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\begin{document}
\begin{center}\textbf{{\Large 
Jump-Diffusion Stock Return Models in Finance:  
\\[1ex] 
Stochastic Process Density with Uniform-Jump Amplitude 
}\\[2ex]
\parbox[c]{2.6in}{\centering Floyd B. Hanson\\
{\small Laboratory for Advanced Computing\\
University of Illinois at Chicago\\
851 Morgan St.; M/C 249\\
Chicago, IL  60607-7045, USA\\
hanson@math.uic.edu}}
and 
\parbox[c]{2.6in}{\centering J. J. Westman\\
{\small Department of Mathematics\\
University of California \\
Box 951555\\
Los Angeles, CA  90095-1555, USA\\
jwestman@math.ucla.edu}}%endauthors
}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
%
The stochastic analysis is presented for the parameter estimation
problem for fitting a theoretical jump-diffusion model to the
log-returns from closing data of the Standard and Poor's 500
(S\&P500) stock index during the prior decade 1992-2001.  The
jump-diffusion model combines a the usual geometric Brownian motion
for the diffusion and a space-time Poisson process for the jumps
such that the jump amplitudes are uniformly distributed.  The
uniform jump distribution accounts for the rare large outlying
log-returns, both negative and positive in magnitude.  The log-normal,
log-uniform jump-diffusion density is derived, leading to a
jump-diffusion simulator approximation for the case the the log-return
time is a small fraction of a year.  There are five jump-diffusion
parameters that need to be determined, the means and variances for
both diffusion and jumps, as well as the jump rate, given the
average log-return time.  A weighted least squares is used to fit
the theoretical jump-diffusion model to the S\&P500 data optimizing
with respect to three free parameters, with the two other parameters
constrained by the mean and variance of the S\&P500 data.  The
weight distribution derives from stochastic methods.  The ideal
fitted model determines the three free parameters, but the
corresponding simulated results resemble the original S\&P500 data
better.  This stochastic analysis paper is a companion to a
computational methods and portfolio optimization paper at this
conference.  
% 
\end{abstract} 
%
%
\section{Introduction}
\setcounter{equation}{0}
%
A classical model of financial market return process, such as the
Black-Scholes \cite{BS73,Merton90}, is the log-normal diffusion
process, such that the log-return process has a normal distribution.
However, real markets exhibit several deviations from this ideal,
although useful, model.  The market distribution, say for stocks,
should have several realistic properties not found in the ideal
log-normal model:  (1) the model must permit large random fluctuations
such as crashes or sudden upsurges, (2) the log-return distribution
should be skew since large downward outliers are larger than upward
outliers, and (3) the distribution should be leptokurtic since the
mode is usually higher and the tails thicker than for a normal
distribution.  For modeling these extra properties, a jump-diffusion
process with log-uniform jump-amplitude Poisson process is used to
fit the \SPI log-returns.  A reasonable estimation of the parameters
of the log-return process can be made using a weighted least squares
approximation that is an improvement over earlier jump-diffusion
model results of Merton \cite{Merton90} and the authors
\cite{ACC01FM,ACC02FM,CDC02FM}.  The computational issues are principally
discussed in another paper of the authors at this conference
\cite{MTNS02FMB}.
%
%
\section{Density for Jump-Diffusions}
%
Let $S(t)$ be the price of a stock or stock fund satisfies a Markov,
continuous-time, geometric, jump-diffusion stochastic differential
equation (SDE),
%
\begin{equation}
\label{SDE}
dS(t) = S(t)\left[\mu_d dt + \sigma_d dZ(t) + J(Q)dP(t) \right]\;, 
~~ S(0) = S_0\;, ~~ S(t) >0\;,
\end{equation}
%
where $\mu_d$ is the mean return rate, $\sigma_d$ is the diffusive
volatility, $Z(t)$ is a one-dimensional stochastic diffusion process,
$J(Q)$ is a log-return mean $\mu_j$ and variance $\sigma_j^2$ random
jump-amplitude and $P(t)$ is a simple Poisson jump process with
jump rate $\lambda$.  It is assumed that the stock price parameters
$\mu_d$, $\sigma_d^2$, $\mu_j$, $\sigma_j^2$ and $\lambda$ are
constants.   The differential diffusion process with drift $\mu_d
dt + \sigma_d dZ(t)$ is has mean $\mu_d dt$ and $\sigma_d dt$
variance.  The space-time jump process $J(Q)dP(t)$ has mean
$E[J(Q)]\lambda dt$, variance $E[J^2(Q)]\lambda dt$ and $dP(t)$
has the discrete distribution
%
\begin{equation}
\label{PoissonDist}
p_k(\lambda dt) = {\rm Prob}[dP(t)=k] = 
\exp(-\lambda dt) (\lambda dt)^k/k!, ~~ k = 0:\infty\;.
\end{equation}
%
The processes $Z(t)$ and $P(t)$ are pairwise independent, while
$J(Q)$ is also independent except that it is conditioned on the
existence of a jump in $dP(t)$.

