An Empirical Approach for the Variation in Capital Market Price Changes

An Empirical Approach for the Variation in Capital Market Price Changes

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  An Empirical Approachfor 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: [email protected]; Co-Author Email: Email: [email protected] 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
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  An Empirical Approachfor 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
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  An Empirical Approachfor 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
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  An Empirical Approachfor 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,
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  An Empirical Approachfor 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
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  An Empirical Approachfor 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 Approachfor 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 Approachfor 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.
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