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\title{\LARGE \bf
Comparison of Market Parameters for Jump-Diffusion Distributions 
Using Multinomial Maximum Likelihood Estimation}
%%
\author{Floyd B. Hanson and Zongwu Zhu% <-this % stops a space
\thanks{The work is supported by the National Science
Foundation under Grant DMS-02-07081. The content of this  material
is that of the authors and does not necessarily reflect the views of
the National Science Foundation.  Published in  Proc. 42nd IEEE Conference on Decision and Control,
2004, pp.~3919-3924. }% 
\thanks{Department of Mathematics, Statistics, and Computer Science,
M/C 249, University of Illinois, Chicago, IL 60607-7045
{\texttt{\small hanson@math.uic.edu} and \texttt{\small zzhu@math.uic.edu}}%
}% endthanks
}% endauthor
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\begin{document}
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% ZZ/FH 8/29: remove central from moments
\begin{abstract}
Previously, we have shown that the proper method for
estimating parameters from discrete, binned stock log returns is
the multinomial maximum likelihood estimation, and its performance
is superior to the method of least squares.  Useful formulas
have been derived for the  jump-diffusion distributions.  
Numerically, the parameter estimation can be a
large scale nonlinear optimization, but we have used techniques to reduce
the computation demands of multi-dimensional direct search.  
In this paper, three jump-diffusion models using
different jump-amplitude distributions are compared. 
These are the normal, uniform and double-exponential.
The parameters of all three models are fit to the Standard
and Poor's 500 log-return market data, constrained by the data first 
and second moments.
While the results for the skew and kurtosis  moments are mixed,
the uniform jump distribution has superior qualitative performance 
since it produces genuine fat
tails that are typical of market data, whereas the others have exponentially
thin tails. 
However, the log-normal model has a big advantage in
computational time of parameter estimation compared with the others,
while the double-exponential is most costly due to having one more model
parameter. 
%
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%
Despite the great success of Black-Scholes 
% Zhu 8/29 options: Black-Scholes model or Black-Scholes option price formulae% 
model \cite{BS73}
% Zhu 8/29: , % end % FH 8/29 Good.
in option pricing, this pure log-normal diffusion model fails to
reflect the three empirical phenomena: (1) the large random
fluctuations such as crashes and rallies; (2) the non-normal features,
that is, negative skewness and leptokurtic (peakedness) behavior
in the stock log-return distribution;  (3) the implied volatility smile,
that is, the implied volatility is not a constant as in the
Black-Scholes model.

Therefore, many different models are proposed to modify the
Black-Scholes model so as to represent the above three empirical
phenomena.   Merton \cite{Merton76} introduced the jump-diffusion 
model in financial modeling, using a Poisson process for the jump timing
and a log-normal process for the jump-amplitudes to describe the
market crashes and rallies. 
Some models are proposed to incorporate the \textit{volatility
smile}, for example, Andersen, Benzoni and Lund \cite{ABL02} have
made elaborate estimations to fit jump-diffusion models with
log-normal jump-amplitudes,
stochastic volatility and other features.  Some models are proposed to
incorporate the asymmetric features of the stock log-return
distributions.   Recently, Kou \cite{Kou02} proposed a
jump-diffusion model with a log-double-exponential process for
the 
% Zhu 8/29 jump-amplitude--> % FH 8/29 OK
jump-amplitudes.   
% end 
Since crashes and rallies are rare events, so
the Poisson process is reasonable for the timing of jumps.  However,
there is a problem in choosing the log-normal or log-double-exponential
process for the  
% Zhu 8/29 jump-amplitude--> % FH 8/29 OK
jump-amplitudes 
% end 
since the exponentially small tails
of the log-normal  and log-double-exponential distributions are contrary
to the flat and thick tails of the long time financial market log-return data. 
Around the near-zero peak of the log-double-exponential and the log-normal, 
the jumps are small, 
so are not qualitatively different from the continuous diffusion fluctuations.  
%When the jumps are large, then  the density tails are exponentially small,
%but the large jumps of the data are more persistent. 
Moreover, an infinite jump domain is unrealistic, since the jumps should 
be bounded in a real world financial markets and an infinite domain leads
to unrealistic restrictions in portfolio optimization \cite{CDC02FM}.

So, Hanson and Westman \cite{MTNS02FMT} proposed one jump-diffusion
model with log-uniform jump-amplitude.  Most recently, Hanson,
Westman and Zhu \cite{CONMATH04} showed that for IID simulations 
% Zhu 8/29: that % end % FH 8/29 OK
the binned distribution is multinomial.  They estimated the market parameters
for this log-uniform model by subsequent multinomial maximum likelihood method
to fit financial market distributions such as the Standard and
Poor's 500 stock index.   
%In this paper, the value of the skewness
%of the log-uniform model is greatly improved using more accurate 
%computations here.

The main purpose of this paper is to compare the performance of 
three jump-diffusion models whose jump-amplitudes are 
the log-normally, log-uniformly and log-double-exponentially distributed.
% FH 8/22:
%The measures of performance are the skewness, kurtosis and computational costs.

%
% Zhu 8/29: \section{Some Jump-Diffusion Models Theoretcial Results} --> 
\section{Some Theoretical Results}
% end / No need to emphasize it since the reader can understand it by default. % FH 8/29 OK
%
\subsection{Stock Return Process, $\mathbf{S(t)}$}
%
The following stochastic differential equation (SDE) is used to model
the dynamics of the asset price, 
\begin{small}$S(t)$\end{small}: 
%
\begin{small}
\eq
\label{dS(t)}
dS(t) = S(t)\left(\mu_d dt + \sigma_d dW(t) + J(Q)dP(t)\right),%
\nq%
\end{small}% 
%
\noindent{where} $\mu_d$ is the drift coefficient, $\sigma_d$ is the diffusive volatility, 
\begin{small}$W(t)$\end{small} 
is the stochastic diffusion process, 
\begin{small}$J(Q)$\end{small} 
is the Poisson jump-amplitude, 
\begin{small}$Q$\end{small} 
is its underlying Poisson amplitude mark process, 
%
\begin{small}$J(Q)dP(t)$\end{small} 
is just a symbol for the counting 
\begin{small}$\sum_{i=1}^{dP(t)}J(Q_{i})$\end{small}, 
% Zhu 8/29: Q_{i} --> J(Q_{i} % end % FH 8/29 Good correction!
%%
\begin{small}$P(t)$\end{small} is the standard Poisson jump process with
joint mean and variance 
\begin{small}$\E[P(t)] = \lambda{t} = \Var[P(t)]$\end{small}.  