Since the SDE (\ref{SDE}) has a geometric or linear form it can
can be transformed to the simplified log-return form using the
stochastic process chain rule,
%
\begin{equation}
\label{DLNS}
d[\ln(S(t))] = \mu_{ld}dt + \sigma_d dZ(t) + \ln(1+J(Q))dP(t)\;,
\end{equation}
%
where $\mu_{ld}dt = \mu_d - \sigma_d^2/2$ is the log-diffusion
drift and $\ln(1+J(Q))$ is the log-return jump-amplitude.  For
finite log-return jump-amplitude and to avoid complete investment
loss, $J(Q) > -1$, so the underlying random jump mark amplitude $Q
= \ln(1+J(Q))$ on $(-\infty,+\infty)$ is chosen for convenience.

For this paper, we are interested in a uniformly distributed mark
variable $Q$ to account for the exceptionally long negative and
positive tails in financial market distributions, as can seen in
the histogram of the log-returns for \SPI \cite{SP500} daily closings
in the decade from 1992-2001 in Figure~\ref{FigHistAll}.  Since
large jumps in the log-returns seem to be rare events relative to
the background ups and downs modeled by the diffusion process, the
jump-amplitude distribution will be assumed to be uniformly
distributed on $[Q_a,Q_b]$, $Q_a < 0 < Q_b$, with time-independent
density

%
\begin{figure}[!ht]
\begin{center}{\includegraphics[width=3.00in]{mtns02fmtsphis.eps}}
\caption{\small Histogram of log-return of daily closings in the 
\SPI for the decade 1992--2001, using 100 bins.
\label{FigHistAll}
}
\end{center}
\end{figure}
%
\begin{equation}
\label{Uniform}
\phi_Q(q) \equiv \phi^{(u)}(q;Q_a,Q_b)
\equiv \frac{U(q;Q_a,Q_b)}{Q_b-Q_a}\;,
\end{equation}
%
where $U(x;a,b)$ denotes a unit step function on $[a,b]$, such that
%
\eq
\label{JumpMoms}
\mu_j = (Q_a+Q_b)/2 \And \sigma_j^2 = (Q_b-Q_a)^2/12.
\nq
%
Thus, the combined log-normal diffusion, log-uniform jump density
derives from a triad form of random processes $\xi+\eta\cdot\zeta$,
with diffusion  $\xi = \mu_{ld}dt+\sigma_d dZ(t)$, jump-amplitude
$\eta = Q$ and jump-time $\zeta = dP(t)$ processes.   This density
is proven in our time-dependent finance paper \cite{CDC02FM} and
is given here in the modified form,

\begin{theorem}
The probability density for the log--normal diffusion log--uniform
jump--amplitude log--return differential $d[\ln(S(t))]$ specified
in the SDE (\ref{DLNS}) is given by
%
\eq
\label{DLNSDensity}
\phi_{d\ln(S(t))}(x) &=&  
p_0(\lambda dt)\phi^{(n)}(x;\mu_{ld} dt,\sigma_d^2 dt)
\\\nonumber ~&~&
+ \sum_{k=1}^{\infty} p_k(\lambda dt)
\frac{\Phi^{(n)}(x-kQ_b,x-kQ_a;\mu_{ld} dt,\sigma_d^2 dt)}{k(Q_b -Q_a )}
\;,
\nq
%
$-\infty < x < +\infty$, where the Poisson distribution
$p_k(\lambda dt)$ is specified in (\ref{PoissonDist}) and the
normal distribution on $[x,y]$ is
%
\begin{equation}
\label{Phid}
\Phi^{(n)}(x,y;\mu_{ld}dt,\sigma_d^2 dt) \equiv
\int_x^y \phi^{(n)}(z;\mu_{ld} dt,\sigma_d^2 dt)dz
\equiv 
\int_x^y
\frac{\exp(-(z-\mu_{ld}dt)^2/(2\sigma_d^2 dt))}{\sqrt{2\pi\sigma_d^2 dt}}dz\;,
\end{equation}
%
where the integrand is the normal density of the diffusion process
$\xi=\mu_{ld}dt+\sigma_d dZ(t)$ in (\ref{DLNS}).
\end{theorem}
%

