Portfolio Optimization with Jump–Diffusions: Estimation of
Time-Dependent Parameters and Application1
Floyd B. Hanson2 John J. Westman3
Abstract
This paper treats jump-diffusion processes in continuous time, with
emphasis on the jump-amplitude distributions, developing more ap-
propriate models using parameter estimation for the market in one
phase and then applying the resulting model to a stochastic optimal
portfolio application in a second phase. The new developments are
the use of uniform jump-amplitude distributions and time-varying
market parameters, introducing more realism into the application
model, a Log-Normal-Diffusion, Log-Uniform-Jump model.
1. Introduction
The empirical distribution of daily log-returns for actual financial in-
struments differ in many ways from the ideal log-normal diffusion
process as assumed in the Black-Scholes model [1] and other finan-
cial models. The log-returns are the log-differences between two
successive trading days, representing the logarithm of the relative
size. The most significant difference is that actual log-returns ex-
hibit occasional large jumps in value, whereas the diffusion process
in Black-Scholes [1] is continuous. Another difference is that the
empirical log-returns are usually negatively skewed, since the nega-
tive jumps or crashes are likely to be larger or more numerous than
the positive jumps for many instruments, whereas the normal distri-
bution associated with the diffusion process is symmetric. Thus, the
coefficient of skew [2] is negative, �
3
�M
3
=(M
2
)
1:5
< 0, where
M
2
and M
3
are the 2nd and 3rd central moments of the log-return
distribution here. A third difference is that the empirical distribution
is usually leptokurtic since the coefficient of kurtosis [2] satisfies
�
4
�M
4
=(M
2
)
2
> 3, where the value 3 is the normal distribution
kurtosis value and M
4
is the fourth central moment. Qualitatively,
this means that the tails are fatter than a normal with the same mean
and standard deviation, compensated by a distribution that is also
more slender about the mode (local maximum). A fourth difference
is that the market exhibits time-dependence in the distributions of
log-returns, so that the associated parameters are time-dependent.
In 1976, Merton [10, Chap. 9] introduced Poisson jumps with inde-
pendent identically distributed random jump-amplitudes with fixed
mean and variances into the Black-Scholes model, but the ability to
hedge the volatilities was not very satisfactory. Kou [9] uses a jump-
diffusion model with a double exponential jump-amplitude distribu-
tion with mean � and variance 2�, having leptokurtic and negative
skewness properties, although it is difficult to see the empirical jus-
1Work supported in part by the National Science Foundation Grant
DMS-99-73231 and DMS-02-07081 at the University of Illinois at Chicago.
Preprint of paper to appear in Proceedings of 2002 Conference on Decision
and Control, pp. 1-6, Las Vegas, 9-13 December 2002.
2Laboratory for Advanced Computing, University of Illinois at
Chicago,851 Morgan St., M/C 249, Chicago, IL 60607-7045, USA, e-mail:
hanson@math.uic.edu
3Department of Mathematics and Statistics, Miami University, Oxford,
OH 45056, USA and Department of Mathematics, University of California,
Los Angeles, e-mail: westmanjj@muohio.edu
tification for this distribution.
Prior to the Black-Scholes model, Merton [10, Chap. 5-6] analyzed
the optimal consumption and investment portfolio with either geo-
metric Brownian motion or Poisson noise and examined an example
of constant risk-aversion utility having explicit solutions. In [10,
Chap. 4], Merton also examined constant risk-aversion problems.
Hanson and Westman [5] reformulated an important external events
model of Rishel [11] solely in terms of stochastic differential equa-
tions and applied it to the computation of the optimal portfolio and
consumption policies problem for a portfolio of stocks and a bond.
The stock prices depend on both scheduled and unscheduled jump
external events. The computations were illustrated with a simple log-
bi-discrete jump-amplitude model, either negative or positive jumps,
such that both stochastic and quasi-deterministic jump magnitudes
were estimated. In [6], they constructed a jump-diffusion model with
marked Poisson jumps that had a log-normally distributed jump-
amplitude and rigorously derived the density function for the log-
normal-diffusion and log-normal-jump stock price log-return model.
In [7], this financial model is applied to the optimal portfolio and
consumption problem for a portfolio of stocks and bonds including
computational results.
