Page 1 of 18
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 07
August 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 82
Formulation of Stochastic Differential Equation of
Stock Market Model
1E.O.Ogbaji, 2E.S.Onah,3T.Aboiyar and 4A.R.Kimbir
1Department of Mathematics and Statistics, Federal University Wukari
(Ogbajieka@yahoo.com)
2,3,4 Department of Mathematics/Statistics/Computer Science University Of Agriculture, Makurdi
Abstract
We formulated four compartments stochastic
differential equations for stock markets from
schematic diagram of stock market interaction.
In stock markets, return on investment is the
most important single factor to investors.
Stock investors desire to know the behaviour
of return on investment and its volatility at a
particular period (either short or long
period).The four compartments include stock
price, return on investment, return on
investment volatility and stock price volatility.
The formulation was a modification and
extension of Heston’s model of two
compartments (Stock price and its volatility).
In this research work, the formulation follow
geometric Brownian motion model. We
showed the existence and uniqueness of the
solution. We conclude that the formulated
model can be use to show the real application
of stock markket in four compartment.
Key words: stochastic differential equations,
stock price, return on investment, return on
investment volatility and stock price volatility.
1.0 INTRODUCTION
The concept of Stochastic Differential
Equation (SDE) has been initiated by Einstein
in 1905, [56]. In his article he presented a
mathematical connection between microscopic
random motion of particles and the
macroscopic diffusion equation. Later it has
been seen that the stochastic differential
equation (SDE) model plays a prominent role
in a range of application areas such as physics,
chemistry, mechanics, biology,
microelectronics, economics and finance.
Earlier the SDEs were solved by using Ito
integral as an exact method which is discussed
in, [41]. But using exact method it is noticed
that there occur some difficulty to study
nontrivial problems and hence approximation
methods are used. In this context various
authors have given their contribution in these
field but we have mentioned which are directly
related to this problem. In [55] defined general
Runge-Kutta approximations for the solution
of stochastic differential equations and there
was given an explicit form of the correction
term. This work was carried out and then [37]
are discussed about the numerical solutions of
stochastic differential equation in detail. [49]
added discrete time strong and weak
approximation methods for the numerical
Page 2 of 18
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 07
August 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 83
methods to find the solution of stochastic
differential equations. Next, [29] gave a major
contribution in this field to solve the
approximate solutions of stochastic differential
equations and discussed few problems. Further
[30] investigated nonlinear stochastic
differential equations numerically. They
presented two implicit methods for Ito
stochastic differential equations (SDEs) with
Poisson-driven jumps. The first method is a
split-step extension of the backward Euler
method and the second method arises from the
introduction of a compensated, martingale,
form of the Poisson process. In this context
different authors have tried for various other
diffusion and application based problems. [28]
solved stochastic point kinetic reactor
problem. They modelled the point stochastic
reactor problem into ordinary time dependent
stochastic differential equation and studied the
stochastic behaviour of the neutron flux.
[36] considered fuzzy sets space for real line
and the existence and uniqueness of the
solution is obtained. The solution is
investigated by taking particular conditions
which are imposed on the structure of
integrated fuzzy stochastic processes such that
a maximal inequality for fuzzy stochastic Ito
integral holds. Next, [46] proposed an
approach to solve FSDE which does not
contain any notion of fuzzy stochastic Ito
integral and the method was based on the
selections of sets. Further, [41] presented the
existence and uniqueness of solutions to the
FSDEs driven by Brownian motion and the
continuous dependence on initial condition
and stability properties are established.
2.0 Formulation of Stochastic Differential
Equation of Stock model
In stock markets, return on investment is an
index that shows the growth or otherwise of
every business. Stock investors desire to know
the behaviour of return on investment and its
volatility at a particular period (either short or
long term). Investors like to know how much
volatility or risk that they are exposed to
before they can invest in a stock of company.
Over the years researches that have been carry
out considered interest rate. This informed
investors on percentage charged on money
loan out but not the total growth of a company.
In our research, we formulated a model that
Page 3 of 18
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 07
August 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 84
included return on investment and its volatility
that inform investors on the totality growth of
a company. This resulted in the modification
and extension of Heston model from two to a
four compartment stochastic differential
equation of stock market in three ways.
1. By replacing interest rate with rate of
reinvestment of return on investment
2.By formulating return on investment and it
volatility model
3. By assuming return on investment and it
volatility follows a random process
2.1 The Heston Model
In this section, we present the Heston model
and the parabolic partial differential equation
derivation of the model and show how the
solution of the model satisfies the derived
parabolic partial differential equation. The
following assumptions were made by Steve
Heston;
(i) The interest rate
is a constant.
(ii)The stock price St follows a Black-Scholes
type of stochastic process.
But with a stochastic variance Vt that follows a
Cox,Ingersoll and Ross(CIR) process. The
model is given as:
dSt= Stdt Vt
StdWt
dVt=x(
Vt
)dt +
Vt
dZt
(2.1)
dWtdZt= dt
The parameters and variables of (2.1) above
are defined as below:
is the drift coefficient of the stock price,
is the long-term mean of variance,
x is the rate of mean reversion,
is the volatility of volatility,
St and Vt are the stock price and volatility
process respectively
To take into account the leverage effect, stock
returns and implied volatility are negatively
correlated.Wt and Zt are correlated Wiener
process, and the correlation coefficients is
,[62].
Definition of Variables and Parameters
Here,we shall explain the variables and
parameter used in the Schematic diagram of
stock market interaction ;
