The chances of an investor in the stock market depends mainly on some certain decisions in respect to equilibrium prices, which is the condition of a system competing favorably and effectively. This paper considered a stochastic model which was latter transformed to non-linear ordinary differential equation where stock volatility was used as a key parameter. The analytical solution was obtained which determined the equilibrium prices. A theorem was developed and proved to show that the proposed mathematical model follows a normal distribution since it has a symmetric property. Finally, graphical results were presented and the effects of the relevant parameters were discussed.

1. An Empirical Approach for the Variation in Capital Market Price Changes
IJSM
An Empirical Approach for the Variation in Capital Market
Price Changes
F.N. Nwobi1
, and *P.A. Azor1,2
1
Department of Statistics, Imo State University, Owerri, Nigeria
1,2
Department of Mathematics & Statistics, Federal University, Otuoke, Bayelsa State, Nigeria
*Corresponding Author Email: azor.promise@gmail.com; Co-Author Email: Email: fnnwobi@yahoo.co.uk
The chances of an investor in the stock market depends mainly on some certain decisions in
respect to equilibrium prices, which is the condition of a system competing favorably and
effectively. This paper considered a stochastic model which was latter transformed to non-linear
ordinary differential equation where stock volatility was used as a key parameter. The analytical
solution was obtained which determined the equilibrium prices. A theorem was developed and
proved to show that the proposed mathematical model follows a normal distribution since it has
a symmetric property. Finally, graphical results were presented and the effects of the relevant
parameters were discussed.
Keywords: Equilibrium Price, Volatility, Stock Price, Differential Equation and Symmetric Characteristics
INTRODUCTION
As seen in Davis (2005,2006), a differential equation is an important tool for harnessing different components into a simple
system and analysing the inter-relationships that exist between these components might remain independent of each
other.
Differential Equations are one of the most frequently used tools for mathematical modelling in engineering and sciences.
Generally, dynamics of a changing process can be modelled into an ordinary differential equation or a partial differential
equation, depending on the nature of the problem. These equations may take various forms like Ordinary differential
equation(ODE), where partial differential equation (PDE) is an equation involving a function of several variables and at
least, one of tis partial derivatives and sometimes a combination of interacting equations of Ordinary and partial differential
equation. If some randomness is allowed into stochastic differential equation (SDE) for example; environmental effects are
allowed into some of the coefficients of a differential equation, a more realistic mathematical model of the problem or
situation can be obtained by introducing one of the key parameters to differential equation.
Nevertheless, a stock represents a share in the ownership of an incorporated company. Investors buy stocks in the hope
that it will yield dividends that grow in value. Thus, market price is the current price at which an asset or service can be
bought or sold. Economic theory contends that the market converges at a point where the forces of supply and demand
meet Dmouj(2006). Market price of stock is the most recent price at which the stock was traded. It is the result of traders
and dealers interacting with each other in a market. Many scholars have used differential equation in modelling financial
concepts. For instance, Osu(2010) considered a stock market price fluctuations. Bassel equation was applied which
determined the equilibrium price. Ugbebor et al. (2001) considered a stochastic model of price changes at the floor of stock
exchange where the equilibrium price and the market growth rate of shares were determined.
However, Tian-Quan and Yun(2009) developed a set of simultaneous differential equations of stock prices of the share in
both A and H stock markets. These set of simultaneous nonlinear differential equations were solved by iteration method
via a proof by a g-contraction mapping theorem. Amadi et al.(2012) worked on the application of non-linear first order
ordinary differential equation on stock market prices. They modelled two competing growth rates and its carrying capacity
on stock market prices which was chosen based on minimum variance criteria. They discovered within the trading period
Research Article
Vol. 8(1), pp. 164-172, May, 2022. © www.premierpublishers.org. ISSN: 2375-0499
International Journal of Statistics and Mathematics

2. An Empirical Approach for the Variation in Capital Market Price Changes
Int. J. Stat. Math. 165
that resolves to sustain its consumers are limited. Duroyaye & Uzome(2020) considered the nonlinear first order ordinary
differential equation model in stock market analysis, the considered two competing growth rates and its carrying capacity.
The analysis shows a continuous growth in the stock strength within the period. Moreso there have been some works with
considerable extensions such as Osu & Okorofor(2007) and Osu et al.( 2009)
Previous studies have therefore modelled equilibrium prices using different approaches. In particular, some studies, for
instance Osu (2010) stated cases such as linear and quadratic, where Bessel solutions were obtained.
