$
and let, for fixed $\tau\in [0, T]$
\begin{equation}
D(\tau) = \lambda_{co}(\tau) \left( \max_{x \in \mathbb{R}^n} \hat{\lambda}_{i j}(\tau) \bar{D}_i(x, \tau) \bar{D}_j(x, \tau) \right)^{1/2}
\label{symmdrift}
\end{equation}
\subsection*{Zero-th order term}
The next two input functions of the one dimensional problem are defined in terms
of those in the multi-dimensional problem as follows:
\begin{equation}
C(\tau) = \min_{x\in \mathbb{R}^n} r(\tau, x)
\label{zeroorder}
\end{equation}
\subsection*{$\epsilon$ family of inhomogeneous terms}
\[
F_{\epsilon} = (- \mathcal{L}_S (\psi_{\epsilon}))^{\sharp, k}
\]
where $\mathcal{L}_S$ is given by (\ref{spatial}) ) and $f^{\sharp, k}$
denotes the Schwartz symmetrization of $f$, defined in \S \ref{symm}.
\section{Main Results}\label{mainresults}
Our main results are the following comparison results.
\begin{theorem}[Purely time dependent bounds for $v_k$)] \label{thm3}
Let $v_k(x, \tau) $, for $n\geq 1$ be the unique solution in
$L^2([0, T] ; V_0^k]$ with $v_t\in L^2([0, T]; L^2_k]$, of
(\ref{smoothinhomog}) on a large ball $B_k$, where $\psi$ is the
payoff of a put or call on an index or a spread, and let $W_k=
W_k^{\epsilon}$ be the unique weak radial solution, in the same space,
of the following problem:
\begin{equation} \label{mainequation}
\begin{gathered}
(W_k)_{\tau} -\lambda_{co}^2 (\tau) (W_k)_{rr}
-\frac{n-1}{r}\lambda_{co}^2(\tau) (W_k)_r + D(\tau) (W_k)_r +
C(\tau) W_k = F_{\epsilon}\\
\mbox{on } \{W_k > 0\} \\ %\label{comparisoneqntimedept}
(W_k)_r (0, \tau) = 0 \, \quad
W_k(r, 0) = 0
\end{gathered}
\end{equation}
where,
\[
F_{\epsilon} = - (\mathcal{L}_S \psi_{\epsilon})^{\sharp, k},
\]
and where the parameters $\lambda_{co}(\tau)$, $C(\tau)$, $D(\tau)$
were defined in \S \ref{parameters}, in the case $n \geq 2$ and
in the case $n=1$, $\lambda_{co}^2(\tau) = \min_{x\in
\mathbb{R}^n} \sigma(x,\tau)$ ($\sigma$ as in (\ref{defnsigma}) ), then
the following comparison principle holds
\begin{equation} \label{integralbound}
\int_0^V \frac{1}{e(\mu, \tau)} v_k^{*, k}(\mu, \tau) d\mu
\leq \int_0^{V} \frac{1}{e(\mu, \tau)} W_k^{*, k} (\mu, \tau) d\mu
\end{equation}
where, $\mu = \omega_n r^n$, $V \in [0, |B_k|]$ and
\begin{equation}
e(\mu,\tau) = \exp\left(\frac{D(\tau)}{(\lambda_{co}(\tau))^2}\,
\mu^{1/n}\right) \label{defnofe}
\end{equation}
\end{theorem}
Note the {\it coincidence region} for the radial problem corresponds to
the set $\{W_k = 0\} = B_k\setminus \{W_k > 0\}$.
As a consequence of (\ref{integralbound}), we have the following statement.
\begin{corollary} \label{coro1}
Under the same conditions of Theorem \ref{thm3}, we have
\begin{equation}
\max_{B_k} v_k(x, \tau)
\leq \max_{B_k} W_k(x, \tau) \quad 0\leq t \leq T
\label{inequalityformaxima}
\end{equation}
\end{corollary}
For the proof: Divide (\ref{integralbound}) by $V$ and let $V \to 0$.
\begin{corollary}[Purely time dependent bound for original problem] \label{coro2}
Let $u$ be the value of the American option, whose payoff $\psi(x)$ is a
call or put on an index or on a spread, then given $\delta > 0$
arbitrarily small there exists a $k = K(\delta, n, r, d_i,
\sigma_{i, j})$ such that on the ball of radius $\frac{k}{2}$ we
have
\[
|u(x, \tau) - \psi(x)| \leq \max_{x\in B_k} W_k^{\delta} + 2\delta
\]
\end{corollary}
\noindent{\bf Remark} The
dependence on $\delta$ is complicated to express but at its root
is the inequality
\[
|u - \psi| \leq |u - u_{\epsilon}| + |u_{\epsilon} -\psi_{\epsilon} | + |\psi_{\epsilon} - \psi|
\]
where $u_{\epsilon}$ is the solution of
\eqref{weightedVIregularity} with $\psi_{\epsilon}$ replacing by
$\psi$. The next step is to estimate $u_{\epsilon} -
\psi_{\epsilon}$ on $B_{k/2}$ by $v_{k}$ , the solution on $B_k$
of \eqref{smoothinhomog}, using the results of Appendix 1, and
then to estimate $v_k$ by $W_k$ using the result of Theorem
\ref{thm3}.
The results above can be complemented with the following result.
\begin{theorem}[time and state dependent bounds for $v_k$] \label{thm4}
Let $v_k(x,\tau) $, for $n\geq 1$ be the solution of (\ref{smoothinhomog})
on a large ball $B_k$ , let $ z = ( \sum_{i=1}^n |w_i| e^{x_i}
e^{ -(r - d_i) \tau} + 1)$ and let $W_k^z\geq 0 $ be the solution
of the spatially one dimensional equation
\begin{align*}
&(W_k^z)_{\tau} - \lambda_{co}^2 (\tau) V_{rr}^z -
\frac{n-1}{r}\lambda_{co}^2(\tau) V_r^z + D^z(\tau) (W_k^z)_r -
C^z(\tau) W_k^z \\
&= (-\frac{\mathcal{L}_S \psi_{\epsilon}}{z})^{\sharp, k} \quad
\mbox{in the region} \quad \{W_k^z > 0\} \\
& \frac{\partial W_k^z}{\partial r}(0, \tau) =0, \quad
W_k^z(r, 0) = 0
\end{align*}
where the coefficients $D^z(\tau)$ and $C^z(\tau) $ are defined by
the same algorithm (\ref{symmdrift}) and (\ref{zeroorder}) as in
section 3, but now applied to the new effective drift and zero-th
order terms
\begin{align*} (\bar{D}_{i}(x, \tau) + \frac{2}{z}\sigma_{ij} \frac{\partial z}{\partial x_j})
\end{align*}
and new zero order term
\begin{align*}
\frac{1}{z}({\mathcal L}_S z + C(x,\tau))
\end{align*}
and new forcing term
\begin{align*} F_{\epsilon}^{z} = (\sum_{i=1}^n |w_i| e^{x_i} e^{- d_i t} +
1) F_{\epsilon} \end{align*}
then we have the same estimates as in Theorem \ref{thm3}, with
$v_k^z = \frac{v_k}{z}$ replacing $v_k$ and with $W_k^z$ replacing
$W_k$.