%
\subsection{Stock Log-Return Process, $\mathbf{ln(S(t))}$}
%
The stock log-return $\ln(S(t))$ can be transformed to a simpler 
jump-diffusion stochastic differential equation (SDE) upon 
% Zhu 8/29: use--> % FH 8/29 OK
the use 
% end 
of the stochastic chain rule \cite{Book04},
%
\begin{small}
\eq
\label{d[ln(S(t))]}
d[\ln(S(t))] = \mu_{ld}dt + \sigma_d dW(t) + QdP(t),%
\nq%
\end{small}% 
%
\noindent{where}  
\begin{small}$\mu_{ld} \equiv \mu_d - 0.5\sigma_d^2$\end{small} 
is the log-diffusive (ld) drift and for simplicity the log-jump-amplitude 
is taken as the mark
\begin{small}$Q = \ln(J(Q)+1)$\end{small}.

%
\subsection{Log-Normal Jump Distribution}
%
Let the density of the jump-amplitude mark $Q$ be normal
%
\begin{small}
\eq
\label{phi_normalQ(q)}
\phi_Q(q) = \phi_n(q;\mu_j, \sigma_j^2),%
\nq%
\end{small}% 
%
\noindent{where} 
\begin{small}$\phi_n(q;\mu_j, \sigma_j^2)$\end{small} 
is the normal density with
mean $\mu_j$ and variance $\sigma_j^2$.  The log-normal jump-amplitude
jump-diffusion model was used in \cite{Merton76}, \cite{ABL02}, 
\cite{ACC02FM} and others.

For the density for this jump-diffusion model with log-normal
jump-amplitude,  Hanson and Westman \cite{ACC02FM} proved,
with corrections given here in terms of the distribution truncated to the second order
approximation in terms of the Poisson distribution,
% 
the following corollary:
\\
\textbf{Corollary:} The second order approximation to 
$[x_{1},x_{2}]$ bin probability distribution for the linear jump-diffusion 
log-return increment $\D[\ln(S(t))]$ with log-normal jump-amplitude
is given by 
%
\begin{small}
\eq
\label{ThNormal_Phijd(x1,x2)} 
\Phi_{\njd}(x_{1},x_{2})\!\!&\!\!\simeq\!\!&\!\!\frac{\sum_{k=0}^{2}p_k(\lambda\Dt)
\Phi_n(x_{1},x_{2};\mu\!+\!k\mu_j,
%\nonumber\\\nonumber\!&\!&\!
\sigma^2\!+\!k\sigma_j^2)}{\sum_{j=0}^{2}p_j(\lambda\Dt)},%
\nonumber\\*[-1ex]\!&\!&\!~%
\nq%
\end{small}% 
%
\noindent{for} \begin{small}$-\infty < x < +\infty$\end{small}, where 
\begin{small}$\Phi_n(x_1, x_2; \mu, \sigma^2)$\end{small} 
is the normal distribution on the 
\begin{small}$[x_1, x_2]$, $\mu \equiv \mu_{ld}\Dt$, $\sigma \equiv \sqrt{\sigma^2_d\Dt}$,
$p_k(\Lambda) =  e^{-\Lambda}\Lambda^k/k!$\end{small}
is the Poisson distribution with parameter 
\begin{small}$\Lambda = \lambda\Dt$\end{small} 
and $k$ jumps and
\begin{small}$\Dt$\end{small} 
is the corresponding trading time increment in years. 

This corollary is based upon the law of total probability \cite[Chapter 0]{Book04}
resulting in the sum over all $k$ Poisson jumps, the convolution 
theorem \cite{Book04} yielding the density of the log-jump-diffusion
conditioned on there being $k$ IID jumps, and the fact that the convolution
of two normals is also normal \cite{Book04}.  
%The corollary
%is posed as the log-return increment rather than for the infinitesimal,
%because the time between trading data is small but not infinitesimal. 
For the purpose of comparison, we use two terms of the expansion 
to provide more accurate estimations since
we are dealing with small but moderately small time steps.   The additional
contribution of the third order approximation is only 1.5\% whereas the
2nd order approximation contributed 23\% to the 1st.   Note that an
appropriate 2nd order renormalization is used to preserve the distribution
property.
%The scale of these time steps can be magnified by a jump rate that includes many 
%small jumps that are indistinguishable from the fluctuations of the 
%diffusion process. 

%
\subsubsection{Basic Moments $M_i^{(\njd)}$ of Log-Return Increments 
$\mathbf{\D[ln(S(t))]}$ for Log-Normal Jumps for $i=1\!\!:\!\!4$} 
%
%\begin{itemize}
%
%\setlength{\itemsep}{-1ex}
%
%\item 1st moment:
\begin{small}  
\equnn
M_1^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\E[\D[\ln(S(t))]]\!=\!(\mu_{ld}\!+\!\lambda\mu_j)\Dt. 
%\nqunn
%
%\item 2nd moment:  
%\equnn
\\
M_2^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\Var[\D[\ln(S(t))]]
\!=\!(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2))\Dt.
%\nqunn 
%
%\item 3rd moment:  
%\equnn
\\
M_3^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\E\left[(\D[\ln(S(t))]\!-\!M_1^{(jd)})^3\right]
\!=\!(3\sigma_j^2\!+\!\mu_j^2)\mu_j\lambda\Dt.
%\nqunn 
%
%\item 4th moment: %%FH6 reformat 
%\equnn
\\
\hspace*{-1em}M_4^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\E\left[(\D[\ln(S(t))]\!-\!M_1^{(jd)})^4\right]
\\\!&\!=\!&\!(\mu_j^4\!+\!3\sigma_j^4\!+\!6\mu_j^2\sigma_j^2)\lambda\Dt 
\!\!+\!3(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2))^2(\Dt)^2.
\nqunn 
\end{small}
~\\*[-3em]
%
%\end{itemize}
%

%
\subsection{Log-Uniform Jump Distribution}
%
Let the density of the jump-amplitude mark $Q$ be uniform  
% 
\begin{small} 
\eq
\label{phi_uniformQ(q)}
\phi_Q(q) = \frac{H(b-q) - H(a-q)}{b - a},%
\nq%
\end{small}%
%
\noindent{where} $a < 0 < b$ and 
\begin{small}$H(x)$\end{small} 
is the Heaviside unit step function.  The mark 
\begin{small}$Q$\end{small} 
has moments,
% Zhu 8/29:\E_Q[Q]-->\E[Q] and etc.  
\begin{small}$\mu_j \equiv \E[Q] = 0.5(b + a)$, 
$\sigma_j^2 \equiv \Var[Q] = (b - a)^2/12$\end{small}.  
% end Zhu: keep the symbols consistent. % FH 8/29 OK
The original jump-amplitude 
\begin{small}$J$\end{small}
has mean 
\begin{small}$\E[J(Q)] = (\exp(b)-\exp(a))/(b - a) - 1$\end{small}.