The aim of this paper is first to establish a dynamic stochastic model of the capital market price aimed at determining the
equilibrium prices when the economic trend follows series price index function.
In this paper, a nonlinear ordinary differential equation was considered with volatility parameter included in the model. An
assumption of equilibrium price was stated to follow an index series price function. This problem was solved in detail and
a closed form analytical solution for equilibrium price was obtained. We proposed a theorem and proved to show the
effectiveness of the proposed model as it affects equilibrium prices.
The rest of this paper is arranged as follows: Section 2 presents preliminaries, the transformation of BS PDE to ODE is in
Subsection 2.1, Problem Formulation is presented in Section 2.2, method of solution is in Section 2.3, results are presented
in Section 3, discussion of results are in Section 4 and the paper is concluded in Section 5.
Mathematical Preliminaries of Underlying Asset
The following (1) is a Black-Scholes partial differential equation as applied in financial modelling
𝜕𝑉
𝜕𝑡
+
1
2
𝜎2
𝑆2
𝜕2
𝑉
𝜕𝑆2
+ 𝑟𝑆
𝜕𝑉
𝜕𝑆
− 𝑟𝑉 = 0 (1)
2.1 Transformation of BS PDE to Ordinary Differential Equation (ODE)
In this section, we use Cauchy-Euler method for the transformation of BS PDE to ODE. Let
𝑉(𝑆, 𝑡) = 𝑉𝑡(𝑠)𝑒𝜆𝑡
(2)
𝜕𝑉
𝜕𝑡
= 𝑉𝑡(𝑠)𝜆𝑒𝜆𝑡
(3)
𝜕𝑉
𝜕𝑆
=
𝑑𝑉𝑡(𝑆)𝑒𝜆𝑡
𝑑𝑆
(4)
𝜕2𝑉
𝜕𝑆2 =
𝑑2𝑉𝑡(𝑆)𝑒𝜆𝑡
𝑑𝑆2 (5)
Putting (2) − (5) into (1) gives
𝑉𝑡(𝑠)𝜆𝑒𝜆𝑡
+
1
2
𝜎2
𝑆2
𝑑2
𝑉𝑡(𝑠)𝑒𝜆𝑡
𝑑𝑆2
+ 𝑟𝑆
𝑑𝑉𝑡(𝑠)𝑒𝜆𝑡
𝑑𝑆
− 𝑟𝑉𝑡(𝑠)𝑒𝜆𝑡
= 0 (6)
1
2
𝜎2
𝑆2
𝑑2
𝑉𝑡(𝑠)𝑒𝜆𝑡
𝑑𝑆2
+ 𝑟𝑆
𝑑𝑉𝑡(𝑠)𝑒𝜆𝑡
𝑑𝑆
+ 𝑉𝑡(𝑠)𝜆𝑒𝜆𝑡
− 𝑟𝑉𝑡(𝑠)𝑒𝜆𝑡
= 0
Rearranging (6)
⟹ 𝑒𝜆𝑡 [
1
2
𝜎2
𝑆2
𝑑2
𝑉𝑡(𝑠)
𝑑𝑆2
+ 𝑟𝑆
𝑑𝑉𝑡(𝑠)
𝑑𝑆
+ 𝑉𝑡(𝑆)(𝜆 − 𝑟)] = 0 (7)
Let 𝜆 = 0 but 𝑒𝜆𝑡
≠ 0, where 𝜆 is a dummy market value, then (7) becomes
1
2
𝜎2
𝑆2 𝑑2𝑉𝑡(𝑠)
𝑑𝑆2 +
𝑑𝑉𝑡(𝑠)
𝑑𝑆
− 𝑟𝑉𝑡(𝑆) = 0 (8)
An investor observes prices and takes actions in discrete time periods 𝑡 = 0,1,2,. . ., 𝑇, the factors underlying price changes
are very uncertain and are described in probability terms. Uncertainty is captured by a stochastic 𝑥𝑡, 𝑡 = 0,1,2,. . . ., 𝑇,
taking values in a measurable space X .The value of the random parameter 𝑥𝑡 characterizes the state of the world at time
t, Evstigneev and Schenk Hoppe(2001).