\end{theorem}
\begin{corollary}[Time and state dependent bounds] \label{coro3}
Let $u[\psi]$ be the solution of the American option problem,
with payoff $\psi$ that is either a call or put on an index or on a spread,
then under the same conditions as in Theorem \ref{thm4}, for all $x\in B_k$
\begin{equation}
|u(x, \tau)- \psi(x) | \leq \frac{\max_{x\in B_k} (W_k^z)^{\delta} ) +
2 \delta }{\sum_{i=1}^n |w_i| e^{x_i} e^{-(r- d_i)\tau } + 1}
\end{equation}
\end{corollary}
\noindent{\bf Remark}
This second class of bounds can be thought of as deriving a comparison
principle for the price measures in a special set of units,
i.e. choosing a {\it numeraire}.
The particular numeraire used above is convenient but by no means
the only possible one.
\section{Proof of the main results}
\subsection*{Estimates for the elliptic variational inequalities associated to
the time-discretized problems}
In this section we provide estimates for the elliptic variational
inequalities associated to the time-discretized, and regularized
parabolic variational inequalities. Estimates for the elliptic variational inequalities then
lead to estimates for the parabolic one, using the method of Vasquez \cite{Va},
as developed by Ferone and Volpicelli \cite{Fe-Vo}
In carrying over the estimates for the elliptic problem to the parabolic one, we wish to allow the volatility and drift
parameters of the symmetrized problem to be time dependent. This
can however be accommodated by a simple extension of their
argument.
For this, let us further simplify the notation by letting
\begin{equation}
\bar{D}_i^{m, n} = (r - d_i - \sigma_{ii} -
(\sum_{j=1}^n (\sigma_{i j})_{x_j})^{m. n}
\end{equation}
We will lighten the notation by dropping, provided the context is clear, the superscript ``$n, m$'' in the presentation below.
We also will denote by $G^- $, the part of the inhomogeneous term that
results from solving the elliptic variational problem in the
preceding interval, ie.
\begin{equation}
G^- = \frac{1}{\Delta \tau} v_k^{m, n}
\label{secondpartofinhomogeneous}
\end{equation}
In this simplified notation, the localized and regularized problem on the
domain $B_k$ can then be written in the form, find $v_k\in V_0^k$ such that
for all $w_k \in V_0^k$,
\begin{equation}
\label{ellipticproblemsimplified}
\int_{B_k} (\sigma_{i j} (v_k)_{x_i} ( w_k)_{x_j} - \bar{D}_i (v_k)_{x_i} w_k
+ ( r + \frac{1}{\Delta \tau}) v_k \,w_k )
\geq \int_{B_k} (G_{\epsilon}^D + G^- ) w_k
\end{equation}
and we let
\begin{equation}
a_k(v, w)= \int_{B_k} (\sigma_{i j} (v)_{x_i} ( w)_{x_j} - \bar{D}_i (v)_{x_i} w
+ ( r + \frac{1}{\Delta \tau}) v\,w )
\label{ellipticbilinearform}
\end{equation}
We next follow closely the steps in Alvino-Matarasso-Trombetti, 1992,
and make the necessary adjustment to incorporate
the use of the Buseman-Minkowski inequality in the treatment of the
principal part and of the drift terms.
In (\ref{ellipticproblemsimplified}),
use the test function
\begin{equation}
\phi_h(x) = \begin{cases} h & t+ h < v_k(x)\\
v_k(x) - t & t < v_k(x) \leq t + h \\
0 & v_k(x) \leq t
\end{cases}
\end{equation}
with $h\geq 0$ and $t\in ]0, \hbox{sup u}[$. Since $ v_k \pm \phi_h \geq 0$
we can replace the test function $w_k$ in $V_0^k$, by the functions
$v_k\pm \phi_h$. We thus obtain
\begin{equation}
\frac{1}{h} a_k(v_k, \phi_h) = \frac{1}{h} \int_{B_k} ( G_{\epsilon}^D + G^-) \phi_h
\end{equation}
where, in the present simplified notation,
$a_k(v_k, \phi_h)$ is given by (\ref{ellipticbilinearform}).
