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The behaviour of stocks on the Ghana stock exchange is examined to show that stock prices on the exchange are subject to sudden price changes. It is shown that such unexpected events and uncertainties affecting trading on the exchange cannot be modeled solely by the conventional geometric Brownian motion outlined in the Black-Scholes model. A new concise and simpler approach is developed to derive the Jump diffusion model and consequently, its suitability to model stocks on the exchange is emphasized and given rigorous treatment. The model is subsequently used to predict the behaviour of stocks using historical stock prices as input parameters. The simulated stock returns are compared to actual returns to determine the model’s suitability to predict the market. The results show that the jump diffusion model is appropriate in predicting the behaviour of approximately 25% percent of stocks listed on the exchange.

The modeling of stock price behaviour has generally been realized through the use of diffusion processes. The fundamental model for a stock following a diffusion process consists of a deterministic drift and stochastic parts and it’s known generally as Geometric Brownian Motion (GBM). Under GBM the stock returns are lognormally distributed largely ensuring non-negative prices. Although GBM has been successful in modelling stock price processes, it is unable to account for the presence of sudden or occasional jumps, which has been observed in many stock markets. Recent empirical evidence from the Ghana Stock Exchange (GSE) suggests that some stocks are characterized by sudden price fluctuations. These observations have motivated the need to critically examine models used to price the behaviour of stocks on the Exchange. Empirical data from December 2016 to December 2017 show that the Ghana Stock Exchange Composite Index recorded a return of 52.73%. Notably, some stocks posted unusually high returns-Benso Oil Palm Plantation Limited (194.23%), Ghana Oil Company Limited (144.55%), Standard Chartered Bank Limited (106.8%), HFC Bank Limited (85.33%) and Total Petroleum Ghana Limited (76.5%). Such return behaviour implicitly suggests the presence of sudden price changes giving rise to high skewness and kurtosis in the return distribution of the stock. Figures 1-3 show the 5-year return distribution of some stocks listed on GSE. The sudden peaks represent jumps in the stock prices distribution.

Empirical Evidence

Figures 1-3 show the log-return graphs of three stocks listed on GSE; Standard Chartered Bank Ghana, Ghana Commercial Bank and Ecobank Ghana respectively, from January 2013 to December 2018. The log-return graphs show increasing activity in the stock market leading to observed higher number of jumps from 2016 to 2018. A typical example is the stock of Standard Chartered Bank in which there were no jumps from 2013 to middle of 2016. However, from December 2016 to June 2018, there were at least nine jumps. Similar jump patterns are observed in Ecobank stock price as shown in

Skewness S = 1 ( n − 1 ) σ 3 ∑ i = 1 n ( X i − X ¯ ) 3 (1)

kurtosis K = 1 ( n − 1 ) σ 4 ∑ i = 1 n ( X i − X ¯ ) 4 (2)

where σ is the standard deviation of a stock’s return.

For the standard normal density, S = 0 and K = 3 in Equations (1) and (2) respectively. If S and K are different from 0 and 3 then the distribution will be called leptokurtic and will have higher peaks and heavier (fat) tails than those of the normal distribution. For stocks listed on Ghana stock exchange in 2017, the skewness and kurtosis as well as the means and volatilities are given in

The results in

Stock | African Champion | Anglo Gold | Aluworks | Ayrton Drug | Benso | Cal Bank | Clydestone | Camelot | Cocoa Processing | |
---|---|---|---|---|---|---|---|---|---|---|

Skewness | 1.561 | 0.000 | 0.953 | −0.257 | −1.895 | −1.004 | 0.796 | 0.678 | −0.658 | |

Kurtosis | 3.226 | 0.000 | −0.193 | −1.156 | −0.581 | 21.159 | −1.369 | −1.271 | −1.571 | |

Mean | 0.00 | 0.00 | 0.0005428 | −0.0007411 | 0.004387 | 0.001428 | 0.00 | −0.00035 | 4.51E−19 | |