Division of both sides of (8) by
𝜎2
2
gives
𝑆2
𝑑2
𝑉𝑡(𝑠)
𝑑𝑆2
+
2
𝜎2
𝑑𝑉𝑡(𝑠)
𝑑𝑆
−
2𝑟
𝜎2
𝑉𝑡(𝑆) = 0
Setting
2𝑟
𝜎2 = 1 gives

3. An Empirical Approach for the Variation in Capital Market Price Changes
Nwobi and Azor 166
𝑆2
𝑑2
𝑉𝑡(𝑠)
𝑑𝑆2
+
𝑑𝑉𝑡(𝑠)
𝑑𝑆
− 𝑉𝑡 = 0 (9)
We assume that stock price is a deterministic function of the stock price itself, so that the stock price is still the only source
of uncertainty (see Osu,2010).
Divide both sides of (9) by S gives
𝑆𝑑𝑉𝑡(𝑠)
𝑑𝑆2
+
𝑑𝑉𝑡(𝑠)
𝑑𝑆
−
𝑉𝑡(𝑠)
𝑆
= 0 (10)
To determine the value of an economic asset, like stock; we need to take full cognizance of its aspect of random variability
such as value of the underlying asset and its price, the variance and standard deviation of the asset at a particular time.
Now, suppose S is the unit price of the underlying asset, 𝑉𝑡 the value of the asset at time t and standard deviation σ is
the volatility of the underlying asset.
Assuming the dividends are declared at time 𝑡, thus we define the index price function to be of the form
𝑉𝑡(𝑆)
𝑆
= 𝑆𝜎2
𝑉𝑡
which is known as the aggregate intrinsic value of stock (see, Osu et al., 2009).
Now we replace
𝑉𝑡(𝑠)
𝑆
with 𝑆𝜎2
𝑉𝑡 in (10), giving the differential equation of the form
𝑆
𝑑2
𝑉𝑡(𝑠)
𝑑𝑆2
+
𝑑𝑉𝑡(𝑠)
𝑑𝑆
− 𝑆𝜎2
𝑉𝑡 = 0 (11)
Problem Formulation
Here, consider the problem of an investor who at the beginning of an investment period is faced with series of decisions
on the optimal choice of investment that will maximize profit at equilibrium. Assuming such equilibrium price follows a
series of an index price function such that
𝐺𝑟𝐴: 𝑊ℎ𝑒𝑟𝑒 𝐴 = (
𝐾𝑆
𝑟𝑡
)
2
+ (
𝐾𝑆
𝑟𝑡
)
4
+ (
𝐾𝑆
𝑟𝑡
)
6
+ (
𝐾𝑆
𝑟𝑡
)
8
+ ⋯
So, describing this type of series needs stock volatility as a key and relevant parameter in the model. Therefore, volatility
increases the chances that the stock will improve adequately.
Following the method of Bunonyo et al(2016), we replace the Right Hand Side of (11) with
−𝑃 − 𝐺𝑟𝐴 (12)
where P is a constant term and Gr is the value of stock variables or quantities
Equating (11) to (12) gives a non-homogenous differential equation in (13)
𝑆
𝑑2
𝑉𝑡(𝑠)
𝑑𝑆2
+
𝑑𝑉𝑡(𝑠)
𝑑𝑆
− 𝑆𝜎2
𝑉𝑡 = −𝑃 − 𝐺𝑟𝐴 (13)
