text:
\section{Introduction}

The landmark works of Black and Scholes (1973) and Merton (1973)
have created a new field in quantitative finance. In the
Black-Scholes/Merton framework, the price of an underlying asset is
often modeled as a diffusion process. With a no-arbitrage argument,
the price of a derivative contract written on the asset can be
determined by solving an initial boundary value problem of a linear
partial differential equation (PDE). In the classical Black-Scholes
model, the volatility of the underlying asset, $\sigma$ is assumed
to be constant. In order to explain the empirical phenomenon of
implied volatility smirk, see e.g., Zhang and Xiang (2008),
researchers propose to use volatilities defined by deterministic
functions of the underlying asset price and time. The corresponding
PDE is often called the generalized Black-Scholes equation.
Analytical formulas of the problem for the general case is not
available. However, the problem of reducibility and solvability of
the generalized Black-Scholes equation has been studied by Carr {\it
et al.} (1999, 2002, 2006), Bouchouev (1998), Li and Zhang (2004),
and Zhang and Li (2012). Haven (2005) suggestions a solution
technique for obtaining analytical solutions to the generalized
Black-Scholes equation via an adiabatic approximation to the
Schr\"odinger PDE.

In 1891, a famous mathematician, Sophus Lie, pointed out that, if an
ordinary differential equation (ODE) is invariant under a
one-parameter Lie group of transformations,  the order of the ODE
can be reduced constructively. The method of finding similarity
reductions of a given PDE by using the Lie group method of
infinitesimal transformation (sometimes called the method of
group-invariant solutions) was originally developed by Lie (1891),
see Olver (1993) for the recent developments. Bluman and Cole (1969)
proposed a generalization of Lie's method which is called the
nonclassical method of group-invariant solutions. The method was
further generalized by Olver and Rosenau (1986). A common feature of
these methods is to determine Lie point transformations of a given
PDE, i.e., transformations that depend only on the independent and
dependent variables, see equation (\ref{LieTrans}). After that, Lie
group analysis was widely applied in solving differential equations
in fluid mechanics and quantum mechanics.
Nowadays Lie symmetry software packages are widely used in solving PDEs.
Reviews and comparative studies of some of the earlier computer algebra packages
have been carried out by Hereman (1997) and Butcher {\it et al.} (2003).
More recently, Rocha Filho and Figueiredo (2011) presented the new MAPLE package SADE
for the determination of symmetries and related properties of systems of differential
equations. Vu {\it et al.} (2012) presented the new MAPLE symmetry package DESOLVII,
an upgrade of DESOLV, which included the functionality to determine higher classical
symmetries for both ordinary and partial differential equations. However, currently,
in the situation that coefficient functions contain arbitrary functions,
software packages for symmetry analysis cannot handle.

In fluid mechanics and quantum mechanics, the determination of the
symmetry group of Fokker-Planck equations has a long history. Finkel
(1999) completely classified the symmetries of the Fokker-Planck
equation and constructed group-invariant solutions for a physically
interesting family of Fokker-Planck equations in the case of two
spatial dimensions, namely
\begin{equation}\label{eqFP}
    u_t(x,y,t)-\frac{1}{2}\Delta u(x,y,t)+M(x,y,t)u=0,
\end{equation}
where $u$ is a dependent variable and $M(x,y,t)$ is a potential
function. Building on Finkel's result, Laurence and Wang (2005)
found some closed-form fundamental solutions for a special family of
Fokker-Planck equations. They showed how these results can be
applied in finance to yield exact solutions for special affine and
quadratic two factor term structure models.
In this paper, we not only show how to
generate a series of new solutions with a given solution by using
the last set of equations in Appendix A, but also perform similarity
reductions of different cases.

In quantitative finance, Lo and Hui (2001) presented Lie-algebraic
method for the valuation of financial derivatives with
time-dependent parameters based upon the Wei-Norman theorem. Lo and
Hui (2002) extended their Lie-algebraic approach for the valuation
of multi-asset financial derivatives in a lognormal framework with
time-dependent parameters (drift, standard-deviation, correlation),
involving also stochastic short-term interest rates. Lo and Hui
(2006) proposed also a Lie-algebraic model for pricing more complex
derivatives like moving barrier options with time-dependent
parameters in a CEV framework. The difference between Lo and Hui's
Lie-algebraic approach and our Lie symmetry approach is as follows.
In our Lie symmetry approach, we obtain similarity reduction by
using a one-parameter invariant group of partial differential
equations. Lie algebras are by-products after we obtain the vector
fields in equation (\eqref{symmetries}). However, Lo and Hui (2001)
start from a Lie algebra, which is elevated to a group via an
exponential mapping. Carr, Laurence and Wang (2006) performed the
classification of driftless time and state dependent diffusions that
are integrable in closed form via Lie's equivalence transformations.
However, the Lie symmetry analysis and similarity reduction of the
generalized Black-Scholes equation (with general volatility
function) are not available yet.

In this paper, we try to solve the problem of option pricing based
on the theory of the Fokker-Planck equation. The $2-$dimensional
generalized Black-Scholes equation, arising from option pricing, can
be transformed into the $2+1-$dimensional Fokker-Planck equation. We
demonstrate how to apply Lie symmetry analysis and similarity
reduction to solve the option pricing problem for volatility as a
function of underlying asset price.

Compared with Lo and Hui's approach, our Lie symmetry approach is
more systematic. The main purpose of this paper is to demonstrate
the methodology by using a state-dependent volatility, $\sigma(S)$.
If the volatility is state- and time-dependent, $\sigma(S,t)$, then
the potential function, $M$, in equation (9) is also a function of
time, i.e., $M(x,y) \to M(x,y,t)$. Our approach can be used to
handle the case in principle as shown by equation (12). The
application to the case of state-and-time-separable volatility,
$\sigma(S,t) = \sigma_1(S) \sigma_2(t)$, will be reported in a
subsequent research\footnote{Lo and Hui's (2001, 2006)
time-dependent CEV, $\sigma(t) S^{\beta/2}$, is a special case of
state-and-time-separable volatility.}.  For the case of only one CEV
process, the parameter $\alpha$ in our Section 5.2 can take any
non-negative value, while Lo and Hui (2001, 2006) focus on $0 \le
\beta < 2$, which is equivalent to our $0 \le \alpha < 1$.

This paper is organized as follows. Section 2 discusses how to
transform a typical option pricing problem into the Fokker-Planck
equation like \eqref{eqFP}. Section 3 applies the Lie symmetry
analysis to the equation. Section 4 presents the similarity
reductions of different cases. Section 5 provides a few exact
solutions of both $2-$dimensional and $1-$dimensional generalized
Black-Scholes equation. Finally, section 6 concludes.

\section{Typical option pricing problem}

In the Black-Scholes (1973)/Merton's (1973) framework, the prices of
two stocks, $S_1$, $S_2$, are modeled by two pure diffusion
processes
\begin{equation}\label{dp}
    dS_i = \mu_i S_i dt + \tilde \sigma_i(S_i) S_i dB_i, \quad \quad
    (i=1,2),
\end{equation}
where $\mu_i$ is the drift, $\tilde \sigma_i(S_i)$ is the volatility
of the stock $i$, and $B_i$ ($i=1,2$) are standard Brownian motions.
The correlation coefficient between $B_1$ and $B_2$ is $\rho$. The
correlation makes it harder to convert the equation to $2+1$
dimensional Fokker-Planck equation like \eqref{eqFP}. In this paper,
we only consider the case where the volatility, $\sigma$, is a
deterministic function of the stock price, and leave the general
case of time dependence for future research. Standard no-arbitrage
theory shows that the price of a European style option,
$c(S_1,S_2,t)$, satisfies the following generalized Black-Scholes
equation
\begin{equation}
  \frac{\partial c}{\partial t} +\frac{1}{2}\sigma_1^2(S_1)
  \frac{\partial^2c}{\partial S_1^2}+
  \rho\sigma_1(S_1)\sigma_2(S_2)\frac{\partial^2c}{\partial S_1 \partial S_2}
  +\frac{1}{2}\sigma_2^2(S_2)\frac{\partial^2c}{\partial S_2^2}
  +rS_1\frac{\partial c}{\partial S_1}+rS_2\frac{\partial c}{\partial
  S_2}-rc=0,
  \label{eq0} \end{equation}\begin{equation}
  c(S_1,S_2,T)=C(S_1,S_2), \label{eq1}
\end{equation}
where correlation coefficient $\rho$ and interest rate $r$ are
assumed to be constant; $C(S_1,S_2)$ is the payoff function of the
option on the maturity date ($t=T$). For brevity, we have used
$\tilde \sigma_i(S_i)S_i = \sigma_i(S_i)$.

In the general case, analytical formulae of the problem \eqref{eq0}
and \eqref{eq1} cannot be obtained. Practitioners rely on numerical
methods such as finite difference, binomial trees, or Monte Carlo
simulation. However, in the way of reducibility and solvability of
the $1+1-$dimensional generalized Black-Scholes equation, Li and
Zhang (2004) determined the boundary condition and the nature of the
eigenvalues and eigenfunctions with Weyl-Titchmarsh theory. The
solution can be written analytically in a Stieltjes integral. Zhang
and Li (2012) provide a systematic way of finding the volatility
function, $\sigma(S)$, for a given solvable potential function.

Analytical solutions for the generalized Black-Scholes equation are
of paramount importance to practitioners as they allow a better
qualitative understanding of the solution behavior. More
significantly, volatility functions are typically fitted to market
data in empirical research. Parametric volatility models that
produce analytical solutions are in very high demand.

For certain volatility functions, e.g., a volatility being a
quadratic function of asset price studied by Z\"uhlsdorff (2001),
the generalized Black-Scholes equation can be transformed into the
standard heat equation, which, in turn, can be solved analytically.
Even for the case where the problem cannot be reduced to the
standard heat equation, it is still possible to solve the problem
analytically for some particular volatility functions. This paper
pushes further along this direction.

With the following transformation
\begin{equation}\label{trans1}
\begin{cases}
   \displaystyle x = \sqrt{\frac{2}{1+\rho}} \bigg(\int_0^{S_2}\frac{1}
   {\sigma_2(S)}dS + \int_0^{S_1}\frac{1}{\sigma_1(S)}dS \bigg),\\
   \displaystyle y = \sqrt{\frac{2}{1-\rho}} \bigg(\int_0^{S_2}\frac{1}
   {\sigma_2(S)}dS - \int_0^{S_1}\frac{1}{\sigma_1(S)}dS \bigg),
\end{cases}
\end{equation}
equations \eqref{eq0} and \eqref{eq1} become
\begin{equation}
    \frac{\partial c}{\partial t}+\frac{1}{2}\bigg(\frac{\partial^2c}
    {\partial x^2} +\frac{\partial^2c}{\partial y^2}\bigg)
    +Q_1\frac{\partial c}{\partial x} + Q_2\frac{\partial c}{\partial y}-rc=0,
    \label{eq2}\end{equation}\begin{equation}
    c(x,y,T)=C(x,y), \label{eq3}
\end{equation}
where
\begin{equation}
\begin{cases}
   \displaystyle Q_1 = \sqrt{\frac{2}{1+\rho}}\bigg(\frac{rS_1}{\sigma_1}
   +\frac{rS_2}{\sigma_2}\bigg)
   -\frac{2}{1+\rho}\bigg(\frac{\sigma_{1x}}{\sigma_1}
   +\frac{\sigma_{2x}}{\sigma_2}\bigg)
   +\frac{2}{\sqrt{1-\rho^2}}\bigg(\frac{\sigma_{1y}}{\sigma_1}
   -\frac{\sigma_{2y}}{\sigma_2}\bigg),\\
   \displaystyle Q_2 = \sqrt{\frac{2}{1-\rho}}\bigg(\frac{rS_1}{\sigma_1}
   +\frac{rS_2}{\sigma_2}\bigg)
   -\frac{2}{\sqrt{1-\rho^2}}\bigg(\frac{\sigma_{1x}}{\sigma_1}
   -\frac{\sigma_{2x}}{\sigma_2}\bigg)
   -\frac{2}{1-\rho}\bigg(\frac{\sigma_{1y}}{\sigma_1}
   +\frac{\sigma_{2y}}{\sigma_2}\bigg),\nonumber
\end{cases}
\end{equation}
$\sigma_{ix}$ and $\sigma_{iy}$ ($i=1, 2$) stand for partial
derivative of $\sigma_i$ with respect to $x$ and $y$ respectively,
$S_i$ is a function of $x,y$, which can be solved by equation
\eqref{trans1}. Consequently, $\sigma_i(S_i)$ converts to
$\sigma_i(x,y)$. For brevity, we have replaced $\sigma_i(x,y)$ by
$\sigma_i$.