For the distribution of the jump-diffusion model with log-uniform 
jump-amplitude, the following corollary follows from the density
in~\cite[Chapter 5]{Book04}, 
\\
\textbf{Corollary:} The  second order approximation to 
$[x_{1},x_{2}]$ bin probability distribution for the linear jump-diffusion
% Zhu 8/29: , % end  % FH 8/29 OK
log-return increment $\D[\ln(S(t))]$ with log-uniform jump-amplitude
is given by 
%
\begin{small}
\eq
\label{ThUniform_Phijd(x1,x2)}
\Phi_{\ujd}(x_{1},x_{2})\!&\!\simeq 
\frac{\sum_{k=0}^{2}p_k(\lambda\Dt)\Phi_{\ujd}^{(k)}(x_{1},x_{2})}{\sum_{j=0}^{2}
p_j(\lambda\Dt)}, 
\nq
\end{small}%
%
\noindent{for}  
\begin{small}$-\infty < x < +\infty$\end{small}, 
where 
\begin{small}$\Phi_{\ujd}^{(0)}(x_1, x_2) \equiv \Phi_n(x_1, x_2; \mu, \sigma^2)$\end{small},
%
\begin{small}
% Zhu 8/29: *, \cdot are deleted. Keep the formulae more consistent. % % FH 8/29 OK
\equnn
\Phi_{\ujd}^{(1)}(x_1, x_2)  
\!&\!\!=\!&\!\frac{1}{b\!-\!a}\biggl( (x_2\!-\!x_1)\Phi(x_2\!-\!b,x_2\!-\!a;\mu,\sigma^{2})
\\\!&\!&\!-(x_1\!-\!b\!-\!\mu)\Phi_n(x_1\!-\!b,x_2\!-\!b;\mu,\sigma^2) 
\\\!&\!&\!+(x_1\!-\!a\!-\!\mu)\Phi_n(x_1\!-\!a,x_2\!-\!a;\mu,\sigma^2) 
\\\!&\!&\!\left.\!+\frac{\sigma}{\sqrt{2\pi}}\!
\left(e^{-\frac{x_{1b}^2}{2}}\!-\!e^{-\frac{x_{2b}^2}{2}}
\!-\!e^{-\frac{x_{1a}^2}{2}}\!+\!e^{-\frac{x_{2a}^2}{2}}\right)\right) ,%
\nqunn%
% end Zhu
\end{small}%
%
\noindent{where}  
\begin{small}$x_{1a} = (x_1\!-\!a\!-\!\mu)/\sigma$, $x_{2a} = (x_2\!-\!a\!-\!\mu)/\sigma$, 
$x_{1b} = (x_1\!-\!b\!-\!\mu)/\sigma$, $x_{2b} = (x_2\!-\!b\!-\!\mu)/\sigma$\end{small},
%
\begin{small}
\equnn
\Phi_{\ujd}^{(2)} (x_1,x_2)\!\!&\!\!=\!\!&\!\!\frac{0.5\sigma^2}{(b\!-\!a)^2}\biggl( 
(x_{2A}^2\!-\!x_{1A}^2)\Phi_n(x_2\!-\!C,x_2\!-\!A;\mu,\sigma^2) 
\\\!&\!&\!+(x_{1B} ^2\!-\!x_{2B} ^2)\Phi_n(x_2\!-\!B,x_2\!-\!C;\mu,\sigma^2)
\\\!&\!&\!+(x_{1A}^2\!+\!1) \Phi_n(x_1\!-\!A,x_2\!-\!A;\mu,\sigma^2)
\\\!&\!&\!+(x_{1B} ^2\!+\!1) \Phi_n(x_1\!-\!B,x_2\!-\!B;\mu,\sigma^2) % Zhu 8/21% bad v
\\\!&\!&\!-2(x_{1C} ^2\!+\!1) \Phi_n(x_1\!-\!C,x_2\!-\!C;\mu,\sigma^2)
\\\!&\!&\!\left.\!+\frac{1}{\sqrt{2\pi}}(2X_{1C,2C}\!-\!X_{1A,2A}\!-\!X_{1B,2B})
 \right),%
\nqunn%
\end{small}%
%
\noindent{where}
\begin{small} 
$A = 2a$, $C = a\!+\!b$, $B =2b$,
$x_{1A} = (x_1\!-\!A\!-\!\mu)/ \sigma$, $x_{2A} = (x_2\!-\!A\!-\!\mu)/ \sigma$, 
$x_{1C}  = (x_1\!-\!C\!-\!\mu)/ \sigma$, $x_{2C}  = (x_2\!-\!C\!-\!\mu)/ \sigma$,
$x_{1B}  = (x_1\!-\!B\!-\!\mu)/ \sigma$, $x_{2B}  = (x_2\!-\!B\!-\!\mu)/ \sigma$,
$X_{1A,2A} = x_{1A}e^{-x_{1A}^2/2}-x_{2A}e^{-x_{2A}^2/2}$, 
$X_{1B,2B} = x_{1B}e^{-x_{1B} ^2/2}-x_{2B}e^{-x_{2B} ^2/2}$, 
$X_{1C,2C} = x_{1C}e^{-x_{1C} ^2/2}-x_{2C}e^{-x_{2C} ^2/2}$.
\end{small}

{For} the distribution (\ref{ThUniform_Phijd(x1,x2)}), the first and second
order term formulas, 
\begin{small}$\Phi_{\ujd}^{(i)}(x_1, x_2)$\end{small} 
for $i\!=\!1\!\!:\!\!2$,
have been reduced by integration by parts to single normal distribution integrals to minimize the computational costs, making it comparable
% Zhu 8/21 that  %  
for the normal jump case in (\ref{ThNormal_Phijd(x1,x2)}).   
Also, the terms are arranged to minimize the effects of catastrophic cancellation.

\subsubsection{Fourth Central Moment of Log-Return Increments
$\mathbf{\D[ln(S(t))]}$ for Log-Uniform Jumps}
%
%\begin{itemize}
%
%\setlength{\itemsep}{-1ex}
%
%\item 1st moment:  
%\equnn
%M_1^{(\ujd)} 
%= (\mu_{ld} + \lambda\mu_j)\Dt.
%\nqunn
%
%\item 2nd moment:  
%\equnn
%M_2^{(\ujd)}
%\!&\!=\!&\! (\sigma_d^2 + 
%\lambda (\sigma_j^2 + \mu_j^2))\Dt.
%\nqunn 
%
%\item 3rd moment:  
%\equnn
%M_3^{(\ujd)}
%\!&\!=\!&\!
%(3\sigma_j^2+\mu_j^2)\mu_j\lambda\Dt.
%\nqunn 
%
%\item 4th moment:  
\begin{small}
\equnn
\hspace*{-1em}M_4^{(\ujd)}
\!\!&\!\!=\!\!&\!\!\left(\mu_j^4\!+\!1.8\sigma_j^4\!+\!6\mu_j^2\sigma_j^2\right)\lambda\Dt
\\\!&\!&\!\!+\!3\left(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2)\right)^2(\Dt)^2.%
\nqunn% 
\end{small}%
%
%\end{itemize}
%
\noindent{Note} that the formulas for the first three moments are the same 
for both log-normal and log-uniform jumps.