Dividing by $h$ and taking the limit
as $h\to 0$, leads in a standard way (see Alvino-Lions-Trombetti
\cite{Al-Li-Tr1})
to the equality (using $w_k = \pm \phi_h$)
\begin{equation} \label{startingpoint}
- \frac{d}{dt}\int_{v_k > t} \sigma_{i j}(x, \tau) (v_k)_{x_i} (v_k)_{x_j}
= \int_{v_k > t} \bar{D}_i (v_k)_{x_i} -
\int_{ v_k > t} (r + \frac{1}{\Delta \tau} ) \, v_k
+ \int_{ v_k > t} (G_{\epsilon}^D + G^-)
\end{equation}
\noindent{\bf Case 1: $n\geq 2$}
The estimates are now carried out in the following steps:
\noindent{\bf Estimate from below of quadratic term, using Minkowski-Buseman
inequality}
We estimate from below
\begin{equation}
-\frac{d}{dt}
\int_{B_k} \sigma_{i j}(x, \tau) (v_k)_{x_i} (v_k)_{x_j}
\end{equation}
From Schwarz's inequality and an argument due to Talenti
\cite[page 711-713]{Ki-Mc},
\begin{equation}
-\frac{d}{dt} \int_{ v_k > t} \sqrt{\sigma_{i j} v_{x_i} v_{x_j} }
\leq \Big\{-\frac{d}{dt} \int_{v_k > t} \sigma_{i j} v_{x_i} x_{x_j} \Big\}^{1/2}
\Big\{-\frac{d}{dt} \mu_{v_k}(t)\Big\}^{1/2}
\label{Schwarz}
\end{equation}
Thus we must estimate below the term
\begin{equation}
-\frac{d}{dt} \int_{ v_k > t} \sqrt{\sigma_{i j} v_{x_i} x_{x_j} }
\label{A}
\end{equation}
The basic idea for doing this is to extend the Minkowski-Buseman
inequality to the setting where the surface $\partial D$ is not a
Lipshitz surface, but rather is the level set of an $H_0^1(B_k)$
surface. Such a generalization was in fact obtained by Amar and
Belletini \cite{Am-Be} and Alvino, Ferone, Lions and Trombetti
\cite{Al-Fe-Tr-Li},
who consider an wider class of functions generalizing those of
bounded variation in the usual metric to the case of the Minkowsky
metric. We illustrate the basic idea here by making stronger
assumptions on $v_k$. Let $v_k$ be a Lipshitz function function
we by applying the co-area formula, adapted to Sobolev functions,
Almgren-Lieb \cite{Alm-Lie} and Ziemer \cite{Zi} that expression (\ref{A})
equals
\begin{equation}
\int_{ v_k =t} \sqrt{\sigma_{i j} \frac{
(v_k)_{x_i}}{|\nabla v_k|} \frac{(v_k)_{x_j}}{|\nabla v_k|}}
\,d\mathcal{H}_{n-1} \label{B}
\end{equation}
so that setting \begin{equation}
\mathbf{\nu} = \frac{\nabla v_k}{|\nabla v_k|}
\end{equation}
Equation (\ref{B}) becomes
\begin{equation} \int_{\{v_k = t\}} \sqrt{ \sigma_{i j} \nu_i \nu_j }
\,d\mathcal{H}_{n-1} \label{C}
\end{equation}
and taking $\partial D = \{ v_k = t\}$
and $Q_{ij} = \sigma_{i j}$ in (\ref{Buseman}) and using the assumption
$\underline{\sigma}(x, \tau) \geq \underline{\Lambda}(\tau)$
we see from the definition (\ref{definitionoflambda}) that
\begin{equation}
\int_{v_k =t} \sqrt{ \sigma_{i j} \nu_i \nu_j }
\,d\mathcal{H}_{n-1} \geq n \omega_n^{1/n} \lambda_{co}(\tau)
|\{ v_k > t\}|^{\frac{n -1}{n}} \label{D}
\end{equation}
The non trivial technical questions involved in justifying these
manipulations for a class of functions (an appropriate
generalization of BV functions) which include those delt with here
are given in Amar and Belletini and in
Alvino-Ferone-Trombetti-Lions.
Combining (\ref{D}) and
(\ref{Schwarz}) we obtain
\begin{equation}
\Big\{-\frac{d}{dt} \int_{ v_k >
t} \sigma_{i j} v_{x_i} v_{x_j}\Big\}^{1/2} \geq n
\omega_n^{1/n} \lambda_{co}(\tau) (\mu_{ v_k > t}(t) )^{1 -
\frac{1}{n}} (-\frac{d}{dt} \mu_{ v_k > t})^{-1/2}
\label{endresultquadratic}
\end{equation}
\noindent{\bf Control of the drift term}
Let
\begin{equation}
\mathbf{\bar{D}}(x, \tau) = ( \bar{D}_1(x, \tau) , \bar{D}_2(x, \tau), \bar{D}_3(x, \tau) )
\label{driftfor1D}
\end{equation}
The term $-\int_{v > t} \bar{D}_i(x, \tau) (v_k)_{x_i}(x, \tau) $ is
estimated as follows. Recall the definition of $\underline{\Lambda}$ and
it's inverse in \S \ref{parameters} and denote the entries of the matrix by
$\lambda_{i j}$ and those of its inverse by $\hat{\lambda}_{i j}$.
Then, using the inequality
$(x, y) \leq (\zeta x , \zeta^0 y)$ for the convex, positive,
homogeneous function $\zeta x =\sqrt{Q(x, x)}$
($ Q x, x = \bar{\lambda}_{i j} x_i x_j$ and its conjugate,
$\zeta^0 y = \sqrt{Q^{-1}(x,x)}$ where $Q^{-1}$ is the quadratic form
associated to the inverse matrix $\underline{\Lambda}^{-1}(\tau)$ of
$\Lambda(\tau)$)
\begin{equation}
\begin{aligned}
& -\int_{v > t} \bar{D}_i(x, \tau) (v_k)_{x_i}(x, \tau) \\
& = \int_{v_k > t} \left(\hat{\lambda}_{i j} \bar{D}_i \bar{D}_j\right)^{1/2}
\left(\lambda_{i j} (v_k)_{x_i} ( v_k)_{x_j}\right)^{1/2} \\
& = \int_t^{+\infty} ds - \frac{d}{ds} \Big\{ \int_{v_k > s }
(\hat{\lambda}_{i j} \bar{D}_i \bar{D}_j)^{1/2}
\left(\lambda_{i j} (v_k)_{x_i} ( v_k)_{x_j}\right)^{1/2}\Big\} \\
&\leq \int_t^{+\infty} ds \Big( - \frac{d}{ds} \int_{v_k > s }
\hat{\lambda}_{i j} \bar{D}_i \bar{D}_j \Big)^{1/2}
\Big( - \frac{d}{ds} \int_{v_k > s } \lambda_{i j}(\tau) (v_k)_{x_i}
( v_k)_{x_j}\Big)^{1/2}
\end{aligned} \label{drift1}
\end{equation}
Recall the definition \eqref{defnV_0^k} of $D(\tau)$ from \S 3 (see (\ref{symmdrift}))