We introduce the following transformation
\begin{equation}\label{trans2}
    c(x,y,t) = e^{\omega(x,y)-r\tau} u(x,y,\tau), \quad \quad \tau = T-t,
\end{equation}
where $\displaystyle \nabla \omega(x,y) = -(Q_1,\,Q_2)$. Here we
need following compatibility condition\footnote{If the condition is
not satisfied, the generalized Black-Scholes equation (\ref{eq0})
will be converted into a general case of Fokker-Planck equation,
instead of the irrotational case studied in this paper. It is
possible to study the solution of the general case of Fokker-Planck
equation by using Lie symmetry approach. The result will be reported
in a subsequent research.}
\begin{equation} \frac{\partial Q_2}{\partial x} = \frac{\partial Q_1}{\partial y}. \nonumber \end{equation}
By a simple calculation, equations \eqref{eq2} and \eqref{eq3}
become $2+1-$dimensional Fokker-Planck equation like \eqref{eqFP}:
\begin{equation}
    \frac{\partial u}{\partial \tau}-\frac{1}{2}\bigg(\frac{\partial^2u}{\partial x^2}
    +\frac{\partial^2u}{\partial y^2}\bigg)+M(x,y)u=0, \label{eq4}
\end{equation}
\begin{equation}
    u(x,y,\tau)|_{\tau=0}=u_0(x,y), \label{eq5}
\end{equation}
where the coefficient $M(x,y)$, regarded as a potential function,
reads
\begin{equation}\label{M}
    M(x,y) = \frac{1}{2}\bigg(\frac{\partial Q_1}{\partial x} +\frac{\partial Q_2}
    {\partial y}+Q_1^2+Q_2^2\bigg).
\end{equation}

Similarly, we can convert the $1-$dimensional generalized
Black-Scholes equation to $1+1-$dimensional Fokker-Planck equation.
Carr, Laurence and Wang (2006) exploit a remarkable intertwining
with the inhomogeneous Burger's equation in the time dependent and
state dependent one dimensional case via point transformations. By
using the separating variable method, Li and Zhang (2004), and Zhang
and Li (2012) transformed the option pricing problem into a
Schr\"{o}dinger equation which is similar to the $1+1-$dimensional
Fokker-Planck equation studied here\footnote{Li and Zhang (2004),
and Zhang and Li (2012) study the pricing of European options
written on a single asset, while we are studying case of two assets.
Their transformation is similar to a single-asset case of ours here
without the drift of risk-free rate.}.

\section{Lie point symmetries}

We now perform Lie symmetry analysis for the $2+1-$dimensional
Fokker-Planck equation. Let us consider a $2+1-$dimensional equation
\begin{equation}\label{sysfu}
    F(u)=u_t(x,y,t)-\frac{1}{2}\Delta u(x,y,t)+M(x,y,t)u.
\end{equation}
and a one-parameter Lie group of infinitesimal
transformation\footnote{More explanation of the treatment and the
meaning of the variables can be found in Chapters 2 and 3 of Olver's
(1993) book.}
\begin{eqnarray}
   & t \rightarrow t + \epsilon T(x,y,t,u),\nonumber\\
   & x \rightarrow x + \epsilon X(x,y,t,u),\nonumber\\
   & y \rightarrow y + \epsilon Y(x,y,t,u),\nonumber\\
   & u \rightarrow u + \epsilon U(x,y,t,u).\label{LieTrans}
\end{eqnarray}
With a small parameter $\epsilon \ll 1$, the vector field associated
with the group of transformations \eqref{LieTrans} can be written as
\begin{equation}\label{vector1}
   \underline{u} = T\frac{\partial}{\partial t}+X\frac{\partial}{\partial x}
                     +Y\frac{\partial}{\partial y}+U\frac{\partial}{\partial u},
\end{equation}
or equivalently in the symmetry form
\begin{equation}\label{sym1}
 \sigma = U-Tu_t-Xu_x-Yu_y.
\end{equation}
We wish to determine all possible coefficient functions $X,\,Y,\,T$
and $U$, so that the corresponding one-parameter group is a symmetry
group of the Fokker-Planck equation. The symmetry equation, i.e. the
corresponding infinitesimal criterion becomes
\begin{equation}\label{syseq2}
   \frac{\partial}{\partial \epsilon} \,F\big(u+\epsilon\sigma \big)
   \bigg|_{\begin{subarray}{1}\epsilon=0 \\ F(u)=0\end{subarray}}=0.
\end{equation}
Based on \eqref{eqFP}, i.e. $F(u)=0$, substituting $u_t$ by
$\frac{1}{2}\Delta u-Mu$ whenever it occurs gives an equation, of which
left hand side is a polynomial with
$u,\,u_x,\,u_y,\,u_{xx},\,u_{yy},\,u_{xy}$ and right hand side is
$0$.  Taking the coefficients of the various monomials in the first
and second order partial derivatives of $u$ in the polynomial be
$0$, we find the determining equations for the symmetry group of the
Fokker-Planck equation.

By solving them, an invariance of equation \eqref{eqFP} under
transformation \eqref{LieTrans} leads to the expressions for the
functions $T,X,Y,U$ of the form (throughout this paper we use
symbolic package MAPLE to perform all calculations)
\begin{equation}\label{vector2}\begin{cases}
   T=f_1,\\
   X=\dfrac{1}{2}\bigg(\dfrac{\partial f_1}{\partial t}\bigg)x+ky+f_2,\\
   Y=\dfrac{1}{2}\bigg(\dfrac{\partial f_1}{\partial t}\bigg)y-kx+f_3,\\
   U=-\bigg[\dfrac{1}{4}\bigg(\dfrac{\partial^2 f_1}{\partial t^2}\bigg)(x^2+y^2)
   +\bigg(\dfrac{\partial f_2}{\partial t}\bigg)x+
   \bigg(\dfrac{\partial f_3}{\partial t}\bigg)y+f_4\bigg]u+g,
\end{cases}\end{equation}
and the compatibility condition
\begin{equation}\label{condition1}
   T_tM+XM_x+YM_y+U_uM=0.
\end{equation}
where $k$ is arbitrary constant, and $f_i,\,\,i=1,\ldots,4$ are
arbitrary functions of $t$, which satisfy the condition
\eqref{condition1}, and $g$ is the solution of the original equation
\eqref{eqFP}. Similar mathematical results were given by Finkel
(1999) and Laurence and Wang (2005) by using a prolongation method.
Nowadays computer algebra packages are widely used in determination of symmetries of
differential equations. However, in this case, $T,X,Y,U$ contain the arbitrary functions of $t$
and the arbitrary function of $x,y,t$, which current software packages for symmetry analysis,
such as DESOLV, DESOLVII and SADE, cannot handle.
Therefore, we will manually use Lie symmetry analysis to deal with the Fokker-Planck equation.

The presence of these arbitrary functions leads to an
infinite-dimensional Lie algebra of symmetries. A general element of
this algebra is written as
\begin{equation}\label{algebra}
   \underline{\emph{v}}=\underline{\emph{v}}_1k+\underline{\emph{v}}_2(f_1)
   +\underline{\emph{v}}_3(f_2)+
   \underline{\emph{v}}_4(f_3)+\underline{\emph{v}}_5(f_4)+\underline{\emph{v}}_6(g).
\end{equation}

Let $\varphi_i$ be arbitrary functions of $t$, $\psi$ and $\phi$ be
arbitrary functions of $x,\,y,\,t$, then
\begin{eqnarray}
   \underline{\emph{v}}_1 & = y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y},
   \nonumber\\
   \underline{\emph{v}}_2(\varphi_i) & = \varphi\frac{\partial}{\partial t}
   +\frac{1}{2}\dot{\varphi_i}x\frac{\partial}{\partial x}
   +\frac{1}{2}\dot{\varphi_i}y\frac{\partial}{\partial y}
   +\frac{1}{4}\ddot{\varphi_i}(x^2+y^2)u\frac{\partial}{\partial u},\nonumber\\
   \underline{\emph{v}}_3(\varphi_i) & = \varphi_i \frac{\partial}{\partial x}
   +\dot{\varphi_i}xu\frac{\partial}{\partial u},\nonumber\\
   \underline{\emph{v}}_4(\varphi_i) & = \varphi_i \frac{\partial}{\partial y}
   +\dot{\varphi_i}yu\frac{\partial}{\partial u},\nonumber\\
   \underline{\emph{v}}_5(\varphi_i) & = \varphi_i u\frac{\partial}{\partial u},\nonumber\\
   \underline{\emph{v}}_6(\psi) & = \psi\frac{\partial}{\partial u}.\label{symmetries}
\end{eqnarray}
The commutation relations between all these vector fields are given
by Table 1.

Table 1:\quad The commutation relations between vector fields.
\begin{itemize}
\item[]\end{itemize}
\hspace*{-45pt}\begin{tabular}{@{}l|lllllll}
 & $\underline{\emph{v}}_1$
 & $\underline{\emph{v}}_2(\varphi_j)$
 & $\underline{\emph{v}}_3(\varphi_j)$
 & $\underline{\emph{v}}_4(\varphi_j)$
 & $\underline{\emph{v}}_5(\varphi_j)$
 & $\underline{\emph{v}}_6(\phi)$\\
\hline
 $\underline{\emph{v}}_1$            & $0$
                                     & $0$
                                     & $\underline{\emph{v}}_4(\varphi_j)$
                                     & $-\underline{\emph{v}}_3(\varphi_j)$
                                     & $0$
                                     & $\underline{\emph{v}}_6(y\phi_x-x\phi_y)$\\
 $\underline{\emph{v}}_2(\varphi_i)$ &
                                     & $\underline{\emph{v}}_2(\varphi_i\dot{\varphi_j}-\dot{\varphi_i}\varphi_j)$
                                     & $\underline{\emph{v}}_3(\varphi_i\dot{\varphi_j}-\frac{1}{2}\dot{\varphi_i}\varphi_j)$
                                     & $\underline{\emph{v}}_4(\varphi_i\dot{\varphi_j}-\frac{1}{2}\dot{\varphi_i}\varphi_j)$
                                     & $\underline{\emph{v}}_5(\varphi_i\dot{\varphi_j})$
                                     & $\underline{\emph{v}}_6(\frac{1}{2}\dot{\varphi_i}(x\phi_x+y\phi_y)+\varphi_i\phi_t)$\\
 $\underline{\emph{v}}_3(\varphi_i)$ &
                                     &
                                     & $\underline{\emph{v}}_5(\varphi_i\dot{\varphi_j}-\dot{\varphi_i}\varphi_j)$
                                     & $0$
                                     & $0$
                                     & $\underline{\emph{v}}_6(\dot{\varphi_i}x\phi-\varphi_i\phi_x)$\\
 $\underline{\emph{v}}_4(\varphi_i)$ &
                                     &
                                     &
                                     & $\underline{\emph{v}}_5(\varphi_i\dot{\varphi_j}-\dot{\varphi_i}\varphi_j)$
                                     & $0$
                                     & $\underline{\emph{v}}_6(\dot{\varphi_i}y\phi-\varphi_i\phi_y)$\\
 $\underline{\emph{v}}_5(\varphi_i)$ &
                                     &
                                     &
                                     &
                                     & $0$
                                     & $\underline{\emph{v}}_6(\varphi_i\phi)$\\
 $\underline{\emph{v}}_6(\psi)$      &
                                     &
                                     &
                                     &
                                     &
                                     & $0$\\
\end{tabular}
\begin{itemize}
\item[] The entry in row $i$ and column $j$ representing $\big[\,\underline{\emph{v}}_i\,,\,\underline{\emph{v}}_j\,\big]$.
\end{itemize}

From Table 1, we see that
${\underline{\emph{v}}_2(\varphi),\underline{\emph{v}}_3(\varphi),
\underline{\emph{v}}_4(\varphi),\underline{\emph{v}}_5(\varphi)}$
constitute a subalgebra. And there exist some types of interesting
subalgebras, For instance, Virasoro algebra and $\omega_\infty$-type
algebra.