with the following boundary conditions;
𝑉𝑡 = 𝜃𝑎 𝑜𝑛 𝑟 = 1 (14)
𝑑𝑉
𝑑𝑆
= 0 𝑜𝑛 𝑟 = 0 (15)
METHOD OF SOLUTION
Using Frobenius method on the LHS of (13)
𝑉𝑡 = ∑ 𝑎𝑛
∞
𝑛=0
𝑆𝑛+𝑐
= 𝑆𝑐 ∑ 𝑎𝑛 𝑆𝑛
∞
𝑛=0
(16)
𝑉𝑡
′
= ∑ 𝑎𝑛
∞
𝑛=0
(𝑛 + 𝑐)𝑆𝑛+𝑐−1
= 𝑆𝑐−1 ∑ 𝑎𝑛(𝑛 + 𝑐)𝑆𝑛
∞
𝑛=0
(17)
𝑉𝑡
′′
= ∑ 𝑎𝑛
∞
𝑛=0
(𝑛 + 𝑐)(𝑛 + 𝑐 − 1)𝑆𝑛+𝑐−2
= 𝑆𝑐−2 ∑ 𝑎𝑛(𝑛 + 𝑐)(𝑛 + 𝑐 − 1)𝑆𝑛
∞
𝑛=0
(18)
Putting (16) − (18) into (12) gives
𝑆. 𝑆𝑐−2 ∑ 𝑎𝑛
∞
𝑛=0
(𝑛 + 𝑐)(𝑛 + 𝑐 − 1)𝑆𝑛
+ 𝑆𝑐−1 ∑ 𝑎𝑛(𝑛 + 𝑐)𝑆𝑛
−
∞
𝑛=0
∑ 𝜎2
𝑎𝑛𝑆𝑛+𝑐+1
∞
𝑛=0

4. An Empirical Approach for the Variation in Capital Market Price Changes
Int. J. Stat. Math. 167
𝑆𝑐−1 ∑ 𝑎𝑛
∞
𝑛=0
(𝑛 + 𝑐)(𝑛 + 𝑐 − 1)𝑆𝑛
+ 𝑆𝑐−1 ∑ 𝑎𝑛(𝑛 + 𝑐)𝑆𝑛
−
∞
𝑛=0
∑ 𝜎2
𝑎𝑛𝑆𝑛+𝑐+1
= 0
∞
𝑛=0
(19)
Collecting like terms in (19) and performing some algebraic expression
𝑉𝑡(𝑆) = 𝑉1 = 𝐴 {1 +
𝜎2
𝑆2
22
+
𝜎4
𝑆4
22 × 42
+
𝜎6
𝑆6
22 × 42 × 62
+
𝜎8
𝑆8
22 × 42 × 62 × 82
+ ⋯ } (20)
Similarly solving for 𝑉2 and also performing some algebraic expressions gives the following:
𝑉𝑡 = 𝑉2 = 𝐵 {𝐼𝑛𝑆 (1 +
𝜎2
𝑆2
22
+
𝜎4
𝑆4
22 × 42
+
𝜎6
𝑆6
22 × 42 × 62
+
𝜎8
𝑆8
22 × 42 × 62 × 82
+ ⋯ )
+ 𝑎0𝑆𝑐 {−
𝜎2
𝑆2
22
−
𝜎4
𝑆4
23 × 42
−
𝜎6
𝑆6
43 × 63
+
𝜎8
𝑆8
23 × 63 × 83
+ ⋯ }} (21)
A linear combination (20) and (21) gives the complete solution
𝑉𝑡(𝑆) = 𝐴 {1 +
𝜎2
𝑆2
22
+
𝜎4
𝑆4
22 × 42
+
𝜎6
𝑆6
22 × 42 × 62
+
𝜎8
𝑆8
22 × 42 × 62 × 82
+ ⋯ }
+𝐵
{
𝐼𝑛𝑆 (1 +
𝜎2
𝑆2
22
+
𝜎4
𝑆4
22 × 42
+
𝜎6
𝑆6
22 × 42 × 62
+
𝜎8
𝑆8
22 × 42 × 62 × 82
+ ⋯ )
−
𝜎2
𝑆2
22
−
𝜎4
𝑆4
23 × 42
−
𝜎6
𝑆6
43 × 63
+
𝜎8
𝑆8
23 × 63 × 83
+ ⋯
}
(22)
Using boundary condition (13) and (14) then setting 𝐵 = 0, 𝑖𝑛 (21) above yields
𝐴 =
𝜃𝑎
𝑦1(1)
(23)
𝑤ℎ𝑒𝑟𝑒 𝑦1(1) = 1 +
𝜎2
𝑆2
22
+
𝜎4
𝑆4
22 × 42
+
𝜎6
𝑆6
22 × 42 × 62
+
𝜎8
𝑆8
22 × 42 × 62 × 82
+ ⋯
which is the complementary solutions
𝑉𝑡(𝑆) =
𝜃𝑎
𝑦1(1)
{1 +
𝜎2
𝑆2
22
+
𝜎4
𝑆4
22 × 42
+
𝜎6
𝑆6
22 × 42 × 62
+
𝜎8
𝑆8
22 × 42 × 62 × 82
+ ⋯ } (24)
Solving the Right-Hand Side (RHS) of (12) we have the following
Let 𝑉
𝑝(𝑆) = 𝐴0 + 𝐴1𝑆2
+ 𝐴2𝑆4
+ 𝐴3𝑆6
+ 𝐴4𝑆8
(25)
𝑉
𝑝
′
(𝑆) = 2𝐴1𝑆 + 4𝐴2𝑆3
+ 6𝐴3𝑆5
+ 8𝐴4𝑆7
(26)
𝑉
𝑝
′′
(𝑆) = 2𝐴1 + 12𝐴2𝑆2
+ 30𝐴3𝑆4
+ 56𝐴4𝑆6
(27)
Putting (25) – (27) into (12) gives
⇒ (4𝐴1 − 𝜎2
𝐴0)𝑆 + (16𝐴1 − 𝜎2
𝐴1)𝑆3
+ (36𝐴3 − 𝜎2
𝐴2)𝑆5
+ (64𝐴4 − 𝜎2
𝐴3)𝑆7
− (𝜎2
𝐴4)𝑆9
≡ −𝑃 − 𝐺𝑟 (1 + (
𝐾𝑆
𝑟𝑡
) 2
+ (
𝐾𝑆
𝑟𝑡
) 4
+ (
𝐾𝑆
𝑟𝑡
) 6