and note that we then clearly have
\begin{equation}
\Big( - \frac{d}{ds} \int_{v_k > s } \hat{\lambda}_{i j}
(\bar{D}_i \bar{D}_j \Big)^{1/2}
\leq \frac{D(\tau)}{\lambda_{co}(\tau)} ( -\mu_{v_k}'(s) )^{1/2}
\end{equation}
so that (\ref{drift1}) can be written
\begin{equation}
\begin{aligned}
&|\int_{v_k > t} \bar{D}_i(x, \tau) (v_k)_{x_i}(x, \tau) d V | \\
&\leq \frac{D(\tau)}{\lambda_{co}(\tau)} \int_t^{+\infty}
( -\mu_{v_k}'(s) )^{1/2}
\Big\{ -\frac{d}{ds}\int_{v_k >s} \lambda_{i j}(\tau) (v_k)_{x_i}
(v_k)_{x_j}\Big\}^{1/2}
\end{aligned}\label{drift2}
\end{equation}
We use (\ref{endresultquadratic}), rewritten in the form
\begin{equation}
\frac{ -\mu_{v_k}'(s))^{1/2} \mu_{v _k}(s)^{\frac{1-n}{n}}}
{n \lambda_{co}(\tau)}
\Big\{- \frac{d}{ds} \int_{v_k > s } \lambda_{i j}(\tau) (v_k)_{x_i} (
v_k)_{x_j}\Big\}^{1/2} \quad \geq 1
\label{1islessthan}
\end{equation}
and multiply (\ref{drift2}) under the integrand in by (\ref{1islessthan}),
so our final estimate of the drift term is
\begin{equation}
\begin{aligned}
& \big|\int_{v_k > t} \bar{D}_i(x, \tau) (v_k)_{x_i}(x, \tau) \big| \\
& \leq \frac{D(\tau)}{n \omega_n^{1/n} (\lambda_{co}(\tau))^2
}\int_t^{+\infty} \{(-\mu_{v_k}'(s)) (\mu_{v
_k}(s))^{\frac{1-n}{n}} (-\frac{d}{ds} \int_{v_k > s} \lambda_{i
j}(\tau) (v_k)_{x_i} ( v_k)_{x_j} ) \} ds
\end{aligned} \label{drift3}
\end{equation}
To estimate the zero-th order term in
(\ref{ellipticproblemsimplified}) use properties i)(b) and v) and
vi) of the rearrangement to get
\begin{equation} - \int_{v_k > t} ( r +
\frac{1}{\Delta\tau}) v_k \leq -\int_{v_k^{*,k} > t} (r_{*} +
\frac{1}{\Delta\tau}) v_k^{*,k} \leq -\int_{v_k^{*,k} > t} (
C(\tau) + \frac{1}{\Delta\tau}) v_k^{*, k},
\label{zeroorder1}
\end{equation}
where the last inequality follows from the definition (\ref{zeroorder})
of $C(\tau)$
For the inhomogeneous term, we record the following inequality, which
is needed in carrying over these estimates
to the parabolic case, using the argument in Ferone-Volpicelli (see p.563-565)
\begin{equation}
\int_{v_k > t} e^{-1}(\mu, \tau) (G_{\epsilon}^D + G^-)
\leq \int_{v_k^{*,k} > t} e^{-1}(\mu, \tau) ((G_{\epsilon}^D)^{*,k} + (G^-)^{*,k} ),
\label{inhomo1}
\end{equation}
where $e$ is defined below (see
(\ref{defnofe})). The inequality follows immdediately from
Property (vii) of the rearrangement, since
$e^{-1}= \exp(-\frac{D(\tau)}{ \lambda^2_{co}(\tau)}) \mu^{-1/n}$
is, for fixed $\tau$, a decreasing function of $\mu$.
Combining the various inequalities, we thus have arrived at the
inequality
\begin{align*}
& - \frac{d}{ds} \int_{v_k > s} \lambda_{i j}(\tau) (v_k)_{x_i} (v_k)_{x_j}\\
&\leq \int_{v_k > s} \sigma_{i j}(x, \tau) (v_k)_{x_i} (v_k)_{x_j} \\
&\leq \frac{D(\tau)}{n (\lambda_{co}(\tau))^2}\int_t^{+\infty} -(\mu_{v_k}'(s)
(\mu_{v _k}(s))^{\frac{1-n}{n}} \left( - \frac{d}{ds} \int_{v_k > s}
\lambda_{i j} (v_k)_{x_i} ( v_k)_{x_j}\right) \\
&\quad + \int_{v_k^{*,k} > t}(G_{\epsilon}^D)^{*,k} + (G^-)^{*,k} )
- (C(\tau) + \frac{1}{\Delta\tau}) v_k^{*,k}
\end{align*}
Now we make use of the following form of Gronwall's inequality.
When $v$ satisfies
\begin{equation}
v(t) \leq g(t) +\int_t^{+\infty} h(\tau) v(\tau) d\tau
\label{Gronwall1}
\end{equation}
and $v$ is zero at $+\infty$, then
\begin{equation}
v(t) \leq - \int_t^{+\infty} g'(\tau) e^{ \int_t^{\tau} h(s) ds}\,.
\label{Gronwall2}
\end{equation}
Applying this inequality, with
\begin{gather}
v(t) = - \frac{d}{ds} \int_{v_k > s} \lambda_{i j} (v_k)_{x_i} (v_k)_{x_j},\\
g(t) = \int_0^{\mu_{v_k}(t)} (G_{\epsilon}^D)^{*,k} + (G^-)^{*,k} )
- (C(\tau) + \frac{1}{\Delta\tau}) v_k^{*,k}
:= \int_0^{\mu_{v_k}(t)} (\mathcal{R}_v(s)) ds, \label{choiceofg}\\
h(t) = \frac{D(\tau)}{n (\lambda_{co}(\tau))^2} (-\mu_{v_k}'(t))
(\mu_{v _k}(t))^{\frac{1-n}{n}},
\label{choiceofh}
\end{gather}
we obtain
\begin{equation} \label{exponential}
\begin{aligned}
v(t) & \leq \int_t^{+\infty} (-\mu_{v_k}'(s) \mathcal{R}(s)
\exp\left( - \frac{D(s)}{n (\lambda_{co}(\tau))^2} (\int_t^s
(\mu_{v_k}(s))^{\frac{1-n}{n}} \mu_{v_k}'(u) du)\right) ds \\
&= \exp(\frac{D(\tau)}{
(\lambda_{co}(\tau))^2})\,(\mu_{v_k}(t))^{1/n}) \\
&\quad\times \int_t^{+\infty}
\exp(- \frac{D(\tau)}{ (\lambda_{co}(\tau))^2}
(\mu_{v_k}(s))^{1/n} (-\mu_{v_k}'(s))) \mathcal{R}_v (s) ds
\end{aligned}
\end{equation}
which introducing the notation
\begin{equation}
e(\mu, \tau) = \exp\Big(\frac{D(\tau)}{ \omega_n^{1/n}
\lambda_{co}(\tau))^2}\Big) \mu^{1/n} \label{defnofe1}
\end{equation}
and making a change of variables leads to
\begin{equation}
v(t) \leq e(\mu, \tau) \int_0^{\mu(t)} e^{-1}(\mu
,\tau)\,\mathcal{R}_v(\mu) d\mu \label{finalestimatedrift}
\end{equation}
Combining this estimate again with (\ref{endresultquadratic}), we
obtain
\begin{equation} (-\mu_{v_k}'(t))^{-1} \leq \frac{1}{(n
\omega_n^{1/n})^2 \lambda_{co}(\tau)^2} \mu_{v_k}^{2 -
\frac{2}{n}}(t)\; e(\mu, \tau) \int_0^{\mu_{v_k}(t)} e^{-1}(\mu)
\mathcal{R}_v(\mu) d\mu\,.