%
\subsection{Log-Double-Exponential Jump Distribution}
%
Let the density of the jump-amplitude mark $Q$ be double-exponential  
%
\begin{small}
\eq
\label{phi_dblexp_Q(q)}
\phi_Q(q) = \frac{p_1}{\mu_1}\exp\left(\frac{\dps q}{\dps \mu_1}\right)I_{\{q<0\}}
\!+\!\frac{p_2}{\mu_2}\exp\left(\frac{\dps -q}{\dps \mu_2}\right)I_{\{q\geq 0\}},%
\nq%
\end{small}%
%
\noindent{where} $\mu_1>0$ and $\mu_2>0$ are one-sided means, and $0<p_1<1$
represents the probability of downward jumps while $p_2 = 1-p_1$ is the probability
of upward jumps.  The set indicator function is $I_{\{\mathcal{S}\}}$
for set $\mathcal{S}$.
The mark $Q$ has moments,
%
\begin{small}
\equnn
\mu_j\!\!&\!\!=\!\!&\!\!\E_Q[Q]\!=\!-p_1\mu_1\!+\!p_2\mu_2 ,
\\
\sigma_j^2\!\!&\!\!=\!\!&\!\!\Var_Q[Q] 
%= p(2-p)\mu_1^2 \!+\! 2pp_2\mu_1\mu_2 \!+\!(1-p^2)\mu_2^2
%=-\mu_j^{2}\!+\!2(p\mu_1^{2}\!+\!p_2\mu_2^2)
\!=\!p_1\left((\mu_j\!+\!\mu_1)^2\!+\!\mu_1^2\right)
%\\\!&\!&\!
\!+\!p_2\left((\mu_j\!-\!\mu_2)^2\!+\!\mu_2^2\right).%
\nqunn %
\end{small}%
%
\noindent{Similar} to Corollary \ref{ThUniform_Phijd(x1,x2)}, we get the following corollary: 
\\  
\textbf{Corollary:}  The  second order approximation to 
$[x_{1},x_{2}]$ bin probability distribution for the linear jump-diffusion, 
log-return increment $\D[\ln(S(t))]$ with log-double-exponential jump-amplitude
is given by 
%
\begin{small}
\eq
\label{ThDblExp_Phijd(x1,x2)}
\Phi_{\dejd}(x_{1},x_{2})\!&\!\simeq 
\frac{\sum_{k=0}^{2}p_k(\lambda\Dt)\Phi_{\dejd}^{(k)}(x_{1},x_{2})}{\sum_{j=0}^{2}
p_j(\lambda\Dt)},%
\nq%
\end{small}%
%
\noindent{for} \begin{small}$-\infty < x < +\infty$\end{small}, where
\begin{small}
\equnn
\Phi_{\dejd}^{(0)}(x_1, x_2)\!&\!\equiv\!&\!\Phi_n(x_1, x_2; \mu, \sigma^2),
\\
\Phi_{\dejd}^{(1)}(x_1,x_2)
\!&\!=\!&\!\Phi_{n}(x_1,x_2;\mu,\sigma^2) 
\!+\!p_1(\rho_{x_2,\nu_1}-\rho_{x_1,\nu_1}) 
\\
\!&\!&\!+ p_2(\rho_{x_1,\nu_2}-\rho_{x_2,\nu_2}),% 
\nqunn%
%\end{small}%
%
%\noindent{where} 
%\begin{small}$\mu = \mu_{ld}\Dt$, $\sigma = \sqrt{\sigma_d^2\Dt}$\end{small}, 
%
%\begin{small}
\equnn 
\nu_1 &=& \mu-0.5\sigma^2/\mu_1, ~~ \nu_2 = \mu+0.5\sigma^2/\mu_2,  
\\
\rho_{x_2,\nu_1} &=&e^{+(x_2-\nu_1)/\mu_1}\Phi_{n}(-x_2;-\mu+\sigma^2/\mu_1,\sigma^2),
\\
\rho_{x_1,\nu_1} &=& e^{+(x_1-\nu_1)/\mu_1}\Phi_{n}(-x_1;-\mu+\sigma^2/\mu_1,\sigma^2),
\\
\rho_{x_1,\nu_2} &=&e^{-(x_1-\nu_2)/\mu_2}\Phi_{n}(x_1;\mu+\sigma^2/\mu_2,\sigma^2),
\\
\rho_{x_2,\nu_2} &=& e^{-(x_2-\nu_2)/\mu_2}\Phi_{n}(x_2;\mu+\sigma^2/\mu_2,\sigma^2),
\nqunn
%\end{small}
%
%\noindent{and}
%
%\begin{small}
\equnn
\Phi_{\dejd}^{(2)}(x_1,x_2)
\!\!&\!\!=\!\!&\!\!\Phi_{n}(x_1,x_2;\mu,\sigma^2) 
\\
\!&\!&\!+\mu_1\left(\left(p_{12}%
\!+\!p_{11}\left(\mu\!-\!\frac{\sigma^2}{\mu_1}\!+\!\mu_1\!-\!x_2\right)\right)\!%
\rho_{x_2,\nu_1}\right. 
\\
\!&\!&\!\left.\!-\!\left(p_{12}%
\!+\!p_{11}\left(\mu\!-\!\frac{\sigma^2}{\mu_1}\!+\!\mu_1\!-\!x_1\right)\right)\!%
\rho_{x_1,\nu_1}\right) 
\\
\!&\!&\!+\mu_2\left(\left(p_{12}%
\!-\!p_{22}\left(\mu\!+\!\frac{\sigma^2}{\mu_2}\!-\!\mu_2\!-\!x_1\right)\right)\!%
\rho_{x_1,\nu_2}\right. 
\\
\!&\!&\!\left.\!-\!\left(p_{12}
\!-\!p_{22}\left(\mu\!+\!\frac{\sigma^2}{\mu_2}\!-\!\mu_2\!-\!x_2\right)\right)\!%
\rho_{x_2,\nu_2}\right)
\\
\!&\!&\!+\frac{\sigma}{\sqrt{2\pi}}(\mu_2p_{22}-\mu_1p_{11})
\left(e^{-z_1^2/2}- e^{-z_2^2/2}\right),% 
\nqunn%
%end{small}%
%
%\noindent{where}
%\begin{small}
$p_{11} = (p_1/\mu_1)^2$, $ p_{22} = (p_2/\mu_2)^2$, 
$ p_{12} = 2p_1p_2/(\mu_1\!+\!\mu_2)$, 
\\
$z_1=(x_1\!-\!\mu)/\sigma$, $ z_2=(x_2\!-\!\mu)/\sigma$\end{small}.\\*[-0.5ex] 
%for $-\infty < x < +\infty$,
%and where $\mathbf{\Dt}$ is the corresponding trading time increment. 