In the theorem there is no mention that $dt$ is the infinitesimal
of time, since it can be used for small but non-infinitesimal time
increments $\Delta{t}$ as needed in the financial markets.  In the
\SPI the average time between closings is $\Delta{t}=0.003967$
years, so $(\Delta{t})^2=0.00001574$ is negligible in comparison
to $\Delta{t}$, if that would be sufficiently accurate.  Hence,
the two-term asymptotic form of (\ref{DLNSDensity}) will be used:

\begin{corollary}
%
As $\Delta{t} \rightarrow 0^+$, (\ref{DLNSDensity}) can be 
asymptotically approximated as
%
\eq
\label{AsDLNSDensity}
\phi_{\Delta\ln(S(t))}(x) &\sim& \phi\jdd(x)
\\\nonumber ~&\equiv&
(1-\lambda \Delta{t})
\phi^{(n)}(x;\mu_{ld}\Delta{t},\sigma_d^2\Delta{t})
+ \lambda \Delta{t} \frac{\Phi^{(n)}(x-Q_b,x-Q_a;\mu_{ld}\Delta{t},
\sigma_d^2\Delta{t})}{Q_b-Q_a}
\;,
\nq
neglecting $O((\Delta{t})^2)$.
%
\end{corollary}

Eq.~(\ref{AsDLNSDensity}) is consistent with the usual zero-or-one
jump definition of the infinitesimal Poisson distribution given in
full form by (\ref{PoissonDist}), such that there are zero jumps
with probability $(1-\lambda\Delta{t})$ and one jump with probability
$\lambda\Delta{t}$.  Note that in (\ref{AsDLNSDensity}) the zero-jump
density is just the diffusion density, while the one-jump density
can be called the \textit{secant-normal} density since it is the
ratio of the difference in normal distributions divided by the
difference in arguments.  Eq.~(\ref{AsDLNSDensity}) is also consistent
with the small time form of the log-return in (\ref{DLNS}), such
that
%
\eq
\label{SDeltaE}
\Delta{\ln(S(t))} = \int_t^{t+\Delta{\tau}}d\ln(S(\tau))
\sim
\mu_{ld}\Delta{t} + \sigma_d \Delta{Z(t)} + Q\Delta{P(t)}\;,
\nq
%[
provided the parameters are constant and higher order jumps are
neglected, with $\Delta{P(t)}$ playing the role of an indicator
function for either zero or one jump.  Eq.~(\ref{SDeltaE}) can also
for the jump-diffusion simulations using $\sqrt{\Delta{t}}$ times
a normal random number generator for $\Delta{Z(t)}$, a standard
uniform generator on $[0,1]$ partitioned into $[0,\lambda \Delta{t}]$
for one-jump and $(\lambda \Delta{t},1]$ for no-jump in $\Delta{P(t)}$,
and a uniform generator on $[Q_a,Q_b]$ for simulating $Q$ provided
a one jump is selected by the simulation of $\Delta{P(t)}$.
%)
%
\section{Jump--Diffusion Parameter Estimation\label{ParmEst}}
%
For financial market modeling purposes, it is necessary to have an
estimate of the parameters of the market distribution.  For the
log-normal diffusion, log-uniform jump-amplitude jump-diffusion
theoretical model, there is a set of five parameters, $\{\mu_d,
\sigma_d^2, \mu_d, \sigma_d^2, \lambda\}$, assuming the time-step
$\Delta{t}$ is known.  The object of this paper is to estimate
these parameters by fitting the theoretical model to the decade
worth of log-returns of the \SPI from 1992 to 2001 portrayed in
$N\bin = 100$ histogram of Figure~\ref{FigHistAll}, subject to some
constraints to keep the parameter estimation computationally
reasonable.   There are a total of 2522 daily closings $S_i\spd$,
so that there are $N\spd = 2521$ log-returns, $\Delta(\ln(S_i\spd))
\equiv \ln(S_{i+1}\spd) - \ln(S_i\spd)$.  The constraints used are
matching the decade mean $M_1\spd \simeq 4.015\times 10^{-4}$ and
variance $M_2\spd \simeq 9.874\times 10^{-5}$.  Relative to the
normal distribution, the higher order moment coefficients are
$\eta_3\spd \equiv M_3\spd/(M_2\spd)^{1.5} \simeq -0.2913$ for
skewness and $\widehat{\eta}_4\spd \equiv M_4\spd/(M_2\spd)^2-3
\simeq 4.804$ for kurtosis, subtracting three for the unshifted
normal kurtosis coefficient.