In this paper, the log-normal-diffusion, log-uniform-jump problem
is treated. In Section 2, the jump-diffusion density is rigorously de-
rived using a modification of our prior theorem [6]. In Section 3, the
time dependent parameters for this log-return process are estimated
using this theoretical density and the S&P500 Index daily closing
data for the prior decade. In Section 4, the optimal portfolio and
consumption policy application is presented and then solved. Con-
cluding remarks are given in Section 5.
2. Log-Return Density for Log-Normal-Diffusion,
Log-Uniform Jump
Let S(t) be the price of a single financial instrument, such as a
stock or mutual fund, that is governed by a Markov, geometric jump-
diffusion stochastic differential equation (SDE) with time-dependent
coefficients,
dS(t) = S(t) [�
d
(t)dt+ �
d
(t)dZ(t) + J(t)dP (t)] ; (1)
with S(0) = S
0
; S(t) > 0, where �
d
(t) is the appreciation return
rate at time t, �
d
(t) is the diffusive volatility,Z(t) is is a continuous,
one-dimensional Gaussian process, P (t) is a discontinuous, one-
dimensional standard Poisson process with jump rate �(t), and asso-
ciated jump-amplitude J(t) with log-return mean �
j
(t) and variance
�
2
j
(t). The stochastic processes Z(t) and P (t) are assumed to be
Markov and pairwise independent. The jump-amplitude J(t), given
that a Poisson jump in time occurs, is also independently distributed.
The stock price SDE (1) is similar in our prior work [6, 7], except
that time-dependent coefficients introduce more realism here.
The continuous, differential diffusion process dZ(t) is standard,
so has zero mean and dt variance. The symbolic notation for
the discontinuous space-time jump process, J(t)dP (t), is better
defined in terms of the Poisson random measure, P(dt; dq), by
the stochastic integral, J(t)dP (t) =
R
Q
b
J(t; q)P(dt; dq), where
Q = q is the Poisson spatial mark variable for the jump ampli-
tude process, and bJ(t; q) is the kernel of the Poisson operator
J(t), such that �1 < bJ(t; q) <1 so that a single jump does not
make the underlying non-positive. The infinitesimal moments of
the jump process are E[J(t)dP (t)] = �(t)dtR
Q
b
J(t; q)�
Q
(q; t)dq
and Var[J(t)dP (t)] = �(t)dt
R
Q
b
J
2
(t; q)�
Q
(q; t)dq, neglecting
O
2
(dt) here, where �
Q
(q; t) is the Poisson amplitude mark den-
sity. The differential Poisson process is a counting process with
the probability of the jump count given by the usual Poisson distri-
bution, p
k
(�(t)dt) = exp(��(t)dt)(�(t)dt)
k
=k!, k = 0; 1; 2; : : :,
with parameter �(t)dt > 0.
Since the stock price process is geometric, the common multiplica-
tive factor of S(t) can be transformed away yielding the SDE of
the stock price log-return using the stochastic chain rule for Markov
processes in continuous time,
d[ln(S(t))] = �
ld
(t)dt+ �
d
(t)dZ(t) + ln(1 + J(t))dP (t); (2)
where �
ld
(t) � �
d
(t)� �
2
d
(t)=2 is the log-diffusion drift and
ln(1 +
b
J(t; q)) the stock log-return jump-amplitude is the logarithm
of the relative post-jump-amplitude. This log-return SDE (2) will be
the model that will used for comparison to the S&P500 log-returns.
Since bJ(t; q) > �1, it is convenient to select the mark process to
be the jump-amplitude random variable, Q = ln
�
1 +
b
J(t;Q)
�
, on
the mark space Q = (�1;+1). Though this is a convenient mark
selection, it implies the time-independence of the jump-amplitude,
so bJ(t;Q) = bJ
0
(Q) or J(t) = J
0
. Since market jumps are rare
and the tails are relatively flat, a reasonable approximation is uni-
form jump-amplitude distribution with density �
Q
(q; t) on the finite,
time-dependent mark interval [Q
a
(t); Q
b
(t)],
�
Q
(q; t) �
H(Q
b
(t)� q)�H(Q
a
(t)� q)
Q
b
(t)�Q
a
(t)
; (3)
where H(x) is the Heaviside, unit step function. The density
�
Q
(q; t) yields a mean E
Q
[Q] = �
j
(t) = (Q
b
(t) +Q
a
(t))=2 and
variance Var
Q
[Q] = �
2
j
(t) = (Q
b
(t)�Q
a
(t))
2
=12, which define
the basic log-return jump amplitude moments. It is assumed that
Q
a
(t) < 0 < Q
b
(t), to make sure that both negative and positive
jumps are represented, which was a problem for the log-normal
jump-amplitude distribution in [7]. The uniform distribution is
treated as time-dependent in this paper, so Q
a
(t), Q
b
(t), �
j
(t) and
�
2
j
(t) all depend on t. The difficulty in separating out the small
jumps about the mode or maximum of real market distributions is
explained by the fact that a diffusion approximation for small marks
can be used for the jump process that will be indistinguishable from
the continuous Gaussian process anyway.