+ (
𝐾𝑆
𝑟𝑡
) 8
+ ⋯ )
⇒ 4𝐴1 − 𝜎2
𝐴0 = −𝑃 − 𝐺𝑟𝐴 (28)
16𝐴2 − 𝜎2
𝐴1 = −𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 2
(29)
36𝐴3 − 𝜎2
𝐴2 = −𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 4
(30)
64𝐴4 − 𝜎2
𝐴3 = −𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 6
(31)
−𝜎2
𝐴4 = 1𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 4
(32)
From (28) – (32) yields
𝐴0 = −
1
𝜎2
[−4𝐴1 − 𝑃 − 𝐺𝑟𝐴], 𝐴1 = −
1
𝜎2
[−16𝐴2 − 𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 2],
𝐴2 = −
1
𝜎2
[−36𝐴3 − 𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 4], 𝐴3 = −
1
𝜎2
[−64𝐴4 − 𝐺𝑟 (
𝐾𝑆
𝑟𝑡
) 6],
𝐴4 = −
1
𝜎2
𝐺𝑟 (
𝐾𝑆
𝑟𝑡
)
8
To get the constant 𝐶,

5. An Empirical Approach for the Variation in Capital Market Price Changes
Nwobi and Azor 168
𝑉
𝑐(𝑆) = {1 +
𝜎2
𝑆2
4
+
𝜎4
𝑆4
(2 × 4)2
+
𝜎6
𝑆6
2 × 4 × 6)2
+
𝜎8
𝑆8
2 × 4 × 6 × 8)2
+ ⋯ }
= {𝑐 +
𝑐𝜎2
𝑆2
4
+
𝑐𝜎4
𝑆4
(2 × 4)2
+
𝑐𝜎6
𝑆6
2 × 4 × 6)2
+
𝑐𝜎8
𝑆8
2 × 4 × 6 × 8)2
+ ⋯ }
𝑉(𝑆) = 𝑉
𝑐 + 𝑉𝑃 which is the general solution
𝑉(𝑆) = {𝑐 +
𝑐𝜎2
𝑆2
4
+
𝑐𝜎4
𝑆4
(2 × 4)2
+
𝑐𝜎6
𝑆6
2 × 4 × 6)2
+
𝑐𝜎8
𝑆8
2 × 4 × 6 × 8)2
+ ⋯ } + {𝐴0 + 𝐴1𝑆2
+ 𝐴2𝑆4
+ 𝐴3𝑆6
+ 𝐴4𝑆8}
Collecting like terms gives a general solution
𝑉(𝑆) = 𝑐 + 𝐴0 + (
𝑐𝜎2
4
+ 𝐴1)𝑆2
+ (
𝑐𝜎4
(2 × 4)2
+ 𝐴2)𝑆4
+ (
𝑐𝜎6
(2 × 4 × 6)2
+ 𝐴3)𝑆6
+ (
𝑐𝜎8
(2 × 4 × 6 × 8)2
+ 𝐴4)𝑆8
(33)
Differentiating (33) with respect to S gives
𝑉′(𝑆) = 2 (
𝑐𝜎2
4
+ 𝐴1)𝑆 + 4 (
𝑐𝜎4
(2 × 4)2
+ 𝐴2)𝑆3
+ 6 (
𝑐𝜎6
(2 × 4 × 6)2
+ 𝐴3)𝑆5
+ 8 (
𝑐𝜎8
(2 × 4 × 6 × 8)2
+ 𝐴4)𝑆7
(34)
Differentiating (34) with respect to S gives
𝑉′′(𝑆) = 2 (
𝑐𝜎2
4
+ 𝐴1) + 12 (
𝑐𝜎4
(2 × 4)2
+ 𝐴2)𝑆2
+ 30 (
𝑐𝜎6
(2 × 4 × 6)2
+ 𝐴3)𝑆4
+ 56 (
𝑐𝜎8
(2 × 4 × 6 × 8)2
+ 𝐴4)𝑆6
(35)
where 𝐴0, 𝐴1,𝐴2, 𝐴3 𝑎𝑛𝑑 𝐴4 are constants
Symmetric Characteristic of the Model
The solution of stock quantity or variable follows a normal distribution since it has symmetric property. Hence the solution
will be subjected to analysis in order to ascertain the symmetric characteristics of stocks. Therefore, we state the theorem
as follows:
THEOREM 1 (Symmetric characteristics): The solution (33) is symmetrical about the centre of the curve. That is
𝑑𝑉
𝑑𝑆 𝑆=0
Proof
To show that the moment of stock variable is symmetrical.