\end{equation}
Arguing as in Talenti \cite[pp. 711-713]{Ta1}, the last inequality can be
expressed in terms of the decreasing rearrangement
\begin{equation}
(v_k^{*,k})' \leq \frac{1}{(n \omega_n^{1/n})^2 \lambda_{co}^2}
\mu^{\frac{2}{n}- 2 }(t) \; e(\mu, \tau) \int_0^{\mu_{v_k}(t)}
e^{-1}(\mu, \tau) \mathcal{R}_v(\mu) d\mu,
\end{equation}
with $ 0 < \mu < |B_k|$.
With this differential inequality in hand, comparison arguments
for one dimensional differential inequalities, may be used,
as discussed in Alvino-Matarasso-Trombetti, to establish Theorem \ref{thm3}, the key
point being that when $\lambda_{co}(\tau)$ and $D(\tau)$ are defined
as in \S 3, the differential inequality becomes an equality.
The interested reader is refered to the proofs
Lemma 2.3 p. 275 and Theorem 3.2 p. 277 of Alvino-Materasso-Trombetti
\cite{Al-Ma-Tr}. \smallskip
\noindent{\bf Case 2: $n=1$}
The one dimensional case lends itself to the sharpest estimates.
Beginning from (\ref{startingpoint}), after using the same test function we have
\begin{equation} \label{quadraticpart}
- \frac{d}{dt}\int_{v_k > t} \sigma_{i j}(x, \tau) (v_k)_{x}^2
= \frac{d}{dt}\int_{v_k > t} \bar{D}_i (v_k)_{x} -
\int_{ v_k > t} (r + \frac{1}{\delta}{\tau}) v_k
+ \int_{ v_k > t} (G_{\epsilon}^D + G^-)
\end{equation}
Letting $\lambda(\tau)= \min_{x\in \mathbb{R}} \bar{\sigma}(x, \tau)$,
and by Shwarz's inequality we get
\begin{equation}
- (\lambda (\mu_{v_k}')^{-1}) \frac{d}{dt} \int_{ v_k > t} |(v_k)_x|^2 \leq -
\frac{d}{dt}\int_{v_k > t} \sigma_{i j}(x, \tau) (v_k)_{x}^2
\end{equation}
Combining this with the sharp relation
\begin{equation}
- \frac{d}{dt} \int_{v_k > t} |(v_k)_x| = \mathcal{M}(v_k)(t),
\end{equation}
where $\mathcal{M}(v_k)$ is the multiplicity function of $v_k$, we arrive at
the lower bound
\begin{equation}
\mathcal{M}^2(v_k(t)) \lambda(\tau) (- \mu_{v_k}'(t))^{-1}
\end{equation}
for the left hand side 0f (\ref{quadraticpart}). Note that since
$v_k$ is zero on the boundary, $\mathcal{M}(v_k)(t) \geq 2$ and is
equal to $2$ for all $t$ if and only if the superlevel sets $\{v_k
> t\}$ are all equivalent to intervals. Deriving these claims
rigorously for BV functions is a bit delicate. A beautiful
presentation thereof appears is Talenti \cite[pp.102-105]{Ta2}.
We estimate the drift term,
\begin{equation}
\big|\int_{v_k > t} \bar{D}(x, \tau) (v_k)_x \big| \leq
\max_{\mathbb{R}} |\bar{D}(x, \tau)| ) \int_{v_k > t} |(v_k)_x|
\end{equation}
and let $D(\tau) = \max_{\mathbb{R}} |\bar{D}(x, \tau)|$.
In the remaining estimates the one dimensional Schwarz
symmetrization (which coincides in the case $n=1$ with the symmetrically
decreasing rearrangement) is used
and by the same estimates as in the multidimensional case
we arrive at the result that $v_k$ may be estimated above
by the (even in $x$) solution of the one dimensional problem
\begin{gather*}
(W_k)_{\tau}( |x|, \tau) - \lambda(\tau) (W_k)_{xx}(|x|, \tau)
+ D(\tau) (W_k)_x (|x|, \tau) + C(\tau) W_k( |x|, \tau)\\
= F_{\epsilon}(|x|, \tau)\quad \in \{ W_k > 0\} \\
W_k\geq 0\quad\mbox{in}\quad B_k \\
W_k(|x|, 0) = 0 \quad x \in B_k \\
\frac{\partial W_k}{\partial |x|}( 0, \tau) = 0 \quad 0 < \tau \leq T
\end{gather*}
where $F_{\epsilon}(x, \tau) = - (\mathcal{L}_S \psi_{\epsilon})^{\sharp, k}$,
as claimed.
\section{Explicit form for regularized inhomogeneous term}
For basket and spread options, in the dimensionless variables, we have
\begin{gather*}
\psi_C = \eta(\Phi_C ), \quad \Phi_C = \sum (w_i e^{x_i} - 1),\\
\psi_P = \eta(\Phi_P), \quad \Phi_P = (1 - \sum w_i e^{x_i})\,.
\end{gather*}
We approximate $\eta$ by a smooth function defined below
and let
\begin{equation}
\psi_{\epsilon}^C = \eta_{\epsilon}(\Phi_C)\,, \quad \psi_{\epsilon}^P= \eta_{\epsilon}(\Phi_P)
\label{defpsieps}
\end{equation}
In the case of calls we consider the upper approximations
$\eta_{\epsilon}^+:\mathbb{R} \to \mathbb{R}$ to the function $\eta$
\footnote{Note that such smoothings of the function $\eta$
are well known, but we exploit the specific form of the smoothing
below to use the smoothing in conjunction with the
symmetrization}
\begin{equation}
\frac{d^2}{dy^2} \eta_{\epsilon}^+(y)=
\frac{1}{\epsilon} \,\hbox{\bf 1 }_{\{-2\epsilon < y < -\epsilon \}}
\end{equation}
Thus the derivative $(\eta_{\epsilon}^+)'$ grows from zero to $1$ on the
interval $-2\epsilon, -\epsilon$. Similarly, we define $\eta_{\epsilon}^-$ so that
\begin{equation}
\frac{d^2}{dy^2} \eta_{\epsilon}^- = \frac{1}{\epsilon}\hbox{\bf 1}_{\{\epsilon < y < 2\epsilon\}}
\end{equation}
and then in all of the above
examples we replace $\eta$ by $\eta_{\epsilon}^+$ in the case of calls
and by $\eta_{\epsilon}^-$ in the case of puts. Since $\eta_{\epsilon}^{\pm}$
thus defined are Lipschitz with Lipschitz constant equal to $1$, it is immediate that for
these regularizations, we have the property
\begin{equation}
|\psi_{\epsilon}^D - \psi^D | \leq \epsilon
\end{equation}
and therefore we may use the comparison principle mentioned in \S 2.
In addition the first derivatives of $\psi_{\epsilon}$ are well behaved
but the second derivatives are not and this will be explored below.