Furthermore, we find that the transform $[\underline{\emph{v}}_1,
\underline{\emph{v}}_6(\psi)] =
\underline{\emph{v}}_6(y\psi_x-x\psi_y)$ is invariant, if $M(x,y)$
satisfies the type $C \cdot (x^2+y^2)$, where $C$ is an arbitrary constant.
In other words, if $g$ is a solution of the Fokker-Planck equation like this,
then $yg_x-xg_y$ is another solution of the same equation.

Moreover, we get a series of transformations of the solution. New
solutions can be generated through them with a known solution. The
one-parameter groups generated by $\underline{\emph{v}}_i$ and the
transformations are included in the Appendix A for the readers with
an interest in the details of applying the theory.

\section{Similarity reductions}

After determining the infinite-dimensional algebra of symmetries,
the similarity variables can be found by solving the characteristic
equations
\begin{equation}\label{reduct1}
   \frac{dt}{T}=\frac{dx}{X}=\frac{dy}{Y}=\frac{du}{U}.
\end{equation}

By solving the ordinary differential equations \eqref{reduct1}, we
can obtain integration constants  $\xi,\,\eta,\,P$. Substituting
$\xi,\,\eta,\,P$ for $x\,,y\,,t\,,u$ in original equation
\eqref{eqFP}, we can reduce the equation from $2+1-$dimensional to
$2-$dimensional finally. This process is called similarity
reductions.

Since there are many arbitrary functions in $T, \,X, \,Y, \,U$, it
is hard to solve the equations \eqref{reduct1} in the general case.
Likewise, it is also hard to solve them by substituting generators
\eqref{symmetries}. Finkel (1999) completely classified the
symmetries of the Fokker-Planck equation based on the compatibility
condition \eqref{condition1}. For simplicity, he has dropped out the
two trivial infinitesimal symmetries $\partial_t$ and $u\partial_u$
in his classification result. It means that constant terms are omitted
in the forms for $f_1$ and $f_4$.

In this subsection, we will list some cases in details for reductions as
the classification done by Finkel, which are helpful for the following subsections.
Other cases of the Fokker-Planck equation for reductions are included in Appendix B.

\begin{itemize}
\item {\bf Case 1.1a}
\end{itemize}
\begin{equation}\label{case1.1a}
\begin{cases}
   M = \dfrac{C_0}{x^2}+by+c_0,\quad C_0\neq0,\\
   f_1 = \delta_2t^2+\delta_1t,\quad k=0,\quad f_2=0,\quad f_3 = \dfrac{b\delta_2t^3}{2}
   +\dfrac{3b\delta_1t^2}{4}+\beta_1t+\beta_0,\\
   f_4 = \dfrac{b^2\delta_2t^4}{8}+\dfrac{b^2\delta_1t^3}{4}+
         \big(\dfrac{b\beta_1}{2}+c_0\delta_2\big)t^2+(\delta_2+c_0\delta_1+b\beta_0)t.
\end{cases}
\end{equation}

We have the similarity variables $\xi, \,\eta, \,P$,
\begin{equation}
\begin{cases}
   \xi = & \dfrac{x}{\sqrt{\delta_2t^2+\delta_1t}}, \\
   \eta = & \dfrac{2y\delta_1^2-b\delta_1^2t^2+4(2\beta_0\delta_2-\beta_1\delta_1)t
            +4\beta_0\delta_1}{2\delta_1^2\sqrt{\delta_2t^2+\delta_1t}}, \\
   P = & u \cdot \exp\Big\{\dfrac{1}{\delta_1^4}\{\eta\delta_1^2\sqrt{\delta_2t^2+\delta_1t}
         (\delta_1^2bt+2\delta_1\beta_1-4\delta_2\beta_0)+
         \delta_1 \ln(\delta_2t+\delta_1)(\delta_1^3+2\delta_1\beta_0\beta_1-2\delta_2\beta_0^2) \\
       & +\beta_0\delta_1 \ln(t)(2\beta_0\delta_2-2\beta_1\delta_1)+
         \dfrac{1}{3}b^2\delta_1^4t^3+2b\delta_1^2(\delta_1\beta_1-2\delta_2\beta_0)t^2 \\
       & +[8\beta_0^2\delta_2^2-8\beta_0\beta_1\delta_1\delta_2-2b\beta_0\delta_1^3
         +\delta_1^4c_0+2\beta_1^2\delta_1^2+\dfrac{1}{2}\delta_1^4\delta_2(\xi^2+\eta^2)]t\}\Big\}.
\end{cases}\label{xieta1.1a}
\end{equation}

and the reduced PDE becomes
\begin{equation}\label{eq1.1a}
   \delta_1^2\xi^2(P_{\xi\xi}+P_{\eta\eta})+\delta_1^3\xi^3P_{\xi}+\delta_1^3\xi^2\eta P_{\eta}
   +(4\delta_2\beta_0^2\xi^2-4\delta_1\beta_0\beta_1\xi^2-2C_0\delta_1^2)P=0.
\end{equation}

We get the solution by the method of separation of variables
\begin{eqnarray}
   P = {\it F_{1}}(\xi){\it F_{2}}(\eta),
\end{eqnarray}
where ${\it F_1}(\xi)$ and ${\it F_2}(\eta)$ is
\begin{equation*}\begin{cases}
   {\it F_1}(\xi) = & \frac{e^{-\frac{\delta_1 \xi^2}{4}}}{\sqrt{\xi}}
                      \bigg[C_1 {\rm WhittakerM}\Big(\frac{c_1}{2\delta_1}-\frac{1}{4},\frac{\sqrt{8C_0+1}}{4},\frac{\delta_1 \xi^2}{2}\Big)
                      +C_2 {\rm WhittakerW}\Big(\frac{c_1}{2\delta_1}-\frac{1}{4},\frac{\sqrt{8C_0+1}}{4},\frac{\delta_1 \xi^2}{2}\Big)\bigg],\\
   {\it F_2}(\eta) = & \frac{e^{-\frac{\delta_1 \eta^2}{4}}}{\sqrt{\eta}}
                       \bigg[C_3 {\rm WhittakerM}\Big(\frac{c_1}{2\delta_1}-\frac{2\beta_0\beta_1}{\delta_1^2}+\frac{2\delta_2\beta_0^2}{\delta_1^3}
                       -\frac{1}{4}, \frac{1}{4},\frac{\delta_1 \eta^2}{2}\Big) \\
                     & +C_4 {\rm WhittakerW}\Big(\frac{c_1}{2\delta_1}-\frac{2\beta_0\beta_1}{\delta_1^2}+\frac{2\delta_2\beta_0^2}{\delta_1^3}-\frac{1}{4},
   \frac{1}{4},\frac{\delta_1 \eta^2}{2}\Big)\bigg],
\end{cases}\end{equation*}
where $c_1,\,C_1,\,C_2,\,C_3,\,C_4$ are arbitrary constants.

\begin{itemize}
\item {\bf Case 1.2b}
\end{itemize}
\begin{equation}\label{case1.2b}
\begin{cases}
   M = \dfrac{C(\theta)}{r^2}+cr^2+c_0,\quad \,c\neq0,\\
   f_1 = \delta_1e^{2\sqrt{2c}t}+\delta_2e^{-2\sqrt{2c}t},\quad k=0,\quad f_2=f_3=0,\\
   f_4 = \big(\sqrt{2c}+c_0\big)\delta_1e^{2\sqrt{2c}t}-\big(\sqrt{2c}-c_0\big)\delta_2e^{-2\sqrt{2c}t}.
\end{cases}
\end{equation}
where $C(\theta)\neq (c_1\cos\theta+c_2\sin\theta)^{-2}$, $C^{'}(\theta)\neq 0$, and $r=\sqrt{x^2+y^2}$.

We have the similarity variables $\xi, \,\eta, \,P$,
\begin{equation}\label{xieta1.2b}
\begin{cases}
   \xi = & \dfrac{xe^{\sqrt{2c}t}}{\sqrt{\delta_1e^{4\sqrt{2c}t}+\delta_2}}, \\
   \eta = & \dfrac{ye^{\sqrt{2c}t}}{\sqrt{\delta_1e^{4\sqrt{2c}t}+\delta_2}}, \\
   P = & e^{-c_0t}{\sqrt{\delta_1e^{4\sqrt{2c}t}+\delta_2}} u \cdot \exp \Big\{\sqrt{\dfrac{c}{2}}
         \big[(\delta_1e^{2\sqrt{2c}t}-\delta_2e^{-2\sqrt{2c}t})(\xi^2+\eta^2)-2t\big]\Big\},
\end{cases}
\end{equation}

and the reduced PDE becomes
\begin{equation}\label{eq1.2b}
  (\xi^2+\eta^2)(P_{\xi\xi}+P_{\eta\eta})+2[4c\delta_1\delta_2(\xi^2+\eta^2)^2-C(\theta)]P=0.
\end{equation}

With the transformation $\xi=\varrho \cos\theta,\,\eta=\varrho \sin\theta$, \eqref{eq1.2b} becomes
\begin{equation}\label{eq21.2b}
   \varrho^2P_{\varrho\varrho}+\varrho P_{\varrho}+P_{\theta\theta}+2[4c\delta_1\delta_2\varrho^4-C(\theta)]P=0.
\end{equation}

We can get the solution by the method of separation of variables
\begin{eqnarray}
   P = {\it F_{1}}(\varrho){\it F_{2}}(\theta),
\end{eqnarray}
where ${\it F_1}(\varrho),\,{\it F_2}(\theta)$ is the solution of
\begin{equation}\label{ODE1.2b}\begin{cases}
   \dfrac{d^2{\it F_{1}}(\varrho)}{d\varrho^2}+\dfrac{1}{\varrho}\dfrac{d{\it F_{1}}(\varrho)}{d\varrho}
   +\big(8c\delta_{1}\delta_{2}\varrho^{2}-\dfrac{c_{1}}{\varrho^{2}}\big){\it F_{1}}(\varrho)=0,\\
   \dfrac{d^2{\it F_{2}}(\theta)}{d\theta^2}+(c_{1}-2C(\theta)){\it
   F_{2}}(\theta)=0,
\end{cases}\end{equation}
where $c_{1}$ is an arbitrary constant. Given $C(\theta)$, the ODE systems \eqref{ODE1.2b} can be solved directly.

\begin{itemize}
\item {\bf Case 1.4b}
\end{itemize}
\begin{equation}\label{case1.4b}\begin{cases}
   M = \dfrac{C_0}{r^2}+cr^2+ax+by+c_0,\\
   f_1 = \delta_1e^{2\sqrt{2c}t}+\delta_2e^{-2\sqrt{2c}t},\quad k=0,\quad f_2=f_3=0,\\
   f_4 = \big(\sqrt{2c}+c_0\big)\delta_1e^{2\sqrt{2c}t}-\big(\sqrt{2c}-c_0\big)
   \delta_2e^{-2\sqrt{2c}t},
\end{cases}\end{equation}
where $r=\sqrt{x^2+y^2}$. If $\delta_1\neq0,\,\delta_2\neq0$,
then $a=b=0$ should be held to make $M$ satisfy the compatibility condition \eqref{condition1}.
Obviously, this is the simplification of \textbf{Case 1.2b}.
We have the similarity variables $\xi, \,\eta, \,P$ same as \eqref{xieta1.2b}, then the reduced PDE becomes
\begin{equation}\label{eq1.4b}
   (\xi^2+\eta^2)(P_{\xi\xi}+P_{\eta\eta})+2[4c\delta_1\delta_2(\xi^2+\eta^2)^2-C_0]P=0.
\end{equation}

With the transformation $\xi=\varrho \cos\theta,\,\eta=\varrho \sin\theta$,
we can get the solution by the method of separation of variables
\begin{eqnarray}
   P = {\it F_{1}}(\varrho){\it F_{2}}(\theta),
\end{eqnarray}
where ${\it F_1}(\varrho)$ and ${\it F_2}(\theta)$ is
\begin{equation*}\begin{cases}
   {\it F_1}(\varrho) = & C_1 J\Big(\dfrac{\sqrt{c_1}}{2},\sqrt{2\delta_1\delta_2c}\varrho^2\Big)
                          +C_2 Y\Big(\dfrac{\sqrt{c_1}}{2},\sqrt{2\delta_1\delta_2c}\varrho^2\Big)\\
   {\it F_2}(\theta) = & C_3 \sin(\theta\sqrt{c_1-2C_0})+C_4 \cos(\theta\sqrt{c_1-2C_0}),
\end{cases}\end{equation*}
where $c_1,\,C_1,\,C_2,\,C_3,\,C_4$ are arbitrary constants, and
$J(\nu, z)$ and $Y(\nu, z)$ are the Bessel functions of the first
and second kinds, respectively.