%
\subsubsection{Third and Fourth Moments of Log-Return Increments
$\mathbf{\D[ln(S(t))]}$ for Log-Double-Exponential Jumps} %Yan
%
%\begin{itemize}
%
%\setlength{\itemsep}{-1ex}
%
%\item 1st moment:  
%\equnn
%M_1^{(\dejd)}
%%FH6\!&\!\equiv\!&\!\E[\D[\ln(S(t))]]\\
%\!&\!=\!&\! (\mu_{ld} + \lambda\mu_j)\Dt; 
%\nqunn
%
%\item 2nd moment:  
%\equnn
%M_2^{(\dejd)}
%%FH6\!&\!\equiv\!&\!\Var[\D[\ln(S(t))]]\\
%\!&\!=\!&\! (\sigma_d^2 + 
%\lambda (\sigma_j^2 + \mu_j^2))\Dt;
%\nqunn 
%
%\item 3rd moment: 
%FH% left/right 
\begin{small}
\equnn
M_3^{(\dejd)}
\!\!&\!\!=\!\!&\!\!6(p_2\mu_2^3\!-\!p_1\mu_1^3)\lambda\Dt;
%\nqunn 
%
%\item 4th moment:  
%\equnn
\\
M_4^{(\dejd)}\!\!&\!\!=\!\!&\!\!24(p_2\mu_2^4\!+\!p_1\mu_1^4)\lambda\Dt
%\\\!\!&\!\!&\!\!
\!+\!3(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2))^2(\Dt)^2.%
\nqunn% 
\end{small}%
%
%\end{itemize}
%
\noindent{The} first and second moments are the same for all three models.

%
\subsection{Skewness and Kurtosis}
%
Negative skewness and leptokurtosis are considered to be general properties
of financial market distributions.  
% FH 8/22:
% The negative skewness and leptokurtosis are considered as benchmarks
% to compare the three jump-diffusion models.  
Therefore,
% Zhu 8/29: it is % end % FH 8/29 Good
$M_3^{(\jd)}$ and $M_4^{(\jd)}$ are needed to get the
theoretical skewness and kurtosis coefficient for these three
models to sufficient accuracy for a satisfactory comparison.
%
\begin{itemize}
%
\setlength{\itemsep}{0.25ex}
%
\item Skewness coefficient: 
\begin{small}$\beta_3^{(\jd)}
\equiv M_3^{(\jd)}/\left(M_2^{(\jd)}\right)^{1.5}$.\end{small}
\item Kurtosis coefficient: 
\begin{small}$\beta_4^{(\jd)} 
\equiv M_4^{(\jd)}/\left(M_2^{(\jd)}\right)^2$.\end{small} 
%
\end{itemize}
Sometimes, the kurtosis is represented as the excess kurtosis coefficient
by subtracting three from the kurtosis coefficient so
that the excess is zero for the normal distribution.

%
\section{Parameter Estimations} % Zhu 8/29: 'Jump-Diffusion' deleted % % FH 8/29 OK
%
The basic point of view, here, is that the financial markets are 
considered to be a moderate size simulation of one of these three 
jump-diffusion processes. 
%
\subsection{Empirical Data}  
%
We use Standard and Poor's 500 (S\&P500) stock index in the decade
1992-2001 \cite{SP500} as the sample of the financial market since it  
is in general viewed as one big mutual fund so that it is less dependent on 
the peculiar behavior of any one stock.   
%
Let \begin{small}$n^{(\sp)} = 2522$\end{small} 
be the number of daily closings \begin{small}$S^{(\sp)}_s$\end{small}
for \begin{small}$s = 1\!\!:\!\!n^{(\sp)}$\end{small}, 
such that there are \begin{small}$ns = 2521$\end{small} 
log-returns,
%
\begin{small}
\eq
\label{LogReturns}
\D\left[\ln\left(S^{(\sp)}_s\right)\right] 
\equiv  \ln\left(S^{(\sp)}_{s+1}\right) 
- \ln\left(S^{(\sp)}_s\right)\,%
\nq%
\end{small}%
%
%\noindent{for} $s = 1\!\!:\!\!ns$ log-returns, 
\noindent{with} empirical average values:
%
\begin{itemize}
%
\setlength{\itemsep}{-0.1ex}
%
\item Mean:
\begin{small}  
%\equnn
$M_1^{(\sp)}%
%= \frac{\sum_{s=1}^{ns}\D\left[\ln\left(S^{(\sp)}_s\right)\right]}{ns}
\!\simeq\!4.015\mbox{e-4}$.
%\nqunn
\end{small}
%
\item Variance:  
%
\begin{small}
%\equnn
$M_2^{(\sp)}% 
%\!&\!=\!&\!\frac{\sum_{s=1}^{ns}
%\left(\D\left[\ln\left(S^{(\sp)}_s\right)\right]
%\!\!-\!\!M_1^{(\sp)}\right)^2}{ns\!\!-\!\!1}
%\\
%\!&\!\simeq\!&\!
\!\simeq\!9.874\mbox{e-5}$.  
%
%\nqunn
\end{small}
%
\item Skewness coefficient: 
%
\begin{small}
%\equnn
$\beta_3^{(\sp)}%
%\equiv M_3^{(\sp)}/\left(M_2^{(\sp)}\right)^{1.5}
\!\simeq\!-0.2913 < 0$,%
%\nqunn%
\end{small}%
%
where $\beta_3^{(n)} = 0$ is the normal distribution
value and $M_3^{(\sp)}$ is the 3rd central log-return moment of the data.
\\*[-0.5em]
%
\item Kurtosis coefficient:  
%
\begin{small}
%\equnn
$\beta_4^{(\sp)}% 
%\equiv M_4^{(\sp)}/\left(M_2^{(\sp)}\right)^2 
\!\simeq\!7.804 > 3$,%
%\nqunn%
\end{small}%
%
where $\beta_4^{(n)} = 3$ is the normal distribution
value and $M_4^{(\sp)}$ is the 4th central log-return moment of the data.
% 
\end{itemize}
%