The distinguishing feature of real markets are the thicker tails
that are longer on the negative side compared to normal distributions,
leading to negative skew and larger kurtosis coefficients.  Hence,
it is important that the fitting of the distributions be sufficiently
weighted so that the tails are sufficiently detectable.  In our
papers \cite{ACC02FM,CDC02FM}, an unweighted least squares was used
which resulted in the negative tails over-dominating the positive
tails.  Here, we use a weighted least squares or $\Chi^2$ fit (see
for instance the summaries in \cite{NRC}),
%
\eq
\label{WTLS}
{\Chi}^2 = \sum_{i=1}^{N\bin}\omega_i\cdot\left(f_i\jdd - f_i\spd\right)^2,
\nq
% 
where $\omega_i$ is the weight of the $i$th bin, $f_i\spd$ is the
$i$th empirical \SAP bin frequency data and $f_i\jdd$ is the $i$th
theoretical jump-diffusion bin frequency corresponding to the same
sample size $N\spd = 2521$.  An estimate of the weights corresponding
to a errors in measurements is not easy to get, but we will use
the following theoretical result to be proved elsewhere:

\begin{theorem}
%(
If $f_i\jds = \sum_{j=1}^N U(\Delta{S_j\jds};x_i,x_{i+1}^-)$ for
$i=1:N\bin$ are the frequencies of the $i$th bin $[x_i,x_{i+1})$
and $\Delta{S_j\jds}$ is the $j$th jump-diffusion simulation, using
$N$ samples, as prescribed for (\ref{SDeltaE}), then the bin
frequency expectation and variance are
%]
%
\eq
\mu_{f_i\jds} = \E\left[f_i\jds\right] =  f_i\jdd \And
\sigma_{f_i\jds}^2 = \Var\left[f_i\jds\right] 
= N\cdot\left(1-\left.f_i\jdd\right/N\right)^2 f_i\jdd,
\nq
%
respectively, where the $i$th expected bin frequency after $N$ 
simulations is
$$
f_i\jdd = N\cdot\int_{x_i}^{x_{i+1}} \phi_i\jdd(x)dx.
$$
\end{theorem}

The bin weights are chosen as the theoretical values,
\eq
\omega_ i = \left(1/\sigma_{f_i\jdd}^2\right)
\left/\sum_{j=1}^{N\bin}\left(1/\sigma_{f_j\jdd}^2\right)\right.\;,
\nq
for $i = 1:N\bin$ bins, normalized to a unit sum for convenience
of small minima.  The problem is reduced to a 3-dimensional global
minimization for the transformed parameter set $\{Q_a,Q_b,\lambda\Delta{t}\}$
subject to constraints,
%
\eq
M_1\jdd = \mu_{ld}\Delta{t} + \mu_j\lambda\Delta{t} = M_1\spd
\And
M_2\jdd = \sigma_d^2\Delta{t} + (\sigma_j^2+\mu_j^2)\lambda\Delta{t} 
=  M_2\spd\;,
\nq
%
serving as eliminants of $\mu_{ld}\Delta{t}$ and $\sigma_d^2\Delta{t}$,  with
the jump-moments definition (\ref{JumpMoms}) of $\mu_j$ and
$\sigma_j^2$ relating them to $Q_a$ and $Q_b$ (in rare case,
non-negativity must be enforced on the variances).  The global
minimizer \textit{Golden Super Finder (GSF)} \cite{GSF02}, developed
for financial problems in \cite{ACC02FM,CDC02FM}, was used to estimate the
fit (\ref{WTLS}).  This method is an extensive modification of the
method of golden section search (see \cite{NRC}) and is described
more in \cite{MTNS02FMB}. The final parameter results are
%
\eq
\mu_d &\simeq& 0.06386\;, ~~ \sigma_d^2 \simeq  0.005513\;,
%\nonumber\\
\mu_j \simeq 0.0007624\;, ~~ \sigma_j^2 \simeq  0.0003679\;, 
~~ \lambda \simeq 55.46\;,
\nq
%
with minimum $\Chi_{\min}^2 \simeq 2.621\times{10}^{-5}$ with a 
relative value-location hybrid stopping criterion of $5\times{10}^{-3}$
in a total of 16 GSF-iterations.  