\section{Case studies}

We now study a few cases, most of which are not well known in the
financial literature. Our purpose here is to demonstrate the
procedure of producing analytical option pricing formulas with the
method of similarity reduction.

\subsection{$2-$dimensional: Double CEV Model}

In the traditional CEV Model (Cox 1975, Cox and Ross 1976, Schroder
1989), $\sigma(S)=\sigma S^{\alpha}$. Base on their work, we try to
build a Double CEV Model, which has two assets. Assuming
\begin{equation}
   \sigma_i(S_i)=\sigma_i S_i^{\alpha_i},
\end{equation}\label{dcev}
where $\sigma_i>0$, $\alpha_i >0$. From \eqref{trans1}, we have
\begin{equation}\label{dcevtrans1}
\begin{cases}
   \displaystyle x = \sqrt{\frac{2}{1+\rho}}\big(\dfrac{S_1^{1-\alpha_1}}{\sigma_1(1-\alpha_1)}
                     +\dfrac{S_2^{1-\alpha_2}}{\sigma_2(1-\alpha_2)}\big),\\
   \displaystyle y = \sqrt{\frac{2}{1-\rho}}\big(\dfrac{S_2^{1-\alpha_2}}{\sigma_2(1-\alpha_2)}
                     -\dfrac{S_1^{1-\alpha_1}}{\sigma_1(1-\alpha_1)}\big)),
\end{cases}
\end{equation}
and the following transformation \eqref{trans2}, where
\begin{equation*}
\begin{cases}
  Q_1 = & \dfrac{r(1-\alpha_1)}{2}\Big(x-\sqrt{\frac{1-\rho}{1+\rho}}y\Big)
          +\dfrac{r(1-\alpha_2)}{2}\Big(x+\sqrt{\frac{1-\rho}{1+\rho}}y\Big) \\
        & -\dfrac{\alpha_1}{1-\alpha_1}\dfrac{1}{\Big(\frac{1+\rho}{2}x-\frac{\sqrt{1-\rho^2}}{2}y\Big)}
          -\dfrac{\alpha_2}{1-\alpha_2}\dfrac{1}{\Big(\frac{1+\rho}{2}x+\frac{\sqrt{1-\rho^2}}{2}y\Big)}, \\
  Q_2 = & \dfrac{r(1-\alpha_2)}{2}\Big(\sqrt{\frac{1+\rho}{1-\rho}}x+y\Big)
          -\dfrac{r(1-\alpha_1)}{2}\Big(\sqrt{\frac{1+\rho}{1-\rho}}x-y\Big) \\
        & -\dfrac{\alpha_1}{1-\alpha_1}\dfrac{1}{\Big(\frac{\sqrt{1-\rho^2}}{2}x-\frac{1-\rho}{2}y\Big)}
          -\dfrac{\alpha_2}{1-\alpha_2}\dfrac{1}{\Big(\frac{\sqrt{1-\rho^2}}{2}x+\frac{1-\rho}{2}y\Big)},
\end{cases}
\end{equation*}
and
\begin{equation*}
   \omega(x,y)=2\Big(\frac{\alpha_1ln(x-y)}{1-\alpha_1}+\frac{\alpha_2ln(x+y)}{1-\alpha_2}\Big)+
               \frac{r}{4}\big[\alpha_1(x-y)^{2}+\alpha_2(x+y)^{2}-2(x^{2}+y^{2})\big]+C_0,
\end{equation*}
where $C_0$ is an arbitrary constant.

Equation \eqref{eq0} becomes
\begin{equation}
   u_t(x,y,t)-\frac{1}{2}\Delta u(x,y,t)+M(x,y,t)u=0,
\end{equation}
where
\begin{equation}
   M(x,y,t) = \frac{48(x^2+y^2)}{(x^2-y^2)^2}+r^2(x^2+y^2)-18r.
\end{equation}
For brevity we have taken $\rho=0$$,\,\alpha_1=\alpha_2=2$.\\
(If $\rho\neq0$, the compatibility condition \eqref{condition1} is
also satisfied with the following $f_i,\,C$ and $g$. Moreover, the
similarity variables and the solution of the reduced PDE can be
obtained. Here, taking $\rho=0$ is just for brevity.)

Obviously, the function $M$ belongs to the {\bf Case 1.2b}.
Therefore, we take
\begin{equation*}\begin{cases}
   f_1 = \delta_1e^{2\sqrt{2}rt}+\delta_2e^{-2\sqrt{2}rt},\quad k=0,\quad f_2=f_3=0,\\
   f_4 = \big(\sqrt{2}r-18r\big)\delta_1e^{2\sqrt{2}rt}-\big(\sqrt{2}r+18r\big)\delta_2e^{-2\sqrt{2}rt}.
\end{cases}\end{equation*}

We have the similarity variables $\xi, \,\eta, \,P$,
\begin{equation}
\begin{cases}
   \xi = & \dfrac{xe^{\sqrt{2}rt}}{\sqrt{\delta_1e^{4\sqrt{2}rt}+\delta_2}}, \\
   \eta = & \dfrac{ye^{\sqrt{2}rt}}{\sqrt{\delta_1e^{4\sqrt{2}rt}+\delta_2}}, \\
   P = & e^{-18rt}{\sqrt{\delta_1e^{4\sqrt{2}rt}+\delta_2}} u \cdot \exp \Big\{\dfrac{1}{\sqrt{2}}r
         \big[(\delta_1e^{2\sqrt{2}rt}-\delta_2e^{-2\sqrt{2}rt})(\xi^2+\eta^2)-2t\big]\Big\},
\end{cases}\label{dcevtrans}
\end{equation}

and the reduced PDE becomes
\begin{equation}\label{dceveq1}
   (P_{\xi\xi}+P_{\eta\eta})+8r^2\delta_1\delta_2(\xi^2+\eta^2)P-\frac{96(\xi^2+\eta^2)}{(\xi^2-\eta^2)^2}P=0.
\end{equation}

With the transformation $\xi=\varrho \cos\theta,\,\eta=\varrho \sin\theta$, \eqref{dceveq1} becomes
\begin{equation}\label{dceveq2}
   \varrho^2P_{\varrho\varrho}+\varrho P_{\varrho}+P_{\theta\theta}+8r^2\delta_1\delta_2\varrho^4P-\frac{96P}{\cos^22\theta}=0.
\end{equation}

The solution can be written as
\begin{eqnarray}
   P = {\it F_{1}}(\varrho){\it F_{2}}(\theta),\label{dcevsoution}
\end{eqnarray}
where ${\it F_1}(\varrho)$ and ${\it F_2}(\theta)$ are
\begin{equation*}\begin{cases}
   {\it F_1}(\varrho) = & C_1 J\Big(\dfrac{\sqrt{c_1}}{2},\sqrt{2\delta_1\delta_2}r\varrho^2\Big)
                          +C_2 Y\Big(\dfrac{\sqrt{c_1}}{2},\sqrt{2\delta_1\delta_2}r\varrho^2\Big)\\
   {\it F_2}(\theta) = & \dfrac{(2 \cos(4\theta)-2)^{\frac{3}{4}}}{\sqrt{ \sin(4\theta)}}\bigg\{
                         C_3(\frac{ \cos(4\theta)+1}{2})^{\frac{1}{2}+\frac{\sqrt{97}}{4}} {\rm Hypergeom} \Big(
                         \big[\frac{3+\sqrt{97}+\sqrt{c_1}}{4},\frac{3+\sqrt{97}-\sqrt{c_1}}{4}\big],\\
                       & \big[1+\frac{\sqrt{97}}{2}\big],\frac{\cos(4\theta)+1}{2}\Big)
                         +C_4(\frac{\cos(4\theta)+1}{2})^{\frac{1}{2}-\frac{\sqrt{97}}{4}} {\rm Hypergeom} \Big(
                         \big[\frac{3-\sqrt{97}+\sqrt{c_1}}{4},\frac{3-\sqrt{97}-\sqrt{c_1}}{4}\big],\\
                       & \big[1-\frac{\sqrt{97}}{2}\big],\frac{\cos(4\theta)+1}{2}\Big)\bigg\},
\end{cases}\end{equation*}
where $c_1,\,C_1,\,C_2,\,C_3,\,C_4$ are arbitrary constants, and
Hypergeom is generalized hypergeometric function.

We can get the original solution $c$ of generalized Black-Scholes equation \eqref{eq0} through substituting $P$ with the
transformation \eqref{dcevtrans}, \eqref{trans2} and \eqref{trans1}.

For $1-$dimensional generalized Black-Scholes equation, similarity
reduction method can be used to reduce the PDE to an ODE which is
easier to solve. Except the time dependent cases, we can also use
this method to reduce all equations Carr, Laurence and Wang (2006)
transformed, which are associated with the $1-$dimensional
simplification of {\bf Case 1.4b}.

\subsection{$1-$dimensional: CEV Model}

Assuming
\begin{equation}
   \sigma(S)=\sigma S^{\alpha},
\end{equation}\label{cev}
from the transformation, we know that the corresponding $M(x)$
\begin{equation}
   M=\alpha\sigma^2\big(\frac{\alpha\sigma^2}{(\alpha-1)^2}-\frac{1}{2(\alpha-1)}\big)
   \frac{1}{x^2}
     +\frac{r^2(\alpha-1)^2}{\sigma^4}x^2-2r\alpha-\frac{r(\alpha-1)}{2\sigma^2},
\end{equation}\label{cevM}
is the one dimensional case of {\bf Case 1.4b}. Therefore the solution can be written as
\begin{equation}
   P(\xi) = c_1 \xi^{\frac{1}{4}-\frac{c_0}{\sqrt{2c}}}.
\end{equation}\label{Scev}

We can get the original solution $c$ by substituting $P$ with the transformation \eqref{xieta1.2b}, \eqref{trans2} and
\eqref{trans1} ($1-$dimensional form).