%
\subsection{Multinomial Maximum Likelihood Estimation} 
%
In a previous paper \cite{CONMATH04}, the multinomial maximum
likelihood estimation of model parameters is justified for binned 
financial data, but applies to very general binned data.  The main 
idea for this method is the following:
%
\begin{itemize} 
%
\item Step 1: 
% Zhu 8/29: Sample Data is sorted into $nb$ bins --->% FH 8/29 OK
Sort the sample data into $nb$ bins
% end Zhu %
and get the sample 
frequency $f_b^{(sp)}$, for $b=1\!\!:\!\!nb$. 
%
\item Step 2: Get the theoretical jump-diffusion frequency with parameter 
vector $\xv$:
\begin{small}
$$
f_b^{(\jd)}(\xv) \equiv  ns \int_{B_b} \phi^{(\jd)}(\eta ;\xv) d\eta\:,
% Zhu 8/29: \phi^{(\jd)}(\eta ;\xv) no definition, need some changes here??!! %
$$%
\end{small}%
%
\noindent{where} $\phi^{(\jd)}(\eta ;\xv)$ is some jump-diffusion density in $\eta$ and  $B_b$ is the $b$th bin.
% FH 8/29: \noindent{where} and  $B_b$ is the $b$th bin.
%
\item Step 3: Minimize the objective function: 
%
\begin{small}
\eq
\label{EstObjective}
y(\xv) \equiv -\sum_{b=1}^{nb}\left[f_b^{(\sp)}\ln\left(f_b^{(\jd)}(\xv)
\right)\right]\:,%
\nq%
\end{small}%
%
\noindent{where} the negative of the likelihood is minimized, %FH 8/29 Killed another rephrase
corresponding to the minimizing 
% Zhu 8/29: \texttt{fminsearch} function ---->% FH 8/29 OK, but the rest is a bad long sentence.
MATLAB %FH 8/29 
function \texttt{fminsearch} 
% end Zhu % % FH 8/29:
to get the optimal parameters
$\xv^*$ for the three compared models, respectively.
This MATLAB function is an implementation of the Nelder-Mead down-hill
% Zhu 8/29: simplex direct search method in MATLAB.
%The Nelder-Mead method \cite{NelderMead} is used -->
simplex direct search method~\cite{NelderMead}. % in MATLAB
% end Zhu% %FH 8/29
% to get the optimal parameters
%$\xv^*$ for the three compared models, respectively.  
The Nelder-Mead
is usually faster than other optimization methods when it works.  
Some comparisons with our multidimensional golden section search method 
for the financial parameter estimation problem are given in 
\cite{CONMATH04}.
%
\end{itemize} 

%
\subsection{Jump-Diffusion Moment Estimation Constraints}  
%
For the jump-diffusion model with log-normal and log-uniform jump-amplitude,
there are five (5) free jump-diffusion parameters: 
%
$\{\mu_{ld}, \sigma_d^2, \mu_j, \sigma_j^2, \lambda\}$.
%
For the model with log-double-exponential 
jump-amplitude, there are six (6) free jump-diffusion parameters: 
%
$\{\mu_{ld}, \sigma_d^2, \mu_1, \mu_2, p_1, \lambda\}$.
% 
So, to reduce this set to a reasonable number, the
multinomial maximum likelihood estimation is subjected to the 
mean and variance constraints:
%
\begin{small}
\eq
\label{MeanCons}
M_1^{(\sp)} = M_1^{(\jd)}
\AND 
\label{VarCons}
M_2^{(\sp)} = M_2^{(\jd)}\:.% 
\nq%
\end{small}%
%
\noindent{Two} diffusion parameters, $\mu_{ld}$ and $\sigma_d$, are eliminated by
%
\begin{small}
\eq
\label{MeanElim}
\hspace*{-3em}\mu_{ld}\!&\!=\!&\!\left(M_1^{(sp)} - \mu_j\lambda\Dt\right)/\Dt,
\\
\label{VarElim}
\hspace*{-3em}\sigma_d^2\!&\!=\!&\!\max\left[\left(M_2^{(sp)} 
- \left(\sigma_j^2 + \mu_j^2\right)\lambda\Dt\right)/\Dt,\varepsilon\right],%
\nq%
\end{small}%
% 
\noindent{subject} to positivity constraints with sufficiently small  $\varepsilon > 0$.   Although 
$\sigma_d^2$ normally should be positive, but not necessarily for the  first argument of the
$\max$ in (\ref{VarElim}).
%
{F}or the log-normal and log-uniform jump-diffusion model, only three free parameters are left: 
%
\begin{small}$\xv=\{\mu_j, \sigma_j^2, \lambda\}$\end{small}.  
%
{F}or the log-double-exponential jump-diffusion model, four free parameters are left: 
%
\begin{small}$\xv=\{\mu_1, \mu_2, p_1, \lambda\}$\end{small}%,
%
with significantly more computational estimation costs and it is subject to   
an exponential form of catastrophic cancellation unless the one-sided exponentials are
appropriately collected.

\section{Numerical Results, Figures and Discussion} 
%
We use the MATLAB 7.0 \cite{MATLAB} 
% Zhu 8/29: But the reference given is about version 6 (2000)??% end Zhu %
% FH 8/29: I knew that but the version 7 manual was not distributed yet! 
to program our codes. 
The multinomial maximum likelihood estimation given here is used to 
estimate the jump-diffusion parameters.  The numerical optimization
was performed using the  \texttt{fminsearch} function \cite{MATLAB}. 
% Zhu 8/29: , an implementation of the down-hill
%simplex direct search method of Nelder and Mead \cite{NelderMead}
%Zhu: In Section B, we already gave an explanation. % end  FH 8/29 Good
For the normal distribution integrals 
\begin{small}$\Phi_n(x_1,x_2;\mu,\sigma^2)$\end{small}
or  
\begin{small}$\Phi_n(x_1;\mu,\sigma^2)$\end{small}, 
the use of  
%Yan the
the fast and reliable MATLAB complementary error function  \texttt{erfc}
was critical, since more standard integration functions for unscaled arguments
failed small variance $\sigma^2$ tests due to poor detection of the main
probability mass.
For the log-normal and log-uniform model, the same
starting point $\xv_0$ is used. For the log-double-exponential
model, the different starting point $\xv_0$ is used: initial $\mu_1$ and
$\mu_2$ are from an initial estimation of the $\mu_j$ 
and using $p_1 \simeq 0.6 > 0.5$ and the $\lambda\Dt$ value are the same as the 
log-normal or log-uniform.