The basic moments of the stock log-return differential are
M
(jd)
1
� E[d[ln(S(t))]] = (�
ld
(t) + �(t)�
j
(t))dt;
(4)
M
(jd)
2
� Var[d[ln(S(t))]]
=
�
�
2
d
(t) + �(t)
�
�
2
j
(t)(1 + �(t)dt)�
2
j
(t)
��
dt;
(5)
where the O2(dt) term has been retained in the variance, rather than
being neglected as usual, since the discrete return time, dt = �t, the
daily fraction of one trading year (about 250 days), will be small, but
not negligible.
The log-normal-diffusion, log-uniform-jump density can be found
by basic probabilistic methods following a slight modification for
time-dependent coefficients of constant coefficient theorem found
our paper [6],
Theorem: The probability density for the log-normal-diffusion,
log-uniform-jump amplitude log-return differential d[ln(S(t))]
specified in the SDE (2) with time-dependent coefficients is given
by
�
d ln(S(t))
(x) = p
0
(�(t)dt)�
(n)
�
x;�
ld
(t)dt; �
2
d
(t)dt
�
+
P
1
k=1
p
k
(�(t)dt)
k(Q
b
(t)�Q
a
(t))
�
h
�
(n)
�
kQ
b
(t)� x+ �
ld
(t)dt; 0; �
2
d
(t)dt
�
� �
(n)
�
kQ
a
(t)� x+ �
ld
(t)dt; 0; �
2
d
(t)dt
�
i
;
(6)
�1 < z < +1, where p
k
(�(t)dt) is the Poisson distribution and
the normal distribution with mean �
ld
dtr and variance �2
d
dt is
�
(n)
(x;�
ld
dt; �
2
d
dt) =
Z
x
�1
�
(n)
(y;�
ld
dt; �
2
d
dt)dy
associated with d ln(S(t)), the diffusion part of the log-return pro-
cess,
�
�
ld
dt+�
d
dZ(t)
(x) = �
(n)
(x;�
ld
(t)dt; �
2
d
(t)dt) :
The proof, which is only briefly sketched here, follows from the den-
sity of a triad of independent random variables, � + � � � given the
densities of the three component processes �, �, and �. Here, (1)
� = �
ld
(t)dt+ �
d
(t)dZ(t) is the log-normal plus log-drift diffu-
sion process, (2) � = Q = ln(1 + bJ
0
(Q)) is the log-uniform jump-
amplitude, and (3) � = dP (t) is the differential Poisson process.
The density of a sum of independent random variables, as in the
sum operation of � + (� � �), is very well-known and is given by a
convolution of densities �
�+��
(z) =
R
+1
�1
�
�
(z � y)�
��
(y)dy (see
Feller [3]). However, the distribution of the product of two random
variables � � � is not so well-known [6] and has the density,
�
��
(x) = p
0
(�(t)dt)�(x)
+
P
1
k=1
p
k
(�(t)dt)[H(Q
b
(t)�x=k)�H(Q
a
(t)�x=k)]
k(Q
b
(t)�Q
a
(t))
;
(7)
for the log-uniform-jump process. The probabilistic mass at x = 0,
represented by the Dirac �(x) and corresponds to the zero jump
event case. Finally, applying the convolution formula for density of
the sum � + (��) leads to the density for the random variable triad
� + �� given in (6) of the theorem.