The final successful minimum weighted least squares iteration results
are illustrated in Figure~\ref{FigJDth-sim}, with both theoretical and
simulation histograms.  The histogram on the right for the simulations
more closely resembles the \SAP data histogram, the \SAP being a
large realistic simulation.
%
\begin{figure}[!ht]
\begin{center}{
\parbox[c]{3.1in}{\includegraphics[width=3.00in]{mtns02fmtthhis.eps}}
\hspace*{1em}
\parbox[c]{3.1in}{\includegraphics[width=3.00in]{mtns02fmtsmhis.eps}}
\caption{\small Histogram of log-returns from the log-normal
diffusion, log-uniform jump-diffusion model fitted to the \SPI
log-returns for the decade 1992--2001 shown in Fig.~\ref{FigHistAll},
using 100 bins.  The figure on the left is the fitted theoretical
jump-diffusion histogram, while the figure on the right is the
corresponding simulated jump-diffusion histogram using the same
final parameter results and the same number of samples as the 
\SAP.
\label{FigJDth-sim}
}%endcaption
}%endcenter
\end{center}
\end{figure}
%
%
\section*{Conclusions}
%
In this paper, significant progress has been made toward fitting
the theoretical log-normal diffusion, log-uniform jump-diffusion
model to realistic financial market data, here the 1992-2001
log-returns of the \SPI.  The log-uniform jump distribution is a
big improvement over the log-normal jump distribution used in
\cite{ACC02FM}.  The crucial advance was to use a least squares method
with weights and to establishing a method for computing the least
square weights from the theoretical bin frequencies.  In essence,
the \SPI data is treated as a large scale jump-diffusion simulation.

The resulting estimated jump-diffusion parameter set can add more
realism to financial market applications, such as the optimal portfolio
and consumption policy problem treated in a computational companion paper 
\cite{MTNS02FMB} of the authors at this conference.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\*[0.1in]
%
\noindent\textbf{Acknowledgement}:  Work supported in part by the
National Science Foundation Computational Program Mathematics Grant 
DMS--99--73231.\\*[-2em]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%
\bibitem{BS73}
F. Black and M. Scholes, 
``The Pricing of Options and Corporate Liabilities,''
\textit{J. Political Economy}, vol. 81, 637-659, 1973.
%
\bibitem{ACC01FM}
F. B. Hanson and J. J. Westman, 
``Optimal Consumption and Portfolio Policies for Important Jump Events: 
Modeling and Computational Considerations,''
\textit{Proceedings of 2001 American Control Conference}, pp. 4456-4661,
25 June 2001.
%
\bibitem{KU02FM}
F. B. Hanson and J. J. Westman, 
``Stochastic Analysis of Jump--Diffusions for Financial Log--Return 
Processes,'' \textit{Proceedings of Stochastic Theory and
Control Workshop}, Springer--Verlag, New York, pp. 1-15, accepted, March
2002.
%
\bibitem{ACC02FM}
F. B. Hanson and J. J. Westman, ``Optimal Consumption and Portfolio Control
for Jump-Diffusion Stock Process with Log-Normal Jumps,''
{\it Proceedings of 2002 American Control Conference}, pp. 1-6,
08 May 2002, to appear.
%
\bibitem{CDC02FM}
F. B. Hanson and J. J. Westman, 
``Portfolio Optimization with Jump--Diffusions: Estimation and Application,''
\textit{Proceedings of 2002 Conference on Decision and Control}, pp. 1-15,
07 March 2002, submitted for an invited session.
%
\bibitem{MTNS02FMB}
F. B. Hanson and J. J. Westman, 
``Computational Methods for Portfolio and Consumption Policy Optimization
in Log-Normal Diffusion, Log-Uniform Jump Environments,''
\textit{Proceedings of the 15th International Symposium on Mathematical 
Theory of Networks and Systems}, pp. 1-6, August 2002, to appear.
%
\bibitem{GSF02}
F. B. Hanson and J. J. Westman, 
``Golden Super Finder:  Multidimensional Modification
of Golden Section Search Unrestricted by Initial Domain,''
under testing and in preparation, April 2002.
%
\bibitem{Merton90}
R. C. Merton,
\textit{Continuous--Time Finance}, Basil Blackwell, Cambridge, MA, 1990.
% 
\bibitem{NRC}
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, 
\textit{Numerical Recipes in C:  The Art of Scientific Computing},
Cambridge University Press, Cambridge, UK, 1992.
%
\bibitem{SP500}
Yahoo! Finance, 
``Historical Quotes, S\&P 500, Symbol $\widehat{~~}$SPC,''
\textit{URL: http://chart.yahoo.com/}, February 2002.
%
\end{thebibliography}
%
\end{document}