The empirical data used in the estimation are the S\&P500 daily 
closing log-returns from the decade 1992-2001.   
This data is displayed here in Figure~\ref{FigHist92-01} with 100 equally spaced bins.  
The ragged appearance of the
histogram resembles the random simulation of a density using a moderate, but
inadequate sample size.  
The crashes are represented by the extreme negative tails and the rallies by the 
extreme positive tails, but these extreme events are difficult to see since they
are rare events with small frequency counts, i.e., the extreme tails are sparsely
populated, with the extreme negative events more widely separated than the 
extreme positive events.
Note that extremeness is measured by the values of the log-returns on the horizontal
axes.
%
\begin{figure}[!hbtp]
\centering
{\includegraphics[width=3.25in]{cdc04normal17sphistbw.eps}}
% {\includegraphics[width=3.25in]{cdc04normal16f32sphistbw.eps}}
\caption{
Histogram of S\&P500 log-return frequencies for the decade
1992-2001, using 100 centered evenly spaced bins.
}
\label{FigHist92-01}
\end{figure}
%

However, if the histogram frequencies are multiplied by the centered
value of the evenly spaced bin log-returns, then the extreme jumps are clearly visible.  
This moment-histogram is called a \textit{hysteriagram} \cite{MTNS02FMT} since it 
magnifies the  count of the larger jumps and corresponds to the extreme reaction of
investors.
The hysteriagram for the S\&P500 is given in Figure~\ref{FigXHist92-01} 
using the same data as in the previous figure, using the same 100 evenly spaced bins. 
%
\begin{figure}[!hbtp]
\centering
{\includegraphics[width=3.25in]{cdc04normal17spxhistbw.eps}}
% {\includegraphics[width=3.25in]{cdc04normal16f32spxhistbw.eps}}
\caption{
Hysteriagram of S\&P500 log-return frequencies multiplied by the
centered bin log-return values for the decade 1992-2001, using 100  
evenly spaced bins.
}
\label{FigXHist92-01}
\end{figure}
%

Figure~\ref{FigNormalXHist} shows that the log-normal jump-amplitude fitted
model hysteriagram exhibits too thin tails that decay too fast with
the jump magnitude.  
From (\ref{ThNormal_Phijd(x1,x2)}) 
the bin distribution will be a Poisson sum 
of normal distributions, so will have the thin  Gaussian exponential tails.  
% The corresponding histogram for the log-normal,
% not shown here, does not show enough visual detail to sufficiently 
% distinguish it from the other jump-amplitude models.
%
\begin{figure}[!hbtp]
\centering
{\includegraphics[width=3.25in]{cdc04normal17jdxhistbw.eps}}
% {\includegraphics[width=3.25in]{cdc04normal16f32jdxhistbw.eps}}
\caption{
Hysteriagram of the predicted log-returns frequencies multiplied by the
centered bin log-return values for the log-normal jump-amplitude 
jump-diffusion model, 
% Zhu 8/29: using 100 evenly spaced bins using estimated parameters.--->
using 100 evenly spaced bins and the estimated parameters.
% end Zhu % FH 8/29 Good
}
\label{FigNormalXHist}
\end{figure}
%

Figure~\ref{FigUniformXHist} shows that the log-uniform jump-amplitude fitted
model hysteriagram exhibits thicker tails that decay more slowly with
the jump magnitude, especially in the shoulders of the hysteriagram.  
The convolution of the diffusion with multiple jump uniform distributions in (\ref{ThUniform_Phijd(x1,x2)}) help counter the normal distribution
tendency to having exponential thin tails, but not for very large values
of the log-returns.  The positive tails are thicker since the extreme
positive jumps are more closely spaced than the negative jumps.   
The uniform model mainly compensates for the non-normal data with significant
lumps in the shoulders of the jump-diffusion distribution.
%
\begin{figure}[!hbtp]
\centering
{\includegraphics[width=3.25in]{cdc04uniform17b100m0jdxhistbw.eps}}
% {\includegraphics[width=3.25in]{cdc04uniform16f32jdxhistbw.eps}}
\caption{
Hysteriagram of the predicted log-returns frequencies multiplied by the
centered bin log-return values for the log-uniform jump-amplitude
jump-diffusion model, 
% Zhu 8/29: using 100 evenly spaced bins using estimated parameters.--->
using 100 evenly spaced bins and the estimated parameters.
% end Zhu % FH 8/29 Good*2
}
\label{FigUniformXHist}
\end{figure}
%

Figure~\ref{FigDblExpXHist} shows that the log-double-exponential jump-amplitude fitted
model hysteriagram exhibits too thin tails that decay too fast with
the jump magnitude that is very similar to the log-normal jump-amplitude 
model.  
The convolution of normal and exponential distributions in
(\ref{ThDblExp_Phijd(x1,x2)}), like the normal jump-amplitude model, can only
lead to exponential thin tails. 
%
\begin{figure}[!hbtp]
\centering
{\includegraphics[width=3.25in]{cdc04dblexp17b100m0jdxhistbw.eps}}
% {\includegraphics[width=3.25in]{cdc04dblexp16f32jdxhistbw.eps}}
\caption{
Hysteriagram of the predicted log-returns frequencies multiplied by the
centered bin log-return values for the log-double-exponential jump-amplitude
jump-diffusion model, 
% Zhu 8/29: using 100 evenly spaced bins using estimated parameters.--->
using 100 evenly spaced bins and the estimated parameters.
% end Zhu % FH 8/29 Good*3
}
\label{FigDblExpXHist}
\end{figure}
%
% FH 8/24:
% The log-uniform model is a qualitatively, but not necessarily quantitatively, 
% better model for the S\&P500 data, since uniform jump model has thicker tails.

From Table~\ref{TAB1Moments1}, we can have a quantitative estimate of
the derived distribution parameters 
$\{\mu_d$,  $\sigma_d$, $\mu_j$, $\sigma_j$, $\lambda$\}.  
Since the trading days per year are about
250 days, it is not likely that the jumps rate is more than 100
per year because the finance market should be kept stable, so,
$\lambda \simeq 64$ for the log-uniform is more reasonable.  
The near-zero peaks of the normal and double-exponential
lead to jump rates that are double or more than the uniform jump rate. 
Note that the jump rate includes all size jumps, including those
hidden under the log-normal diffusion.
In the table the overall jump mean $\mu_j$ is given for the purpose of
comparison, but for the double-exponential, the negative jump mean is
$-\mu_1 =$ -6.88e-3 and the positive jump mean is $\mu_2 =$ +6.35e-3.  
For the double-exponential, the probability of negative jumps is
$p_1=0.504$ and that for positive jumps is $p_2=0.496$.
The overall jump-diffusion parameters 
$\{\mu_d, \sigma_d, \mu_j, \sigma_j, \lambda\}$  
have somewhat different distributions among the three jump models, with the
diffusive means and volatilities being the closest among the parameters.   
The uniform distribution gives  more weight to the negative jumps with
the largest $-\mu_j =$ 12.18e-4 and largest $\sigma_j \simeq$ 1.52e-2.
%
\begin{table}[!hbtp]
\centering
\caption{
Comparison summary of derived distribution parameters for the log-normal,  
log-uniform and log-double-exponential jump-diffusion models, respectively. 
}
\label{TAB1Moments1}
\begin{tabular}{|c||c|c|c|c|c|}\hline
Model & $\mu_d$ & $\sigma_d$ & $\mu_j$   & $\sigma_j$ & $\lambda$\\\hline\hline
Normal  & 0.191    & 0.088           & -7.09e-4    & 1.19e-2       & 121.           \\\hline
Uniform & 0.184   & 0.100           & -12.18e-4   & 1.52e-2       & 64.0           \\\hline
Dbl-Exp & 0.170   & 0.085           & -3.21e-4     & 0.94e-2        & 202.           \\\hline
\end{tabular}
\end{table}
%

Table~\ref{TAB2Moments2} shows the differences of the variance-normalized 
higher moments of skewness  $\beta_3$ and kurtosis $\beta_4$
between the estimate value and the observed values.  
% Zhu 8/29: The skewness  difference
%is lowest for the log-uniform jump model, while the  kurtosis difference
%is lowest for the log-double-exponential jump model.  ---> FH 8/29 OK
The absolute skewness  difference
is the lowest for the log-uniform jump model, while the  absolute kurtosis difference
is the lowest for the log-double-exponential jump model.  