\noindent{\bf Calculation of the smoothed source term}
The calculation of the action of the operator $\frac{\partial }{\partial \tau}
+ \mathcal{L}_S$ on the payoff $\psi_{\epsilon}$ is more straightforward to carry out
in the original spatial variables $S$. It can then be easily transposed
to the new variables $x$ and new dependent variable
$\frac{u}{K}$.
We have
\begin{gather*}
(\eta_{\epsilon}(\Phi))_{S_i} = \eta_{\epsilon}'(\Phi) \Phi_{S_i} \\
(\eta_{\epsilon}(\Phi))_{S_i S_j}= \eta_{\epsilon}''(\Phi) \Phi_{S_i}\,\Phi_{S_j}
+ \eta'(\Phi)\Phi_{S_i S_j}
\end{gather*}
We let
\[
G^C_{\epsilon}(x) = - \mathcal{L}_{ S}(\eta_{\epsilon}^+(\Phi^C)), \quad
G^P_{\epsilon}(x) = -\mathcal{L}_{S}(\eta_{\epsilon}^-(\Phi^P))
\]
So,
\begin{equation}
\begin{gathered}
G^{D}_{\epsilon} = - S_i S_j \sigma_{i, j} \left(\eta_{\epsilon}'\Phi_{S_i S_j}^D +
\eta_{\epsilon}''\Phi_{S_i}^D\Phi_{S_j}^D\right) + S_i (r - d_i) \;\eta_{\epsilon}'
\Phi_{S_i}^D - r \psi_{\epsilon}^D\\
D^+ =C, \quad D^-= P,\quad
\psi_{\epsilon}^{D^{\pm}} = \eta_{\epsilon}^{\pm}(\Phi^{D^{\pm}})
\end{gathered} \label{defnF^D}
\end{equation}
For the action of the operator $-\mathcal{L}_S$
on $\psi_{\epsilon}^{D^{\pm}}$, we get
\begin{equation}
G^{D^{\pm}}_{\epsilon}(x) = \frac{1}{\epsilon}\sum_{i, j = 1}^n e^{x_i + x_j} w_i w_j
\sigma_{i, j} {\bf 1}_{ \{-2\epsilon < \Phi^{D^+} < -\epsilon\}}
\pm w_i e^{2 x_i} (r - d_i) \eta_{\epsilon}'(\Phi^{D^{\pm}}) - r \psi_{\epsilon}^{D^{\pm}}\,.
\label{regularizedinhomo}
\end{equation}
Note that the only difference in the form of the right hand side
for calls and puts is the call has $`+'$ and the put
a $`-'$ multiplying the term involving a first derivative.
\subsection*{Dependence of the solution on $\epsilon$ and $k$}
With (\ref{regularizedinhomo}) we have in fact
a 2- parameter family of comparison problems. In this section
we address the question as to how the
bounds derived depend on these parameters.
In two important cases we can give an analytical elucidation
of this dependence.
In the other cases the question needs to be investigated
numerically.
The two cases where an analytical elucidation is possible are:
$\bullet$ An index Put option (with or without dividends), with payoff
$\psi^P = (1 - \sum_i w_i e^{x_i})^+$, the dependence of $k$
simplifies considerably. Indeed in this case the
continuation region is, in the original $S$ variables
connected and bounded away from zero where the minimum distance
can be estimated via the lower bound on the volatility
matrix in the lognormal coordinates . This is clear
intuitively and follows from the results in Broadie and
Detemple \cite{Br-De} and Villeneuve \cite{Vi}. Translated to the $x$ variables
this
means that the continuation region is connected and bounded away from
$- \infty$ , i,e. $\min_{i=1, \cdots n}\min_{x\in \mathcal{C}}$ $x_i = -L_1$,
in all directions. Furthermore when any $x_i$ is sufficiently large
the payoff of the option is zero in the complement of the region
$ \mathcal{C}_n=\mathcal{C} \cap \{\sum_i w_i e^{x_i} < 1\}$
and the right hand side in (\ref{regularizedinhomo}) is identically
zero. Let $L_2$ be a constant so large that $\mathcal{C}_n$
is contained in $B(0, L_2)$. The maximum
value of $u - \psi^D$ is not reached on the latter set.
Thus, if we choose $k_0= \max(L_1, L_2)$ we can be sure that
the maximum of $u - \psi$ will be captured in $B_{k_0}$ and there
is no need to choose a larger $k_0$. For fixed $k$ the optimal
bounds as a function of $\epsilon$ must be determined numerically.
$\bullet$ For a call or put option on one asset, as shown in Laurence 2000
(pages 49-51),
it is possible to pass to the limit as $\epsilon \to 0$ and
derive a limiting comparison problem for any fixed $k$.
For a put the same considerations as above apply to find
a reasonable value for $k$. Using put call symmetry as in Detemple 2001
we can then extend the result to calls.
\section{Appendix:
The Relation between the solution on all of $\mathbb{R}^n$ and the solution on $B_k$}
In Bensoussan and Lions, 1982, it is shown that $u_k$, the solution of the variational problem
in Theorem \ref{thm2}, is also a solution of an optimal stopping problem on $B_k$. Let
$T_k^{t, x} = \inf\{s > t, |X_s^{t, x}| > k\}$ then $u_k$ solves
\[
u_k(x, t) = \sup_{\tau\in T_k} E[e^{-\int_t^{\tau\wedge T_k^{t, x}} r(s) ds}
\psi(X_{\tau\wedge T_k^{t, x}}^{t, x})]
\]
On the other hand $u$, the solution on all of $\mathbb{R}^n$, is also a solution of
the problem
\[
u(x, t) = \sup_{\tau\in T^{t, T}} E[ e^{- \int_t^{\tau} r(s) ds}
\psi(X_{\tau}^{t, x})]
\]
So we need to estimate $|u(x, t) - u_k(x, t)|$.
The key to such an estimate is to note that
\begin{align*}
|u(x, t) - u_k(x, t)|
&\leq \sup_{\tau\in T_{t, T}} E\big[
\big|\psi(X_{\tau}^{t, x }) - \psi(X_{T_k}^{t, x}) \big|
\, {\bf 1}_{T_k^{t,x} < \tau}\big] \\
&\leq E\big[2 t[2 \sup_{[0, T]} | \psi(X_s^{t, x})|\, {\bf 1}_{T_k^{t,x} <\tau}
\big] \\
&\leq 2 M \sqrt{E(\exp(2 M \sup_{[t, T]} | X_s^{t, x} |})\sqrt{P(T_k^{t, x} < T)}
\end{align*}
To estimate the difference between $u$ and $u_k$, we thus need
to estimate the two terms
\begin{gather}
E(\exp(2 M \sup_{[t, T]} | X_s^{t, x} | ), \label{aa}\\
P(T_k^{t, x} < T) \label{bb}
\end{gather}
for $x$ restricted to a ball of radius $ R(k) < k$.