\subsection{$1-$dimensional: Exponentially Decreasing Volatility}

Assuming
\begin{equation}\label{edv}
   \sigma(S)=e^{-S},
\end{equation}
from the transformation, we know that the corresponding $M(x)$
\begin{equation}\label{edvM}
   M=\frac{1}{2x^2},
\end{equation}
where for brevity we let $r=0$ \footnote{Similar transformation can
be found in Li and Zhang (2004), and Zhang and Li (2012).}, is the
one dimensional case of {\bf Case 1.1a}. Therefore the solution can
be written as
\begin{equation}
   P(\xi) = \frac{e^{-\frac{\delta_1 \xi^2}{4}}}{\sqrt{\xi}}
   \big[c_1 {\rm WhittakerM}(\frac{\delta_2t}{2\delta_1}-\frac{1}{4},\frac{\sqrt{5}}{4},
   \frac{\delta_1 \xi^2}{2})
   +c_2 {\rm WhittakerW}(\frac{\delta_2t}{2\delta_1}-\frac{1}{4},\frac{\sqrt{5}}{4},
   \frac{\delta_1 \xi^2}{2})\big].
\end{equation}\label{Sedv}
where WhittakerM and WhittakerW are the Whittaker function $M$ and
$W$, respectively.

We can get the original solution $c$ through substituting $P$ with
the transformation \eqref{xieta1.1a}, \eqref{trans2} and
\eqref{trans1} ($1-$dimensional form).

With a proper re-scaling transformation, the exponential decreasing
function volatility function can be converted to
$$ \sigma(S_t) = \sigma_0 S_0 e^{\alpha \left(1-\frac{S_t}{S_0} \right)},$$
where $S_0$ stands for the initial stock price, then
\begin{eqnarray} \tilde \sigma(S_t) &=& \sigma_0 \frac{S_0}{S_t} e^{\alpha
\left(1-\frac{S_t}{S_0} \right)} \nonumber \\
&=& \sigma_0 \left \{ 1 -(1+\alpha) \left(\frac{S_t}{S_0} - 1
\right) + \left( 1 + \alpha + \frac 1 2 \alpha^2 \right)
\left(\frac{S_t}{S_0} - 1 \right)^2 + O \left[ \left(\frac{S_t}{S_0}
- 1 \right) \right]^3 \right \}, \nonumber \end{eqnarray}
where $O(\epsilon)$ is the order of $\epsilon$. The function is
negatively skewed for positive $\alpha$, can be used to produce the
phenomenon of the implied volatility smirk observed by Zhang and
Xiang (2008), see also, Zhang and Li (2012).

\section{Conclusion}

With some transformation, we convert the problem of option pricing
under state-dependent volatility into an initial value problem of
the Fokker-Planck equation with a certain potential. By using the
Lie symmetry analysis and similarity reduction method, we are able
to write the solution analytically.

The study on a few cases demonstrates that our new method can be
used to produce analytical option pricing formulas for certain
volatility functions. A few exact solutions of the corresponding
cases provided in this paper can be regarded as contributions to the
option pricing literature.

The comparison with Finkel (1999), and Laurence and Wang (2005) is
as follows. In terms of the method, Finkel (1999) studied $2+1-$
dimensional Fokker-Planck equation in general by using the
prolongation of vector-field, but he did not discuss the
applications in finance. Laurence and Wang (2005) used the same
method as Finkel's and applied Finkel's results in finance. Our
method presented in Section 3 is more succinct. In terms of the
results, Finkel (1999) provided the vector fields of group
invariants in the symmetry reduction and group invariant solutions
in the particular case of 1.1a.
On the top of Finkel (1999), Laurence and Wang
(2005) provided the group invariant solutions via subgroups
generated by particular subalgebras in cases of 1.1ab, 1.2ab, 1.4ab,
1.5ab, 1.7ab.
We perform similarity reduction, and provide the group invariant
solutions in the cases of 1.3, 1.6 and 1.8ab. In terms of finance
application, Laurence and Wang (2005) only studied the case of the
generalized Black-Scholes equation on a single asset. We point out
that the problem of independent double-CEV can be reduced to case
1.2b. Even for the case of a single asset, the examples in our
Section 5.2, 5.3, were not studied in Laurence and Wang (2005).

In finance, it is an open problem to find a closed form solution for
the option on two correlated CEV assets. The case of independent
double-CEV has been studied in Section 5.1 in this paper. In order
to study the case of correlated double CEV, we need to use the
general case of Fokker-Planck equation. In principle, we can find
generators of invariant groups by using Lie symmetry approach. With
Finkel's (1999) classification, we can then obtain reduced equation
by using similarity reduction method. In quantitative finance, we
are interested in a solution of the generalized Black-Scholes
equation with a particular final condition, i.e., payoff function.
Constructing a solution of relevance in quantitative finance by
using some particular solutions seems not straightforward. This
problem is left for further research.

It is also an interesting topic to explore the application of
current approach to the pricing of path-dependent derivatives.

\section*{Acknowledgement.}
Wenqing Bao has been supported by the Soft Science Research Program
of Shanghai Science and Technology Development Fund (No. 201006007).
Jin E. Zhang has been supported by an establishment grant from
University of Otago.

\pagebreak

redpajama_set_name: RedPajamaArXiv


text:
Mark James ist der Name folgender Personen:

* Mark James (Songwriter) (* 1940), US-amerikanischer Songwriter
 Mark James (Golfspieler) (* 1953), US-amerikanischer Golfspieler

redpajama_set_name: RedPajamaWikipedia


text:
[caption id="attachment_131324" align="alignleft" width="228"] Stephanie May[/caption] When Stephanie May's parents traveled to Granville County from Mexico to work in the tobacco fields in the 1970s, few tobacco warehouses still stood
A Guide to Downtown Oxford
In this town built on tobacco and education, a new generation of residents seeks to preserve their town's history and caring community.
by Jodi Helmer
Modern additions to Oxford's historic downtown — like The Orpheum event venue and Tobacco Wood Brewing Company — make the town a pleasant pit stop off Interstate 85.
photograph by Anagram Photo
Stephanie May photograph by Anagram Photo
When Stephanie May's parents traveled to Granville County from Mexico to work in the tobacco fields in the 1970s, few tobacco warehouses still stood in downtown Oxford.
But evidence of the role that the industry played in shaping the town remained strong, and many of those clues can still be seen today. Some of the most impressive mansions in the historic district were built by tobacco barons. A sign reading "Oxford sells tobacco higher" hangs on the side of the Granville History Building — part of the Granville County Historical Society Museum, where May is executive director. The museum highlights the economic and cultural influence of tobacco. Even new businesses offer a nod to those agricultural roots. All of this, May says, makes the town "a quaint, friendly community that feels like taking a walk in the past."
Granville County Historical Society Museum. photograph by Anagram Photo
Granville County Historical Society Museum. Housed in an 1858 jailhouse, this museum features exhibits on education, military, and agricultural history. The Harris Exhibit Hall, a separate building adjacent to the main museum, showcases the work of local artists. "We want people to jump into history and not just read about it," May says.
The Orpheum at Oxford. This historic Art Deco theater went from abandoned to reimagined. "For many years, people said, 'Something needs to be done to save that building,'" May recalls. Local entrepreneurs stepped in to make it happen. Now, the neon marquee at the restored event venue is a beacon downtown.
The Orpheum at Oxford. photograph by Anagram Photo
Main Street Oasis. Regulars love the intimate atmosphere and dishes like steak, prime rib, and mahi-mahi. "It's the place to go when you want to have a date night," May says.
Strong Arm Baking. photograph by Anagram Photo
Strong Arm Baking. Julia and Thomas Blaine have been making doughnuts, cookies, brownies, breads, and other baked goods using ingredients sourced from local farms since 2014. The couple, regular vendors at several farmers markets, just opened their first brick-and-mortar location downtown.
Sunrise Biscuits. "In every town, there is a place that serves the greatest food you'll ever try in your life," May says. In Oxford, "that is Sunrise Biscuits." The mom-and-pop restaurant serves ham, sausage, steak, chicken, tenderloin, and fatback biscuits, along with other breakfast favorites.
Uptown 101. Enjoy rib eyes, burgers, and comfort-food specials like fried catfish and meatloaf. May calls this modern bistro "the perfect place to wind down in the evening."
Tobacco Wood Brewing Company. Mara Shelton opened the first female veteran-owned brewery in the state. Beers like Bulletstopper and False Motivation are a nod to her and her husband's military experience and local military history. Sandwiches, flatbreads, tacos, and barbecue make this a great spot to grab a bite, too.
Tobacco Wood Brewing Company. photograph by Anagram Photo
Art & Antique Gallery of Oxford. This shop features a wide selection of antiques and collectibles, including a community gallery with a rotating selection of local art. Owner Peter Tocci has helped restore several historic homes and provided them with period furniture and decor.
The Hub on Main. Order craft beer or wine and shop the selection of art, accessories, and gourmet foods. "It is genius to allow someone to have a drink while shopping," May says. Fourteen years ago, the North Carolina Hot Sauce Contest was created here when Julia Stovall Overton hosted an event featuring NC hot sauces in what was then Stovall's Gifts and Collectibles.
Nan's Young Fashions. This family-owned business opened in 1986. "If you are expecting a baby, you have a registry at Nan's," May says. In addition to adorable baby clothing and accessories, the shop also sells monogrammed clothing and gifts.
Since 2006, Oxford has been the home of all things spicy — at least during the Hot Sauce Contest in September. The festival features food trucks, art vendors, hot pepper eating contests, and a hot sauce competition, attracting those who like a little zip in their wine, beer, sauces, and snacks.
A self-guided walking tour of the Main Street and College Street historic districts includes homes dating back as far as 1820. For more information, visit oxfordhpc.org. photograph by Anagram Photo
This story was published on Oct 27, 2020
North Carolina-based journalist Jodi Helmer writes about food, farming, and the environment.

redpajama_set_name: RedPajamaCommonCrawl


Hello I'm Nik, Creative Fine Art Wedding & Lifestyle Photographer based in Bristol. I also cover Bath, Somerset, Dorset & Gloucestershire.
Please take a look around. . .
I'm fully insured & member of the Guild of Wedding Photographers & Society of Wedding & Portrait professionals.
I like to keeping things Fun, light, & relaxed.
Full Day, Half day or for just a couple of hours (if you are having a Registry Office ceremony) , see pricing.
So, feel free to grab yourself a coffee and take a look around. If you like my style and think we can work together - fly me an email.
Welcome to my website. Take a browse around my galleries and wedding packages and get in touch if you have any questions.
Come and chat about your wedding and take a look at my photography products.


text:
Bandar Tinggi is een bestuurslaag in het regentschap Labuhan Batu van de provincie Noord-Sumatra, Indonesië. Bandar Tinggi telt 5522 inwoners (volkstelling 2010).