Using the log-normal jump-diffusion log-return density in (6), the
third and fourth central moments with finite return time dt = �t are
computed, for later use for skew and kurtosis coefficients, respec-
tively, yielding the jump-diffusion higher moments [6],
M
(jd)
3
� E
�
�
d[ln(S(t))]�M
(jd)
1
�
3
�
= 6�
j
(t)(�(t)dt)
2
�
2
j
(t) + (3�
j
(t)�
2
j
(t) + �
3
j
(t))�(t)dt ;
(8)
M
(jd)
4
� E
�
�
d[ln(S(t))]�M
(jd)
1
�
4
�
= 3(�
2
j
(t))
2
(�(t)dt)
4
+ (6�
2
j
(t)�
2
j
(t)
+18(�
2
j
(t))
2
)(�(t)dt)
3
+ (3�
4
j
(t) + 30�
2
j
(t)�
2
j
(t)
+21(�
2
j
(t))
2
+ 6�
2
j
(t)dt�
2
j
(t))(�(t)dt)
2
+(�
4
j
(t) + 6�
2
j
(t)dt�
2
j
(t) + 6�
2
j
(t)�
2
j
(t)
+6�
2
j
(t)�
2
j
(t)dt+ 3(�
2
j
(t))
2
)�dt+ 3(�
2
j
(t))
2
dt
2
:
(9)
3. Jump-Diffusion Parameter Estimation
Given the log-normal-diffusion, log-uniform-jump density (6),
it is necessary to fit this theoretical model to realistic empirical
data to estimate the parameters of the log-return model (2) for
d[ln(S(t))]. For realistic empirical data, the daily closings of
the S&P500 Index during the decade from 1992 to 2001 are
used from data available on-line [13]. The data consists of
n
(sp)
= 2522 daily closings. The S&P500 data can be viewed
as an example of one large mutual fund rather than a single
stock. The data has been transformed into the discrete analog of
the continuous log-return, i.e., into changes in the natural loga-
rithm of the index closings, �[ln(SP
i
)] � ln(SP
i+1
)� ln(SP
i
)
for i = 1; : : : ; n(sp) � 1 daily closing pairs. For the decade,
the mean is M(sp)
1
' 4:015 � 10
�4 and the variance is
M
(sp)
2
' 9:874 � 10
�5
, the coefficient of skewness is
�
(sp)
3
�M
(sp)
3
=(M
(sp)
2
)
1:5
' �0:2913 < 0, demonstrating
the typical negative skewness property, and the coefficient of
kurtosis is �(sp)
4
�M
(sp)
4
=(M
(sp)
2
)
2
' 7:804 > 3, demonstrating
the typical leptokurtic behavior of many real markets.
The S&P500 log-returns, �[ln(SP
i
)] for i = 1 : n(sp) decade data
points, are partitioned into 10 yearly data sets, �[ln(SP(spy)
j
y
;k
)]
for k = 1 : n(sp)
y;j
y
yearly data points for j
y
= 1 : 10 years, where
P
10
j
y
=1
n
(sp)
y;j
y
= n
(sp)
. For each of these yearly sets, the pa-
rameter estimation objective is to find the least sum of squares
of the deviation between the empirical S&P500 log-return his-
tograms for the year and the analogous theoretical log-normal-
diffusion, log-uniform-jump distribution histogram based upon the
same bin structure. Since jumps are rare, 100 centered bins within
the log-return domain [x
a
; x
b
] were used. Since the most ex-
treme log-returns are the same as the most extreme jumps, the
log-return domain is selected to coincide with the time-dependent
uniform distribution domain, i.e., [x
a
(t); x
b
(t)] = [Q
a
(t); Q
b
(t)],
both dimensionless, where Q
a
(t) = min
k
(�[ln(SP
(spy)
j
y
;k
)]) and
Q
b
(t) = max
k
(�[ln(SP
(spy)
j
y
;k
)]) with t = T
j
y
= Year
j
y
+ 0:5,
say, assigning the yearly value to the mid-year with steps of
dt = �T
j
y
, for each j
y
= 1 : 10. For a given t = T
j
y
year,
fixed [Q
a
(t);Q
b
(t)] implies fixed uniform distribution parameters
�
j
(t) = (Q
b
(t) +Q
a
(t))=2 and �2
j
(t) = (Q
b
(t)�Q
a
(t))
2
=12.