The dependence of $R(k)$ on $k$ is clarified below. Details as to how these terms may be estimated
are given in the report by Jaillet-Lamberton-Lapeyre
\cite[pages 107-109]{Ja-La-La0}.
We recall the form of the final estimate and review its dependence
on the constants. For (\ref{aa}) this takes the form
\[
\sup_{(x, t)\in [0, T]\times \{x\in \mathbb{R}^n : |x|\leq k\}} E\big[\exp(2M
\sup_{[t, T]} | X_s^{t, x}|)\big] \leq C < \infty,
\]
where $C$ depends on a pointwise bound on the magnitudes of the drift coefficient
and of the volatility matrix, as well as on $T$ and $R$.
For (\ref{bb}) let
\[
\hat{\sigma} = \sup \Big(\sum_{i=1}^n \sum_{j, k=1}^n
\sigma_{ij}\sigma_{ik}\Big)^{1/2}
\]
i.e. a bound for the trace of $\sigma^* \sigma$.
\begin{equation}
P(T_K^{t, x} < T) = P\Big( \sup_{s\in [t, T]} |X_s^{t, x}| > k \Big)
\end{equation}
and then estimate the right hand side below
\[
P\Big( \sup_{s\in [t, T]} |X_s^{t, x}| > k\Big)
\leq n P\Big\{\sup_{s\in[0, \hat{\sigma}^2 T]} |B_s| > (k - R - D T) \times
\frac{1}{n}\Big\}
\]
where $B_s$ is a standard one dimensional Brownian motion.
Thus we get
\begin{equation}
\begin{aligned}
P(\sup_{s\in[0, \hat{\sigma}^2 T]} |B_s| > k - R - D T)
&\leq 2 P\Big( \sup_{s\in[0, \hat{\sigma}^2 T]} B_s > k - R - D T\Big)\\
&= 2(1 - N(k - R - DT))
\end{aligned}\label{escapefromball}
\end{equation}
where $N$ is the cumulative normal distribution function.
Therefore in order to obtain an estimate for (\ref{bb}) that becomes
small as $k\to\infty$ we must choose $k $ to be large compared to $R$
ie choose $k - R - D T \gg 1$.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1.eps} %amputwithdiv1.eps
\end{center}
\caption{An American put option with parameters
$K = 15$, $r= 0,5 $, $d =.03$. Here $\sigma = 0. 2$
is plotted against spot one year from expiration. The $S$ coordinate of the
point of tangency between the payoff option value
corresponds to the position of the free boundary at that time}
\end{figure}
\begin{figure}[htb] \newpage
\begin{center}
\includegraphics[width=0.6\textwidth]{fig2.eps} %amputwithdivnorm.eps
\end{center}
\caption{The American put option with the same parameters
as in Figure 1, but now in the normalized
logarithmic variable $log(S/K)$ and
normalized payoff $(1 - e^{x})^+$}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3.eps} %amputwithdifference2.eps
\end{center}
\caption{The difference between the
option's value and the payoff's value, in the
normalized variables}
\end{figure}
\subsection*{Acknowledgment}
We thank the Courant Institute for providing a stimulating
atmosphere for carrying out this research, and to Marco
Avellaneda for his helpful suggestions.
\begin {thebibliography}{00}
\bibitem{Al-Li-Tr1}
Alvino, A., Lions, P.L., Trombetti, G.,
{\em Comparison results for elliptic and parabolic equations via Schwarz
symmetrization},
Ann. I.H.P. An. Nonlineaire, \bf{ 7}(1990), 25-50.
\bibitem{Al-Ma-Tr} Alvino, A., Matarasso, S., , Trombetti, G.,
{\em Variational Inequalities and rearrangements}, Rend. Mat. Acad.
Lincei, s {\bf 9}(1992), v 3, 271-285.
\bibitem{Al-Fe-Tr-Li} Alvino, A., Ferone, V., Trombetti, G., Lions, P.L.,
{\em Convex symmetrization and applications}, Anal. Non
Lin\'eaire, {\bf 14}(1997), no. 2, 275-293.
\bibitem{Alm-Lie} Almgren, F., Lieb, E.,
{\em Symmetric decreasing rearrangement can be discontinuous},
J. Am. Math Soc., {\bf 4}(1989), 683-773.
\bibitem{Av-Le-Pa} Avellaneda, M., Levy, A., Paras, A.,
{\em Pricing and hedging derivative securities with uncertain volatilities},
Appl. Math Finance, {\bf{3}(1995), 23-51.
\bibitem{Am-Be} Amar, M., Bellettini, G.,
{\em A notion of total variation depending on a metric with discontinuous
coefficients}, Ann. Inst. H. Poincar\'e Anal. Nonlin\'eaire}
\bf{11}(1994), n0. 1, 91-133.
\bibitem{Av-La} Avellaneda, M., Laurence, P.,
{\it Quantitative Modeling of Derivative Securities: From theory to practice},
Chapman Hall-CRC (1999).
\bibitem{Ba-Ca-Zh} Bakshi, G.S., Cao, C., Zhiwu, C,
{\em Empirical performance of alternative option pricing models},
J. of Finance, \bf{ 52}(1997), No. 5, 2002-2049
\bibitem{Ba1} Bandle, C.
{\em Symmetrizations in parabolic differential equations},
J. Analyse Math., \bf{ 30}(1976), 98-112
\bibitem{Ba-Wh} Barone-Adesi, G., Whaley, R. E.,
{\em Efficient analytic approximation of amaerican option values},
Journal of Finance, {\bf 42}(1987), 301-320.
\bibitem{Be} Bensoussan. A.,
{\em On the theory of option pricing}, Acta. Appl. Math., {\bf{ 2}}(1984), 139-158.
\bibitem{Be-Li} Bensoussan, A., Lions, J.L.,
{\em Applications of variational inequalities in stochastic control},
Studies in Applied Mathematics, {\bf{ 12}}(1982), North Holland, Amsterdam.
\bibitem{Bo-Fe} Bonnesen, Fenchel,
{\it Theory of convex bodies}, Chelsea Press
reprints, (1987). Translated from the German and edited by L. Boron, C.
Christenson and B. Smith. BCS Associates, Moscow, ID.
\bibitem{Br-Sc} Brennan, M.J., Schwartz, E.S.,
{\em The valuation of American put options}, Journal of Finance, {\bf{32}}(1976),
449-462.