From (33)
𝑉(𝑆) = 𝑐 + 𝐴0 + (
𝑐𝜎2
4
+ 𝐴1)𝑆2
+ (
𝑐𝜎4
(2 × 4)2
+ 𝐴2)𝑆4
+ (
𝑐𝜎6
(2 × 4 × 6)2
+ 𝐴3)𝑆6
+ (
𝑐𝜎8
(2 × 4 × 6 × 8)2
+ 𝐴4)𝑆8
Differencing 𝑉(𝑆) with respect to S gives
𝑑𝑉
𝑑𝑆
= 2 (
𝑐𝜎2
4
+ 𝐴1)𝑆 + 4 (
𝑐𝜎4
(2 × 4)2
+ 𝐴2)𝑆3
+ 6 (
𝑐𝜎6
(2 × 4 × 6)2
+ 𝐴3)𝑆5
+8 (
𝑐𝜎8
(2 × 4 × 6 × 8)2
+ 𝐴4)𝑆7
𝑑𝑉
𝑑𝑆
= 2 (
𝑐𝜎2
4
+ 𝐴1)(0) + 4 (
𝑐𝜎4
(2 × 4)2
+ 𝐴2)(0)3
+ 6 (
𝑐𝜎6
(2 × 4 × 6)2
+ 𝐴3)(0)5
+8 (
𝑐𝜎8
(2 × 4 × 6 × 8)2
+ 𝐴4)(0)7
.
That is
𝑑𝑉
𝑑𝑆 𝑆=0
Therefore, the claim is true, which touches around a vertical axis of symmetry
= 0
= 0

6. An Empirical Approach for the Variation in Capital Market Price Changes
Int. J. Stat. Math. 169
RESULTS
This section presents the computational results for the problem transformed in (2) – (5) whose solution is in (33)-
(35) in Section 2.
The tabular solutions in Tables1-6 are obtained by substituting the parameter values C= 55, 𝐴0 = 0.03, 𝐴1 =
0.02, 𝐴2 = 0.023, 𝐴3 = 0.015 𝑎𝑛𝑑 𝐴4 = 0.04 into (33)-(35) in Section 2.
The graphical solutions are obtained using MATHEMATICA to plot the tabular solutions.