Plaats in Noord-Sumatra


text:
Home Events Don't Miss the Merle Haggard Tribute Concert April 6 at Bridgestone
Don't Miss the Merle Haggard Tribute Concert April 6 at Bridgestone
Donna Vissman
In a career that spanned more than 50 years, Merle Haggard racked up an astounding 38 No. 1 country hits and every major award imaginable. Songs like "Mama Tried," "Okie From Muskogee," "I'm a Lonesome Fugitive" and "The Fightin' Side Of Me" continue to inspire new generations of artists and fans alike, and as we approach what would have been a major milestone, the time has come to celebrate his truly remarkable career. He was an iconoclast who refused to bend to country music convention and helped create the indelible Bakersfield sound. He gave voice to outlaws and outsiders and in a way that few, if any, have ever equaled.
Sing Me Back Home: The Music Of Merle Haggard is an event that includes:
Hank Williams Jr.
Alison Krauss,
Ronnie Dunn
The concert event will take place at Bridgestone Arena in Nashville on Thursday, April 6, 2017 at 7p in honor of what would have been Merle's 80th birthday and the one-year anniversary of his passing. This one-night-only concert event taping will bring together fans, friends, and music icons to honor the life and songs of music legend Merle Haggard.
A limited number of tickets for this all-star concert event are currently available at ticketmaster.com, by phone at 800-745-3000, all Ticketmaster locations and the Bridgestone Arena box office.
[scroller style="sc1″ title="More Entertainment " title_size="17″ display="cats" cats="55″ number_of_posts="8″ speed="5000″]
Previous articleGlobal Motorsports Car of the Week: 2015 Lexus LX 570
Next articleI Hate All Criticism and I Shouldn't – Here's Why
Donna is one of those former corporate types (Xerox) who wanted to try something new. She went from marketing to blogger and now Style Editor, and is always on the look out for what's trending in restaurants, new stores, charity events and entertainment. To keep up the pace, Donna is usually found drinking at least one Cold Brew coffee a day or on a busy day make it two.
5 Concerts to Attend this Week
Grand Ole Opry | July 16 & 17
Grand Ole Opry House
6th Annual Gretsch Presents Georgia On My Mind
Greg Koch: Koch Marshall Trio
3rd and Lindsley
Mike Ryan with Gable Bradley
Exit/In
Anders Osborne Solo with Special Guest Marc Ford


text:
/ Bob Gaudio
December 24, 2020 By Joe Marchese 12 Comments
Chris Christian rose to prominence as one-third of the pop trio Cotton, Lloyd and Christian alongside producer Michael Lloyd (The Osmonds, Barry Manilow) and Daryl Cotton (Zoot, Olivia Newton-John). Before long, Christian's songs were being recorded by the likes of Elvis Presley ("Love Song of the Year"), Dionne Warwick ("When the World Runs Out of Love"), and Carpenters ("(Want You) Back in My Life Again"). Christian also produced and wrote songs for artists including B.J. Thomas, Marilyn
Categories: News Formats: CD Genre: Gospel, Holiday, Pop Tags: Bob Gaudio, Charlene, Cheryl Ladd, Chris Christian, Christopher Cross, David Foster, Frankie Valli, Gerry Beckley, Kirk Whalum, Nigel Olsson, The Boone Girls
July 1, 2014 By Mike Duquette Leave a Comment
Bob Marley, Legend: 30th Anniversary Edition (Tuff Gong/Island/UMe) The best-selling reggae album of all time is back with two unreleased studio rarities and, on Blu-ray, a new 5.1 surround mix. CD/BD: Amazon U.S. / Amazon U.K. 2LP: Amazon U.S. / Amazon U.K. Pink Floyd, The Division Bell: 20th Anniversary Edition (Parlophone) The 20th anniversary of the last Pink Floyd album means an Immersion-level box set with a new 5.1 surround sound mix on Blu-ray and bonus vinyl pieces. CD (2011
Categories: News, Release Round-Up Formats: Blu-Ray, Box Sets Tags: Bob Gaudio, Bob Marley, Bon Jovi, Lulu, Pink Floyd, The Grateful Dead, The Jayhawks, The The, Vinyl
June 11, 2014 By Joe Marchese Leave a Comment
Next Friday, June 20, marks the highly-anticipated opening of director Clint Eastwood's film adaptation of the smash 2005 Broadway musical Jersey Boys. With John Lloyd Young reprising his Tony Award-winning performance as Frankie Valli opposite a cast of theatre and film veterans including Christopher Walken as Jersey mobster Gyp DeCarlo, Eastwood's film promises to bring the gritty story of Valli and The Four Seasons (Nick Massi, Tommy DeVito and Bob Gaudio) to an even wider audience than
Categories: News Formats: Box Sets Tags: Bob Gaudio


text:
Finally, the Narrow Water bridge is going ahead as part of Micheál Martin's Shared Island Initiative…
Mick Fealty on June 29, 2021, 10:27 am378 Comments | Readers 5283
In private conversations I often reference the failure to bring this project to pass (despite offers from the EU to cover most of the cost) as a perfect metaphor for one of the key weaknesses in North South relations: a tragic unwillingness to build.
Or within the wider political process that have arisen on foot of the Belfast Agreement: ie, a cataclysmic failure to understand that the core utility of bridges (metaphorical and literal) are the benefits it brings, not the bridge itself.
It is not about augmenting what we have. By conjoining two great landscapes, the Mournes and the Cooleys, it creates a new transversal space in which new and sustainable forms of eco tourism can form: if we can tap the imagination of local people.
Whilst it has not exactly set the news flows in the south alight this morning, it does signal that the Shared Island initiative is more than the sort of empty talk that has characterised the longshore drift in North South relations over the last ten years.
This Narrow Water Bridge has been a longstanding commitment of the Irish Government and was also, like the measures involved in the culture and language strategy, previously pledged under the New Decade New Approach Agreement.
Importantly, all the money is coming from Dublin, with the allocation of…
€3m from the Shared Island Fund for progressing the Narrow Water Bridge project to tender stage, based on a due diligence of the current design/building standards and live planning permission.
Allocation of further funds for the construction of the bridge, following determination of the likely final costs arising and Government approval of the successful tender.
The project originally failed when the then Fine Gael led administration refused to indemnify Louth County Council against capital overruns, leaving it to the then DUP Finance Minister Sammy Wilson to say if Dublin wouldn't, he couldn't.
With construction due to begin 2023, perhaps it will be taken as a signal of intent. Talk alone is cheap. After 20 odd years of examining our collective navels it's good to see someone in politics ready to put their money into concrete ideas.
It comes at a time of great turmoil but also greater political activism on the part of the UK government with regard to Ireland than at any time since Tony Blair left office in 2007, just after the signing of the St Andrews' Agreement.
If either of the great ism's that have dominated (and limited) Northern Ireland's capacity to look positively into its own future are to have a future of their own, sitting back is not an option.
Mick Fealty
Mick is founding editor of Slugger. He has written papers on the impacts of the Internet on politics and the wider media and is a regular guest and speaking events across Ireland, the UK and Europe. Twitter: @MickFealty
Event: Queen's Policy Engagement June Post-Brexit Clinic Wednesday 30 June 2021 at 12.30pm…
"But Violence Worked For the Other Side!" – Loyalism's Dangerous and Dishonest Myth. Part 1…


This is cool, after I move and have my setup I'll post a picture. My console collection is probably just a bit above average, but my PC collection is scary large. And I just packed up by N64 games tonight.
We didn't get owned, we got impressed.
Gotta love the labels on the side of the N64 games (I thought they were Super Famicom games at first). Once I get my crap in proper order I'll post it.
Here's my list off the top of my head.
There's many more I don't remeber, I'll try to list more later.
Click the names of each system to see pictures of it!
Also click the links below each game list to see the games!
56k warning, the images are big to show detail!
Since the games are self contained I feel like I owe you... a while back I took it apart. Check out the guts!
-Brain Age Train Your Brain in Minutes a Day!
There we go, that's all of it! One hundred forty two games not counting the duplicates and my PC games. You can see all of the systems lined up here, check it out because it's cool and something you don't see every day.
For those of you that noticed and were wondering, Super Mario Bros. 2 and Brain Age are missing because I'm a nice guy and lent them to my sister. Also I figured you guys didn't want to see the GameSharks or the Action Replay. Oh, and before I get someone nagging me, I did own Mario 64 but it got stolen.
Here is one last cool thing though, it's a deck of Club Nintendo Hanafuda Cards that were manufactured by (You guessed it!) Nintendo! Some of you may not know this, but when Nintendo was created in 1889 they made Hanafuda cards and are still one of the world's largest Hanafuda card manufacturers! There you go, you learned something new today; and if you already knew that well hats off to you!
I hope you enjoyed looking at my collection as much as I enjoyed taking it all down and snapping some photos (I spent four or five hours doing it and loved it!). Thanks for your time!
I guess I should also add that everything I own works and it works perfectly. Hell, I took apart the NES and reset the pin connector so I wouldn't have to blow into my games any more.
Fixed up the hyper links... I got more games but I'm not going to post them for a while.
well i knew they made playing cards and board games and stuff like that, but i didn't know that still do. they do?
my collection is abysmal compared to the rest of yours, so i don't really want to post it.
I need to go to my local flea market and find some more games. Only my 64 collection is as nearly impressive as yours Pat.
Well, I actually have 2 more than you. And if I kept my Pokemon Snap, Pokemon Stadium and Bug's Life it would be more. I'm targeting for a copy of Majora's Mask and StarFox 64 and I'll be happy with my collection.
I swapped my system's motherboard so now it perfectly works. And I have 5 controllers in the colors of red, yellow, green, grey (newer condition), and an atomic purple.
Anyhow, great collection. If I knew where everything anything is, I'd list them.
You have me beat in everything.... except NES games. We had about 80 NES games. I'm not joking or exaggerating.
I have only owned the NES, GameBoy, GameBoy Color, Nintendo 64, Gamecube, GameBoy Advance SP, Nintendo DS (original... broken... and the LITE!... broken). Oh and a playstation 1 and xbox... but I didn't care for them so much. I only got a couple games for each.
Currently, NES is sitting in a box in my closet. ALL the games except for Legend of Zelda and Tetris are missing. I presume they are in a box in a garage, not having seen the light of day since I packed them when we moved almost 7 years ago. The GameBoy is sitting right here in front of me on the top shelf of my desk. It's games (3 of them. Super Mario Land, Tetris, and Hyper Load Runner) are stored with the NES games I believe. Nintendo 64 is in the same box as the NES, while the games I haven't a clue about their final resting place. Perhaps in my closet. I believe I had 9 N64 games. My GameCube and my games for it are over at my GF's house. She's not allowed to buy consoles of her own. She loves the Zelda Collectors Edition game. GameBoy Advance SP I gave to my friend before he quit school. I have my games in a box in my closet as well. My Nintendo DS, obtained the first day it came out... was split in two the following summer by my sister. I think got a black DS LITE the following November when final fantasy III came out. The following summer my cousin broke one of the hinges, though it still plays games. Just the top screen is very weak and can't hold itself up anymore. Currently, the first DS is under my bed. DS lite I gave to my GF because she loves final fantasy.
Get that hat up high.
I can't believe somebody else owns Sim Isle. I remember never being able to figure that darned thing out. I didn't even know the point of the game. Did you ever figure it out, Martyk? And did you ever beat the last level of Sim Copter?? Im-freaking-possible!
The last level of Sim Copter isn't impossible, though, I do play every city I can, meaning repeating difficulty levels as many times as possible, just so I have enough money to get up to the better copters. Also, I own a copy of Sim Isle too, I remember figuring out how to make a working factory and tourist center on the same island once.
The amount of Snes, nes, and atari games i own outnumbers everything else ive gotten in the past 10 years. Ill take a picture soon as i collect everything.
The one and only JamDunc.


Mmmmmm that looks awesome! I will have to try it. Do you think it would be ok without the pesto?
Where do you find the cheese? I normally don't go for the processed stuff, but the sandwich looks too good.
Sweet!!! Since I have gone the "plant based" route the only thing that I really miss is the cheese. I do live within walking distance from whole foods (much to the demise of my paycheck). I have fresh basil so I think I might try it with that, avacado, and tomato slices. Will see how that turns out. Oh, if you guys are a fan of stout beer WF has a great one by Four in Hand (I think that is the brewing company) that is my new favorite. It is like Giunness but a bit more sweet (the Irish were really on to something with that one, love it!!).
Cheers & beers!! Stout beer that is.