\bibitem{Br-De} Broadie, M. Detemple, J.,
{\em The valuation of American options on multiple assets}, Math Finance,
\bf{ 7}(1997), (3), 241-286.
\bibitem{Br-De-Gh-To} Broadie, M., Detemple, J. , Ghysels, E.,Torres, O.,
{American options with stochastic dividends and volatility: a non-parametric
investigation}, J. Econometrics, \bf{94}(2000) (1-2), 53-92.
\bibitem{Br-De-Gh-To2} Broadie, M., Detemple, J. , Ghysels, E.,Torres, O.,
{\em Non parametric estimation of American options'exercise boundaries
and call prices}, Journal of Economic Dynamics and Control, \bf{ 24}(2000),
1829-1857.
\bibitem{Bu} Buff, R.,
{\em Worst case scenarios for American Options},
Int. J. Theor. Applied Finance, 3 (2000), no. 1, 25-58.
\bibitem{Bus} Buseman, H., {\em A theorem on convex bodies}, Proc.
Natl. Acad. Sci. USA, \bf{ 35}(1949), 27-31.
\bibitem{Carr} Carr, P., Jarrow, R., Myeni, R.,
{\em Alternative characterizations of American put options},
Math. Finance, 2(1992), 87-106.
\bibitem{Ch-Ri} Chong, K. M., Rice, N.M.,
{\it Equimeasurable rearrangements of functions}, Queen's Papers in Pure and
Applied Mathematics (1971), no. 28.
\bibitem{Det} Detemple, J.B., {\it Handbooks in Mathematical Finance:
Topics in Option Pricing, Interest rates and Risk Management}, J.
Cvitanic, E. Jouni and M. Musiela, eds, July 2001: 67-104.
\bibitem{Di-Mo} Diaz, I., Mossino, J., Isoperimetric
{\em Inequalities in the parabolic obstacle problems},
J. Math. Pures Appliqu\'ees, \bf{9}(1992), 71, no. 3, p. 233-266.
\bibitem{El} El Karoui, N.,
{\em Les aspects probabilistes du contr\^ole stochastique},
Lecture Notes in Mathematics, \bf{ 876}(1979), 73-238.
\bibitem{Fe-Vo} Ferone, V., Volpicelli, R.,
{\em Comparison results for solutions of parabolic equations},
Ricerche Matematiche, \bf{42}(1996), no. 1, p. 179-191.
\bibitem{Gu-Mo} Gustafsson, B., Mossino, J.,
{\em Isoperimetric inequalities for the Stefan problem},
Siam J. Mathematical Analysis, \bf{20}(1989), no. 5, 1095-1108.
\bibitem{Ha-Li-Po} Hardy, G.H., Littlewood, J.E., Polya, G.,
{\it Inequalities}, Cambridge University Press, Cambridge (1958).
\bibitem{Ja-La-La0} Jaillet P., Lamberton, D., Lapeyre, B.,
{\em Variational Inequality approach to American options},
Acta Appl. Math., 21(1988), no. 3, 363-289.
\bibitem{Ja-La-La} Jaillet P., , Lamberton, D., Lapeyre, B.,
{\it Analyse Num\'erique des options am\'ericaines},
Cahier du Cerma, (1990) no. 9, 66-126.
\bibitem{Ju-Zh} Ju , N, Zhong, R.,
{\em An approximate formula for pricing American Options},
Journal of Derivatives, Winter 1999, 31-40.
\bibitem{Ka} Kawohl, B.
{\it Rearrangements and convexity of level sets in PDE},
Lecture Notes in Mathematics, 1150, Springer Verlag (1985).
\bibitem{Kar} Karatzas, I.,
{\em On the pricing of American options}, Appl. Math. Optimization,
\bf {17}(1988) , no. 1, 37-60.
\bibitem{Ke} Kesavan, S.,
{\em Comparison results via Schwarz symmetrization-a survey},
Sympos. Math., 35, Cambridge University Press, Cambridge (1994).
\bibitem{Ki-Mc} Kinateder, K., Mc Donald , P.,
{\em Variational Principles for average exit time moments for
diffusions in Euclidean space}, Proc. American Math. Soc.,
\bf{127}(1999), no 9, p. 2767-2772.
\bibitem{Ki-Mc1} Laurence, P.,
{\em Bounds for an american option
on several assets: Index and Spread Options}, (2000). Available at
http://papers.ssrn.com
\bibitem{Lyons} Lyons, T. J.,
{\em Uncertain volatility and the risk-free
synthesis of derivatives}, Applied Mathematical Finance 2(1995), 117-133.
\bibitem{Me} Merton, R.C., {\em Theory of rational option pricing},
Bell J. Econ. Management, {\bf 4}(1973), 141-183.
\bibitem{Mo} Mossino, J., {\it In\'egalit\'es isoperimetriques et
applications en physique}, Travaux en cours, Hermann \'editeur (1984).
\bibitem{Ne} Necati Ozisik, M., {\it Boundary Value Problems of
heat conduction}, Dover Publications (1968).
\bibitem{Po-Sz} Polya, G., Szeg$\ddot{o}$, G., {\it Isoperimetric Inequalities
in Mathematical Physics}, Princeton University Press, Princeton, NJ (1951).
\bibitem{Sc} Schwarz, H.A.,
{\it Gesammelte Mathematische Abhandlungen}, Vol {\bf 2}(1890),
Springer Verlag, Berlin.
\bibitem{St} Steiner, J. {\it Gesammelte Werke}, Vol {\bf 2}(1882), Berlin.
\bibitem{Ta1} Talenti, G.,
{\em Elliptic equations and rearrangements},
Ann. Scuola Norm. Sup. Pisa, \bf{ 3}(1976), p. 687-718.
\bibitem{Ta2} Talenti, G., {\em The standard isoperimetric theorem},
Handbok of Convex geometry, Elsevier Science Publishers (1993).
\bibitem{Va} Vazquez, J.L., {\em Symmetrisation pour $u_t = \Delta \phi(u)$
et applications}, C.R. Acad. Sci. Paris, s\'erie I, Vol {\bf 295}(1983),
p. 71-74 et Vol 296, 455.
\bibitem{Wa} Walter, W., {\it Differential and Integral Inequalities},
Ergebnisse der Mathematik und Ihre Grenzgebiete, Band {\bf 55}(1970).
\bibitem{Vi} Villeneuve, S.,
{\em Exercise Regions of American Options
on several assets}, Finance Stochastics, 3(1999), 295-322.
\bibitem{Zi} Ziemer, W.,
{\it Weakly differentiable functions, Sobolev functions and functions of bounded
variation}, Graduate Texts in Mathematics {\bf 120}(1989),
Springer Verlag, New York.
\end{thebibliography}
\end{document}
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