However, the Poisson jump rate �(t) is still a free parameter for
the jump component of the log-return process. Further to keep the
number of free parameters as small as practical, we require that the
mean and variances of the yearly log-returns be the same for both
empirical and theoretical distributions, i.e.,
M
(spy)
1;j
y
� Mean
n
(sp)
y;j
y
k=1
h
�
h
ln
�
SP
(spy)
j
y
;k
�ii
= M
(jdy)
1;j
y
using (4) and
M
(spy)
2;j
y
� Var
n
(sp)
y;j
y
k=1
h
�
h
ln
�
SP
(spy)
j
y
;k
�ii
= M
(jdy)
2;j
y
using (5), for each j
y
= 1 : 10 years. This, in turn, implies con-
straints on the log-diffusion parameters,
�
ld;j
y
=
�
M
(spy)
1;j
y
� (� dt�
j
)
j
y
�
=�T
j
y
; (10)
�
2
d;j
y
=
�
M
(spy)
2;j
y
� (�dt((1 + �dt)�
2
j
+ �
2
j
)
j
y
)
�
=�T
j
y
; (11)
with �2
d;j
y
> 0 for each j
y
= 1 : 10 years. Of the six parameters
f�
ld;j
y
; �
2
d;j
y
; �
j;j
y
; �
2
j;j
y
; �
j
y
;�T
j
y
g, needed for each year j
y
to
specify the jump-diffusion log-return distribution, only the jump rate
�
j
y
needs to be estimated by least squares. The time step dt = �T
j
y
is the reciprocal of the number of trading days per year, close to 250
days, but varies a little for j
y
= 1 : 10 and has values lying in the
range, [0.003936, 0.004050], used here for parameter estimation.
Thus, we have a one dimensional global minimization problem
for a highly complex discretized jump-diffusion density function
(2). The analytical complexity indicates that a general global
optimization method that does not require derivatives would be
useful. For this purpose, such a method, Golden Super Finder
(GSF) [8], was developed for [7] and implemented in MATLABTM ,
since simple techniques are desirable in financial engineering. The
GSF method is an extensive modification to the Golden Section
Search method [4], extended to multi-dimensions and allowing
search beyond the initial hyper-cube domain by including the
endpoints in the local optimization test with the two golden section
interior points per dimension, moving rather than shrinking the
hypercube when the local optimum is at an edge or corner. The
method, as a general method, is slow, but systematically moves
the search until the uni-modal optimum is found at a interior point
and then approaches the optimum if within the original search
bounds. Additional constraints can be added to the objective
function, such as (10,11). If the diffusion coefficient vanishes,
�
2
d
! 0
+
, then (11) implies a maximum jump count constraint,
max[� � dt] = 0:5(
p
((�
2
j
+ �
2
j
)
2
+ 4�
2
j
�M
2
)� (�
2
j
+ �
2
j
))=�
2
j
.
An additional compatibility constraint, �
j
(t) > 0, does not need
enforcement as long as Q
a
(t) < Q
b
(t) and is not violated here.
The jump-diffusion estimated parameter results in this log-normal-
diffusion, log-uniform-jump amplitude case are summarized in Ta-
ble 1. The jump rate estimates and their variability are summarized
in Table 2. A hybrid value-position stopping criterion with a toler-
ance, tol = 5:e–3 was used, and all yearly iterations converged in at
most 13 iterations each, out of a maximum limit of 20, except for the
year 1999� which exhibited little evidence of the long and flat tails
of other years, with a limiting behavior indicating a zero jump rate
value, �
8
' 2:52e � 4 ' 0:0 for Year
8
= 1999
� when j
y
= 8.
Table 1: Summary of yearly coefficients for Log-Normal-
Diffusion, Log-Uniform-Jump estimated parameters by
least squares (variance of deviation between S&P500 and
jump-diffusion histograms) with respect to the variable
�dt given dt = �T
j
y
and constraints mentioned in the
text.
Year
j
y
�
d;j
y
�
d;j
y
�
j;j
y
�
j;j
y
�
j
y
1992 4.1e-2 7.3e-2 -1.6e-3 9.9e-3 36.
1993 6.7e-2 7.0e-2 -2.6e-3 1.3e-2 15.
1994 -1.5e-2 7.6e-2 -9.1e-4 1.3e-2 22.
1995 3.0e-1 5.8e-2 1.5e-3 9.9e-3 25.
1996 1.7e-1 9.7e-2 -6.0e-3 1.5e-2 17.
1997 2.8e-1 1.5e-1 -1.1e-2 3.5e-2 7.1
1998 2.2e-1 1.5e-1 -1.0e-2 3.5e-2 14.
1999� 1.9e-1 1.8e-1 3.1e-3 1.8e-2 2.5e-4�
2000 -1.2e-1 1.9e-1 -6.8e-3 3.1e-2 14.
2001 -1.1e-1 1.8e-1 -7.9e-4 2.9e-2 15.
In Figure 1 a sample comparison can be made of the empirical
S&P500 histogram on the left for the relatively noisy year of 2000
with the corresponding theoretical jump-diffusion histogram on the
right using the fitted, optimized parameters and the same number of
Table