text:
SEVEN TO ETERNITY #1
Art: Jerome Opena & Matt Hollingsworth
James R: This was a title that looked a must-buy from the off. Any new project from Remender and Opena will always be something special, and the first issue of Seven To Eternity certainly aims for the epic and the spectacular. Set on Zahl, a world where magic is real, Seven To Eternity focuses on Adam Osidis, a man whose family have always resisted the overtures of the powerful Mud King but now finds he is forced to listen to the offer following the death of his father. Firstly, it is absolutely beautiful - Opena is a remarkable artist, and every page of this first issue looks spectacular. Remender repeats the technique he used in the debut issue of Low, immersing us into a fully-realised world but holding back from full explication as to the mechanics of it all. I have to hold up my hand and say that I am not the biggest fan of fantasy or 'magic'-based stories, so this wasn't the instant hit that Black Science was for me on its debut. I will definitely give the second issue a go as the creative team are so good, but this was an interesting rather than a spectacular start. 7/10
Matt C: From the sounds of it – sell outs and swift second printings – this is something of a big hit for Image. We're huge fans of Rick Remender here at the PCG so that response comes as no surprise (he's one of the best writers in the business) but there's that extra element here that's possibly turning more heads than expected. Jerome Opena's past collaborations with Remender have been exceptional but – holy shit! – this is on another level entirely. The detail present, the way each of the panel compositions seems to hold the optimal viewpoint... this is easily Opena's greatest work yet and a real masterclass of comic book art at its finest. Matt Hollingsworth's colours bring an intense intimacy to the images and while the visuals often overshadow the script, Remender is clearly at his world-building best once more. A striking opening to what is likely to be a major buzz series going forwards. 8/10
Writer: Tom King
Artists: Gabriel Hernandez Walta & Jordie Bellaire
James R: Can you escape your destiny? The philosophical puzzle that lies at the heart of King and Walta's remarkable series is a fitting one for the Avengers' thoughtful synthezoid. As the book draws to a conclusion, Tom King has written another magical issue, one that manages to reflect on the question of destiny whilst also surprising us too. The Vision has been gaining plaudits since issue #1, and rightfully so - in a time in which rebooting and relaunching is the Big Two's main tactic, The Vision shows that nothing is as effective as a good story well told and beautifully illustrated. Reading this issue, there's a part of me that's pleased that it is a limited series - it's fair to say that it's been one of the stand-out books of the year, and bringing on a new creative team would personally lessen its impact and wonderfully self-contained plot. Kudos too to King for a nod to Steve Gerber's Omega the Unknown, and a meta-nod at the current ubiquity of superhero movies. Walta's art is great as always - he's as adept at conveying kinetic action as he is as showing the very human emotions of Vision and his family. There's just a coda left for this book, but it's safe to say that in term of superhero comics, it is peerless. 9/10
Writer: Francis Manapul
Art: Francis Manapul
DC $2.99
Matt C: I've not been paying much attention to the DC Universe over the last couple of years so this 'new/old' Superman has passed me by a bit, but I was keen to give this a look as I think Francis Manapul is one of those writer/artists whose scripts can match his visuals. I liked what he did on the New 52 iteration of Flash so I felt this series, combing DC's most familiar and iconic characters, was worth a try. It's a surprising opener because it eschews expected action and adventure for a more intimate look at the fledgling dynamics of this trio, centring it around a dinner party at the Kent household. There are some nice moments, a good deal of wit, and while it doesn't exactly thrill, it's an intriguing start, one with a cliffhanger that will ensure I'll back to see where it's headed. 7/10
Artists: Dean Ormston & Dave Stewart
James R: The thing about Jeff Lemire is that he always gets you in the heart. I love his ability to make you both empathise with his characters and feel their pain and loss. In the latest issue of Black Hammer the plot focuses on the book's Martian Mahunter analogue, Barbalien. Before being trapped in the current dimension, the Martian Mark Markz had sought to integrate with humanity in Spiral City but found that acceptance and tolerance is often a problem for mankind. That also proves to be the case in the small town in which our heroes now reside. As I've mentioned previously, Lemire seems to find an extra gear when working on his creator-owned books, and this is no exception as he portrays Barbalien's isolation and self-loathing perfectly. Dean Ormston's work is great in these pages, mixing the pulpy feel of the flashback/origin sections with the rural, everyday sections to perfection. We are only three chapters in, but Black Hammer has already established itself as one of the highlights of my pull-list, and there's no reason on this - or any parallel Earth - why you shouldn't be picking this up. 8/10


Our over all goal is achieving equal rights, equal opportunities and equal participation of the persons with disabilities in order to make them equal citizens.
To rehabilitate the disable persons residing within limits of Vavuniya district or other districts to enable them to mobilize their full potential and maintain their self esteem through the COMMUNITY BASED REHABILITATION.
To assist the disable to secure suitable jobs in Government and private sectors.
To assist the disable to engage themselves in productive activities.
To provide community health facilities and other services in order to prevent disability among the population.
To organize social, economic and cultural activities among the disable in order to enable them to keep up their moral and to integrate themselves in to the society.
To eliminate any discrimination, whatsoever due to the disabilities and to activate the disable in every possible way.
To assist the disable with any relief assistance that may be required.
Committed and dedicated Executive members and staff.


Between trimming the tree, decorating your home and wrapping gifts, the season of joy can easily turn into an ordeal that's more stressful than merry. Make this holiday season festive and efficient with 20 clever ways to save time, money, and last but definitely not least, sanity. We say simplify the season and cut corners whenever possible.
No need to place a special order to get a professional-looking centerpiece for your holiday table. You can easily make an impressive floral arrangement using a simple $10 bouquet from the grocery store and greens from your yard or neighborhood. You may just put your local florist out of business.
Not everyone is #blessed with a wood burning fireplace, but that doesn't mean you can't make the most of what you've got. Get that old fashion fireplace experience with this simple tutorial. You'll be enjoying crackling embers, glowing orange flames and that authentic scent of burning wood in no time.
This year, don't settle for a pile of holiday cards on the coffee table. Instead, make a mantle-worthy display to showcase the beautiful cards you receive from your family and friends. If you're comfortable using power tools, you can tackle this project in an afternoon, and you'll enjoy the holiday cheer all the more for it.
Decorate your mantel in record time with mini homemade stockings. They add a bright pop of color to a stark fireplace when strung together as garland, as pictured. Bonus: You can also make them as gift card holders for last-minute gifts (or even as ornaments!).
No one will ever guess that you made this designer-looking wreath using a foil baking sheet and gold spray paint. So go ahead, decorate the halls on the cheap. We won't tell anyone your DIY secret. But you'll be so proud of your handmade holiday decor, you'll probably want to announce it to everyone yourself with a perfectly styled Instagram post.
Turn empty wine bottles into Pinterest-worthy home accents with this tutorial. Place the illuminated bottles on your mantle, on your dining room table in lieu of candles, or in an empty fireplace to add a modern twist to your holiday decor.
Celebrating the holidays in a small space? Make your own evergreen stand-in using a tomato cage and ivy bunches. Accent with a festive bow to add extra holiday cheer to your apartment. (This DIY is also ideal for procrastinators who waited until the week of Christmas to get a tree).
Choose your favorite holiday sentiment, such as Joy, Love or Family, and use jewelry wire and beads to create a beautiful yet simple adornment for your mantel, front door or tree. The best part? You can display this homemade home accent all year long.
Calling all apartment dwellers: If you don't have a traditional fireplace to decorate for the holidays, fret not. Make your own cardboard faux fireplace using inexpensive materials from your local hardware store. This cost-effective DIY is easy to assemble and gives you a festive mantle to deck for the holidays.
Turn the back of a closet door into an organized gift wrapping station to keep scissors, wrapping paper and ribbon at your finger tips -- unruly spools of ribbon and unraveled rolls of wrapping paper will be a thing of the past. Santa's elves couldn't wrap a present more efficiently themselves.
When it comes to wrapping a present, it's all about proportions and crisp edges. With these simple tricks, your presents will be looking perfect under the Christmas tree.
Skip the expensive gift tags and opt for these free printable versions. This simple yet modern design will look great tied around the neck of a bottle of wine as a hostess gift, adorning a present under the tree, or atop a homemade gift fit for a foodie.
Make every gift an impressive one by topping it with a homemade fancy bow. It almost doesn't matter what's in the box because the wrapping is so pretty. But if you do want the gift to be as awesome as the wrapping, check out these easy DIY stocking stuffers for everyone on your list.
Don't waste your money on pricey printed paper. Instead, make your own festive gift wrap using just a sprig of pine and white paint. Simply snip a small piece of the Christmas tree to transform a plain roll of wrapping paper into holiday-ready snowflake gift wrap. Now you can spend the gift wrap money you saved on cider and eggnog.
Make a quick trip to your local home improvement store and grab the materials you need to turn a garden trellis into a simple advent calendar. This easy project will only set you back about $10 and the tree-like shape of the trellis doubles as festive decor. We call that a win-win.
If you're entertaining this holiday season, make simple placemats by hot gluing birch wood slices onto neutral placemats. A small succulent Christmas tree on each plate adds the final touch to your festive place settings.
Whether you waited until the last minute to get a Christmas tree for your home or you're looking to impress the in-laws with a great hostess gift, this Christmas boxwood tree can be made in a flash (and on a budget). Simply use floral foam, boxwood cuttings and spare ornaments to make your own.
There's no need to spend over $15 on a festive swag from your local nursery to adorn your front door. All you need to do is gather items from your own yard (or your neighbor's yard if you're on good terms -- or if you're feeling daring) to make your own holiday swag for free.
Decorate your home with a wreath that will not only make your home look and smell great, but that you can also use while you're cooking for the holidays. Make a rosemary wreath for an easy and cost-efficient version, or create a wreath with a medley of herbs to keep all your favorite herbs within arm's reach.
Maybe you forgot to get the neighbors a gift, or perhaps you need a last-minute hostess gift -- whatever the reason, we're not judging you. Chocolate bark is a simple yet impressive-looking dessert that can be made in mass quantities in a time crunch. Plus, it's easy to put your own twist on the recipe to make it your own. May we recommend sweet, salty, nutty chocolate bark?


Oakley Drop Point Review | Get Pumped!
Sunglass Rob and Pumped Tyler are back and better than ever with the all new Oakley Drop Point Sunglasses! Think of a revamped, more sophisticated sibling to the Oakley Canteen. Reporting from inside the museum at Oakley Headquarters (whoa…), and with a little help from their friend Jared (wait, I thought that was Phil), Rob and Tyler show us why you should be pumped about the new Oakley Drop Point Sunglasses, too!
Offered in amazing color combinations, the Oakley Drop Point Sunglasses are a new performance/sun/active/lifestyle frame designed for small to medium faces. Not for a big guy, but maybe a big youth. Packing an 8-base wrap around frame, they are constructed of a lightweight, yet durable material called O Matter™. Meaning they are super-duper comfortable and super-duper hard to break. In fact, to demonstrate how flexible the O Matter™ is, Tyler pulls a temple away from the other like he's pulling a wishbone on Thanksgiving! Not a problem! Being O Matter™ also means the frame doesn't get misshapen over time, is twice as strong as acetate, and yet 25% lighter. Pri-tay cool!
The Oakley Drop Point Sunglasses feature a Three-Point Fit which holds the lenses in precise optical alignment. The temples go straight back for super comfort while eliminating pressure points. And they gently hold on behind the ears instead of hooking back – nice touch! These Oakleys also have anti-fog ventilation and easy lens interchange to help promote that active lifestyle of yours!
According to Sunglass Rob, the Oakley Drop Point Sunglasses are closer to a medium frame than a small frame. From temple to temple they measure an impressive 132mm, with 132mm running down the temple, as well. The rectangular lenses are 60mm wide and 41mm high. They also pack Plutonite® lens material, which filters out 100% of harmful UVA/UVB/UVC rays and blue light. Boasting Unobtainium® nose pads to help with grip, the Oakey Drop Point Sunglasses have a slightly wider than average nose bridge at 18mm.
Need a little Rx-ing to go with those Oakley Drop Points? SportRx is there for you! In fact, it's our specialty 7 days a week. Simply contact us at SportRx.com or give us a jingle on the phone and an expert optician will be happy to help you find some prescription sunglasses! So whether it's mountain biking, soccer or simply a day at the races, Oakley Drop Point Prescription Sunglasses won't let you miss out on any of the action!