On the other hand, the skewness difference is the worst for the log-normal jump model
and the kurtosis difference is the worst for the log-uniform jump model, 
% FH 8/24
but the log-normal is only 5\% lower than the log-uniform.
% The skewness  and kurtosis differences
% are lowest for the log-double-exponential jump model.  
% The skewness difference is almost the same for
% the log-uniform as for the log-double-exponential jump model. 
% On the other hand, the skewness difference is the worst for the log-normal jump model
% and the kurtosis difference is the worst for the log-uniform jump model.
Of course, the numerical calculations of the third and fourth moments are
of doubtful computational reliability for data.
The final multinomial maximum likelihood values using the negative of minimum 
values are essentially the same for all models, since the same stopping criterion
was used. 
%
\begin{table}[hbtp]
\centering
\caption{
The skewness and kurtosis coefficients for the three models are
compared to S\&P500 values, respectively, and Multinomial Maximum Likelihood 
(MML $\simeq -\min[y(\mathbf{x})])$.
}
% FH 8/24:%
\label{TAB2Moments2}
\begin{tabular}{|c||c|c|c|c|c|}\hline
Model    & $\beta_3$ &  $\%$diff. & $\beta_4$ & $\%$diff.  & MML  \\\hline\hline
Normal  & -0.147      & -49.5          & 5.98           & -23.4          & 1107.   \\\hline 
Uniform & -0.219     & -24.7          & 5.57           & -28.7           & 1106.   \\\hline
Dbl-Exp & -0.183     & -37.6          & 6.80           & -12.8           & 1108.  \\\hline\hline
S\&P500 & -0.291     &   0.0           & 7.80           &   0.0            & ---         \\\hline
\end{tabular}
\end{table}
%

Table~\ref{TAB3Perf} shows that 
% Zhu 8/21 % 
%the log-normal and log-uniform
%models take somewhat more iterations and function evaluations, 
the log-normal model takes  somewhat more iterations and function evaluations 
than the log-uniform model does, 
%end 8/21 % 
but the log-normal model parameter estimate takes about 60\% of the time to execute. 
One reason is that the log-normal requires only similar normal distribution 
calculations for each 
% Zhu 8/29:jump-->
jump count % size %FH 8/29 size usually means amplitude
%end Zhu % 
$k$ in (\ref{ThNormal_Phijd(x1,x2)}), while the others
require more complex combinations of  normal distributions, powers and exponentials.
The extra parameter needed for the double-exponential means the
iteration count, the function evaluation count and the timings are much greater.
However, the computational effort for both the uniform and double-exponential
models have been greatly reduced by using integration by parts and more to get single integrals.    
% An added advantage of such a reduction also can improve accuracy and speed of computation. 
If  the parameter estimation is done off-line for an application instead of on-line
% Zhu 8/29: than---> FH 8/29 Good
then 
% 
the computational saving would not be too much of an advantage.
%
\begin{table}[hbtp]
\centering
\caption{
Comparison summary of computational performance measures:
}
% FH 8/24:
\label{TAB3Perf}
\begin{tabular}{|c||c|c|c|c|}\hline
Jump     & Number & Number   & Function & Timings \\
Model     & Parms. & Iters.  & Evals.   & (sec)   \\\hline\hline
Normal  & 3          & 131      & 238       & 4.7    \\\hline
Uniform & 3          & 71       & 128       & 7.8    \\\hline
Dbl-Exp & 4          & 205     & 343       & 21.   \\\hline
\end{tabular}
%
\\*[0.5em]
\begin{small}
Combined Legend for Table~\ref{TAB1Moments1}, Table~\ref{TAB2Moments2} and 
Table~\ref{TAB3Perf}:
%
\begin{itemize}
%
%\item Normal:  Log-normal jump-amplitude. 
%
%\item Uniform:  Log-uniform jump-amplitude.
%
%\item Dbl-Exp:  Log-double-exponential jump-amplitude.
% GY 8/31 Change:
\item Maximum Number of Iterations: 400.%$^*$.
% GY 8/31 Changes:
\item Using same fminsearch tolerances: tolx $=$ 5e-7 \\ and toly $=$ 5e-7. 
%
\item Using a dual G5@2GHz CPU computer processor with MATLAB 7.0. 
%
\end{itemize}
\end{small}
\end{table} 
%
  
%
\section{Summary and Conclusions}
%
From the above theoretical and data analysis, we can get the
following conclusions:
%
\begin{itemize}
%
\item The log-uniform model is the qualitatively best overall among the three models,
in terms of genuinely representing the fat tail property, better approximation to
the empirical skewness and more reasonable jump rates
of real-world market distributions.  
%
\item The log-normal model runs faster than the other two models due
to simpler normal bin integrals. 
On the other hand, the integration by parts technique has been used to reduce the
computational effort for the log-uniform and log-double-exponential models.
However, the deficiencies of the log-normal model demonstrates that
the distribution that is better analytically is not necessarily a better model 
for financial markets.
%  , i.e., finding a better model may be counter to the desire
% to obtained closed form solutions.   
%
\item  The results for the log-normal and log-double-exponential 
jump amplitude models are qualitatively similar, having exponentially small tails
and near-zero peaks in the jump distribution 
making small jumps more likely.  
%
\item However, all three models give reasonable quantitative, although somewhat
mixed, results.
%
\item For the future research and considerations. 
\begin{enumerate}
\item To  improve
the log-uniform and other jump models, 
% Zhu 8/29 the stochastic volatility should be considered.--->
% FH 8/29 does not read properly:  consider the stochastic volatility 
the stochastic volatility will be considered with
% and 
other factors.
% end Zhu % 
% since in the real world the implied volatility curve is not a
% constant, but has a curve with a `smile'.  
%
\item To consider the option price and optimal portfolio
applications, 
% Zhu 8/29 getting --->
% Fh 8/29 ? get the 
% end Zhu % 
approximate solutions to these problems % FH 8/29?
will be obtained.
\end{enumerate}
%
\end{itemize}

%
\bibliographystyle{amsalpha}
\begin{thebibliography}{00}
%
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%
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%Time-Dependent Parameters,'' 
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\bibitem{Book04}
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%\newpage
%
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\end{thebibliography}
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\end{document}
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