Tables 1 – 2: Equilibrium Prices Displaying Different Stock Volatilities
Table 1 Table 2
S Volatility V(S) S Volatility V(S)
0.1 1.25 55.25 0.1 2.5 55.89
0.2 1.25 55.89 0.2 2.5 58.52
0.3 1.25 56.98 0.3 2.5 63.03
0.4 1.25 58.53 0.4 2.5 69.67
0.5 1.25 60.54 0.5 2.5 78.71
0.6 1.25 63.05 0.6 2.5 90.61
0.7 1.25 66.09 0.7 2.5 105.94
0.8 1.25 69.70 0.8 2.5 125.44
0.9 1.25 73.92 0.9 2.5 150.06
1.0 1.25 78.80 1.0 2.5 181.03
1.1 1.25 84.43 1.1 2.5 219.88
1.2 1.25 90.89 1.2 2.5 268.53
Tables 3 – 4: Equilibrium Prices Displaying the First Rate of Change as a Results of Price Increase
Table 3 Table 4
S Volatility 𝑽′(𝑺) S Volatility 𝑽′(𝑺)
0.1 1.25 4.31 0.1 2.5 17.33
0.2 1.25 8.67 0.2 2.5 35.47
0.3 1.25 13.13 0.3 2.5 55.29
0.4 1.25 17.75 0.4 2.5 77.73
0.5 1.25 22.59 0.5 2.5 103.89
0.6 1.25 27.70 0.6 2.5 135.04
0.7 1.25 33.15 0.7 2.5 172.72
0.8 1.25 39.03 0.8 2.5 218.84
0.9 1.25 45.43 0.9 2.5 275.71
1.0 1.25 52.47 1.0 2.5 346.22
1.1 1.25 60.29 1.1 2.5 433.91
1.2 1.25 69.07 1.2 2.5 543.19
Tables 5 – 6: Equilibrium Prices Displaying 2nd
Rate of Changes as a Result of Price Increase
Table 5 Table 6
S Volatility 𝑽′′(𝑺) S Volatility 𝑽′′(𝑺)
0.1 1.25 43.26 0.1 2.5 175.96
0.2 1.25 44.03 0.2 2.5 188.32
0.3 1.25 45.33 0.3 2.5 209.64
0.4 1.25 47.17 0.4 2.5 241.04
0.5 1.25 49.61 0.5 2.5 284.18
0.6 1.25 52.70 0.6 2.5 341.34
0.7 1.25 56.52 0.7 2.5 415.54
0.8 1.25 61.22 0.8 2.5 510.65
0.9 1.25 66.97 0.9 2.5 631.57
1.0 1.25 74.01 1.0 2.5 784.43
1.1 1.25 82.65 1.1 2.5 976.75
1.2 1.25 93.32 1.2 2.5 1217.73

7. An Empirical Approach for the Variation in Capital Market Price Changes
Nwobi and Azor 170
Figure 1: Graphical Solution of the First Equilibrium Price
0.2 0.4 0.6 0.8 1.0 1.2
S
Figure 2: Equilibrium Price with its first Rate of Change
0.2 0.4 0.6 0.8 1.0 1.2
Volatility=1.25
Volatility=2.5
Volatility=1.25
Volatility=2.5
S

8. An Empirical Approach for the Variation in Capital Market Price Changes
Int. J. Stat. Math. 171
0.2 0.4 0.6 0.8 1.0 1.2
S
Figure 3: Equilibrium Price with its second rate of change
DISCUSSION OF RESULTS
In Tables 1 and 2, it can be observed that an increase in stock volatility increases equilibrium price. The two
plots start at a particular point and grow exponentially; which shows the rate of equilibrium price throughout
the trading days. The changes are as a result of volatilities which serve as a guide and an eye-opener to
investors on how to manage their portfolio of investments.
It is clear that increase in stock volatility increases equilibrium price. The changes in the two plots seem to
start at a particular point before it starts deviating differently. These changes are as a result of little changes
in the stock market business which can be detrimental or of good profit margin throughout the trading days.
See Tables 3 and 4 respectively
Tables 5 and 6 are the variation of volatilities against equilibrium price on second rate of change. It shows
that increase in stock volatility increases the equilibrium prices. This remark is beneficial to investors in
trading business in order to maximize profit.
A critical look at the two plots, show upward trends with greater changes in the equilibrium price.
CONCLUSION
In this paper, a non-linear ordinary differential equation with volatility parameter in the model is considered herein. The
proposed system of equation was solved analytically and solution verified graphically. The graphical results showed the
behaviour of the system as follows:
(i) Equilibrium price increases with increasing volatility parameter (ii) increase in stock volatility leads to an increase in
equilibrium price for first rate of change and(iii) Increase in stock volatility increases the equilibrium for second rate of
change. A theorem was developed and proved to show that the proposed model follows a normal distribution and obeys
the concept of financial modelling. However, introducing stochastic term in the system will be another important area to
explore.
Volatility=1.25
Volatility=2.5

9. An Empirical Approach for the Variation in Capital Market Price Changes
Nwobi and Azor 172
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Accepted 18 May 2022
Citation: Nwobi FN, Azor PA (2022). An Empirical Approach for the Variation in Capital Market Price Changes.
International Journal of Statistics and Mathematics, 8(1): 164-172.
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