Community Pancake Breakfast: 9-11 a.m., Queen Creek Town Hall courtyard, 22350 S. Ellsworth Road. Free. Residents are invited to "wake up to what's cooking in town" during this free, fun and casual community outreach program provides residents opportunities to have their opinions heard and encourages them to learn about projects affecting Queen Creek. Town representatives will be on hand to answer questions and display upcoming projects. Topics to be covered include: the town council's five-year goals; plans for the Town Center area; road improvements; sewer and wastewater projects; the approved General Plan; economic development goals and strategies; dust control initiatives; Queen Creek Water service; fire service and public safety; and garbage and recycling services. For more information or to suggest a topic to be covered on a future date, contact Debra Kuffner at Debra.Kuffner@queencreek.org.


text:
Towers of Midnight: Chapter 15
<< Previous Chapter | Next Chapter >>
Author: Bryce al'Mara
Use a Pebble
Chapter Icon: Frayed Weaving
Points of View: Nynaeve, Egwene
3.1 Mysteries
3.2 Character Development
Nynaeve and Naeff check out a Bubble of Evil. Egwene learns all she can about Mesaana.
Nynaeve's Point of View:
Setting: Streets of Tear
Characters: Nynaeve, Naeff, Rand
Nynaeve and Naeff are walking down the street and Naeff sees Myrddraal in the shadows because of the madness he got from the taint. They are heading for All Summers neighborhood where there has been a bubble of evil. People run by them scared looking over their shoulders. They arrive to a line of Defenders who let them by. The area of the city is blander. Nynaeve asks the defenders if they have helped anyone but they tell her they have not let anyone in because it's not safe. Nynaeve tells them to start looking for injured people or Rand won't be happy. As Nynaeve steps forward the paving stone under her feet turns to dust. As she takes more steps she notices all the stones she touches turn to dust. They walk into an inn to find everyone frozen in place not moving. The floor of the inn is gone and they can see the cellar. Nynaeve delves multiple people in the room without touching them and they all show no signs of life. Nynaeve has Naeff create a gust of wind and everything turns to dust and blows away. They do this to the whole part of the city and create a cyclone of the dust. Nynaeve adds fire and it all catches on fire. Naeff says maybe the Fades caused it which makes Nynaeve want to delve him. She asks him to stand still and delves him. She finds the madness in his head nestled in like thorns and pulls them out. After she takes the thorns out the madness disappears and he no longer sees the Fades. Nynaeve sends him on his way and then Rand shows up. He thanks her for helping his people and for helping him. Nynaeve brings up his breaking the seals, asking if it will make the Dark One more powerful. Rand replies it will for a moment but that he has to face him while he is strong. He then tells her not to change like all the other Aes Sedai and to keep her passion. He then tells her to go to Egwene and that he wants her there with him at the Last Battle as one of the three to use Callandor. She tells him that she will be there. She then delves him to see if she can heal his madness but sees that there are thousands of thorns. She tells him she can't heal him but she will heal the other Asha'man before she leaves.
Egwene's Point of View:
Setting: White Tower
Characters: Egwene, Saerin, Yukiri, Seaine
Egwene is having a meeting with Saerin, Yukiri, and Seaine asking them what they have discovered about Mesaana. Yukiri and Seaine have a list of possibilities for how she could have avoided the oath rod. Seaine tells Egwene that there is a possibility that Mesaana had a second oath rod on her that she had an inverted weave of spirit set on so that she could forswear any oaths they made her take. The second possibility they suggest is that she could have sent a lookalike that used the mirror of mists and was put under compulsion and since they weren't dark friends they would have been able to say it. Egwene states that it would have taken a lot of preparation and Saerin says that from what she learned about Mesaana that she was good at preparation. The third possibility is that she used a weave that made it sound like she was saying things she wasn't. So she could say something and it would sound like she was saying the oaths. Egwene tells them to continue looking. Saerin then goes into her report about Mesaana saying she was a researcher that was rejected. That she governed the Aes Sedai that went to the Shadow, leading them into battle at times. Saerein says out of the Forsaken she is the realist, the one that will step in and get her hands dirty. She says she was kind of like an Amyrlin for the Shadow. She then tells them that Mesaana is only second in anger to Demandred. Egwene then sets out to make plans to capture her.
This section contains Notes on this Chapter which may contain spoilers. Please expand to view.
How much of the info they found on Mesaana is true?
Will Nynaeve be able to heal everyone's madness or was she only able to heal Naeff because he didn't have much madness?
What is Egwene planning?
Egwene
She is set on learning everything about Mesaana and capturing her.
She has developed the ability to heal madness at least for some.
She is becoming more and more skilled at Delving.
She does not like not being able to heal everything.
Retrieved from "https://linode.tarvalon.net/index.php?title=Towers_of_Midnight:_Chapter_15&oldid=138003"
Towers of Midnight Summaries


Uncover one of the greatest stories never told.
A scientist born and raised in Southwark, a rebel once silenced for her maverick theories, she made one of the greatest discoveries of our times.
Solve the mystery in reality and augmented reality. The truth is always bigger than you believe.
The event organisers,The Milo Wladek Co., make films and create interactive theatre.


Get results in Glendive, Montana; exclude locationless jobs emailed to you daily!
Class A CDL Experienced Truck Drivers Needed!
© 2019 JobsInTheUS.com - All rights reserved JobsinMT.com, part of the JobsInTheUS.com network, is your trusted local source for finding the most relevant jobs quickly and easily. We help local employers and qualified job seekers in to find the perfect fit.


Researchers and patients are reeling over an Israeli Biotech's claim to have found a cure-all for cancer.
Next-generation medications have made chronic conditions out of previously fatal cancers.
But scientists continue to overlook Africans in genome research.
The animals have 100 times as many cells as humans but rarely get cancer. New gene studies may help explain why.
Doctors say the high-tech health devices are also good at objectively measuring quality of life.


text:
St. Petersburg State University (the Department of Oriental Studies) invites scholars to join the International conference on the 65th Anniversary of Southeast Asia languages studies in our country that will take place on 15-16 September 2020, at the SPbSU in St-Petersburg, Russia. Calls For Papers The systematic studies of mainland and island Southeast Asia indigenous peoples and culture, teaching languages of this region that started in 1955 at the Leningrad State University, led to the appearance of now well-known scientific school, that organically fit into the tradition of oriental studies in St. Petersburg. The format of the upcoming conference includes…
More than 2,500 test their proficiency level in Oriental languages at St Petersburg University
St Petersburg residents are not the only people who want to obtain an international certificate that proves their level of proficiency in Japanese, Korean or Persian. Residents of many other regions in Russia – along with citizens of China, Uzbekistan, Estonia and Latvia – have taken exams at the St…
St Petersburg University strengthens cooperation with partners from Vietnam
The University has been visited by a delegation from the Ho Chi Minh National Academy of Politics. The guests were received by the Rector of St Petersburg University Nikolay Kropachev. He expressed hope for continued joint work on the launching of new academic programmes with the Vietnamese component. 'The University…
Lecture of Sonja Haeussler (Stockholm University, Sweden)
December 9, a lecture by Professor, Head of the Department of Korean Studies at Stockholm University Sonja Haeussler will be held at St. Petersburg State University.Theme of the lecture: 'Im Kwon-Taek's Beyond the years and the Sŏp'yŏnje trilogy' The lecturer will discuss the film Beyond the years (Korean: Ch'ŏnnyŏnhak, "Thousand…
The Minister of Culture of the Republic of Korea visited St Petersburg University during the International Cultural Forum
Category: All news
St Petersburg University has been visited by a South Korean delegation headed by Park Yang-woo, the Minister of Culture. Among the guests were Lee Sok-bae, Ambassador Extraordinary and Plenipotentiary, and Kwon Dong Seok, the Consul General of the Republic of Korea in St Petersburg. The meeting was held by: Elena Chernova,…
The Ambassador of the Republic of Turkey has met with the Rector of St Petersburg University
Mehmet Samsar, Ambassador Extraordinary and Plenipotentiary of the Republic of Turkey, has come to St Petersburg University for a reconnaissance visit. He met with the Rector of St Petersburg University, and also talked with students and lecturers of the Department of Turkic Philology. Nikolay Kropachev, the University rector, thanked the…
Literary and cultural relations between Istanbul and St Petersburg have been discussed at the VIII St Petersburg Cultural Forum
The conference 'Istanbul in Turkish and Russian literature' has been held at St Petersburg University. Famous Turkish and Russian writers – İskender Pala, Hüseyin Kandemir and Ilya Boyashov – took part in the discussion. Associate Professor of St Petersburg University and the Director of the Centre for Contemporary Turkish Studies…
Hagia Sophia and the 'grandfather' of Stonehenge: three exhibitions on Turkish culture are held at St Petersburg University
Three exhibitions dedicated to the cultural heritage of Turkey opened at St Petersburg University as part of the VIII St Petersburg International Cultural Forum. More than two dozen photographs of historical monuments of Turkey and over 150 books are on display in the corridor of the Twelve Collegia biulding. The…
St Petersburg University has partnered up with Atatürk University
St Petersburg University has been visited by a delegation of Atatürk University (Turkey), led by its rector Professor Ömer Çomakli. During the meeting, the parties entered into an agreement on cooperation. This includes the development of the Centre for Contemporary Turkish Studies at St Petersburg University. The agreement with Atatürk…
St Petersburg University is furthering cooperation with one of the top universities in Qatar
St Petersburg University has been visited by a delegation of Hamad bin Khalifa University (Qatar). It was headed by Adam Al Saadi, Director of Public Relations and Student Affairs. The purpose of the visit was to get acquainted with master's and doctoral academic programmes of St Petersburg University, taught in…
Roerich's heritage: an artist, a philosopher, an archaeologist and a graduate of St Petersburg University
On the anniversary of the birth of Nicholas Roerich, St Petersburg University opened an international scientific and practical conference 'Roerich's Heritage'. This year the event was dedicated to the 145th anniversary of the world-famous artist and was held in St Petersburg for the 19th time. Experts from Russia, USA, India,…
St Petersburg University and Shinshu University have signed a framework cooperation agreement
Shinshu University Vice-President for International Relations Kiyoshi Tanaka paid a visit to St Petersburg University. During the visit the framework cooperation agreement was signed. The idea of the partnership agreement was initiated by scientists and researchers of St Petersburg university, who know Shinshu University very well. For example, the head…
President of Peking University Hao Ping has visited St Petersburg University
St Petersburg University has held a meeting between Nikolay Kropachev and President of Peking University Hao Ping. The visit began with a tour of the historic building of the Twelve Collegia. Professor Kropachev showed the distinguished guest the Dmitri Mendeleev Museum Archive, the Petrovsky Hall, and the Modern Sculpture…
Saint Petersburg State University and Nanjing University cordially welcome paper proposals for the 9th International Conference "Issues of Far Eastern Literatures", dedicated to the 380th anniversary of outstanding Chinese man of letters Pu Songling (1640-1715). The conference will be held on June 27 – 29, 2020 in Saint Petersburg and…
St Petersburg University team overtakes Oxford, Yale and Harvard at the World Rowing University Regatta
The World Rowing University Regatta was held in Xi'an (China). The rowing club team of St Petersburg University reached the final of the competition and came seventh. Ilia Kostylev, the coach of the university team, told us that he was pleased with the progress of the athletes and 'how they…
Ankara University and St Petersburg University are enhancing their cooperation
The Consul General of the Republic of Turkey Yunus Belet and the Deputy Dean of the Faculty of Politics of Ankara University Can Giray Özgül paid an official visit to St Petersburg University. The objectives of the meeting were to discuss the prospects for further cooperation between Turkish universities and St Petersburg University. Deputy Rector for International Affairs of St Petersburg University Sergei…
St Petersburg University has partnered with leading Turkish universities
The solemn ceremony took place at the 2nd Rectors' Forum, held as part of the Russian-Turkish public forum. The meeting was attended by representatives of the 36 largest universities of the two countries. Andrey Maksimov, the Chairman of the Committee of Science and Higher Education of St Petersburg, welcomed the…
St Petersburg University – in the top 100 of reputation ranking of the THE on quality of education
With the quality of education St Petersburg University has kept a position in the top 100 of the world reputation ranking of universities made by the Times Higher Education (THE) magazine. The level of teaching in 2019 has enabled St Petersburg University to be ranked the 91st in the corresponding…
Vice Minister of Education of the People's Republic of China has visited the University
St Petersburg University has been visited by a delegation led by Mr Zhong Denghua, Vice Minister of Education of the People's Republic of China (PRC). The purpose of the visit was to discuss the issues of promoting training and research exchange between the University and Chinese universities. Sergey Andryushin, Deputy…