Volatility | 0.00 | 0.00 | 0.0168687 | 0.0116244 | 0.018568 | 0.031976 | 0.00 | 0.005548 | 0.072554 | |

Stock | Ecobank Ghana | Enterprise Group | Ecobank Transnational | Fan Milk | GCB | Guinness | GOIL | Golden Star | Golden Web | |

Skewness | 1.496 | 1.348 | 0.092 | −0.696 | −0.688 | −3.775 | −0.120 | 0.310 | −0.128 | |

Kurtosis | 0.692 | −1.110 | 5.159 | 0.646 | −1.244 | 6.718 | −0.583 | −1.574 | −1.584 | |

Mean | 0.00063555 | 0.00176 | 0.0019106 | 0.0018604 | 0.001376 | 0.000952 | 0.003635 | −0.00011 | 0.00 | |

Volatility | 0.01326269 | 0.018091 | 0.0341023 | 0.0119202 | 0.011604 | 0.010731 | 0.011702 | 0.001723 | 0.00 | |

Stock | HFC | Mechanical Lloyd | Pioneer Kitchenware | Produce Buying | PZ Cussons | Standard Chartered | SIC | Starwin Products | SG-SSB | |

Skewness | −1.139 | −0.465 | −0.174 | −0.183 | −0.001 | 0.270 | −0.157 | −0.175 | −0.664 | |

Kurtosis | −1.021 | 26.342 | −1.974 | −0.527 | −0.022 | −0.026 | 12.595 | −1.302 | 0.092 | |

Mean | 0.00250807 | −0.00372 | 0.00 | −2.031E−18 | −0.00039 | 0.002954 | −0.00074 | 2.71E−18 | 0.001137 | |

Volatility | 0.02179135 | 0.020868 | 0.00 | 0.04573 | 0.006077 | 0.021696 | 0.037589 | 0.084415 | 0.010978 | |

Stock | Sam Woode | Trust Bank | TOTAL | Transaction Solutions | Tullow Oil | Unilever | UT Bank | Mega African | ADB | Access Bank |

Skewness | −1.547 | −0.911 | 5.022 | 2.691 | −7.792 | 0.477 | −0.554 | −0.292 | 0.174 | −0.333 |

Kurtosis | 1.425 | −1.703 | 24.354 | 5.252 | −1.451 | −1.579 | −1.368 | −1.244 | −1.776 | −0.872 |

Mean | 0.00090709 | 0.001208 | 0.0023096 | 0.00 | −0.00183 | 0.001672 | 0.003984 | −1.4E−05 | 0.001715 | −5E−05 |

Volatility | 0.01422711 | 0.023732 | 0.0134325 | 0.00 | 0.015226 | 0.01281 | 0.066734 | 0.00015 | 0.012896 | 0.029841 |

The objective is to use the jump diffusion to model the dynamics of stocks behavior on the exchange. The Jump Diffusion Model is thus developed and used to simulate stock prices. The return statistics are compared with that of the market prices to determine if the model fit the market data. This paper will specifically;

· Develop the Jump Diffusion Model from first principles;

· Use Jump Diffusion Model to simulate the stock price behaviour with data from Ghana Stock Exchange;

· Identify stocks for which the Jump Diffusion Model is a suitable predictive model.

Literature ReviewThe literature on stock price modeling is extensive. The foundations were laid by Bachelier [

Three approaches dominate the research to address the presence of jumps observed in empirical data: jump-diffusion, stochastic volatility and the Constant Elasticity of Variance models. Merton [

In stochastic volatility models the return of the stock price follows the diffusion process but the constant volatility is replaced by a stochastic volatility. Examples include Hull and White [

The Constant Elasticity of Variance (CEV) model was first proposed by Cox and Ross [

To implement simulation of the jump diffusion models, Glasserman [

In a risky stock, the stock price S ( t ) is assumed to follow the lognormal process and is modelled by GBM as

d S ( t ) S ( t ) = μ d t + σ d W ( t ) (3)

where μ is the expected return on the stock, σ is the standard deviation of the return and W ( t ) is the standard Brownian motion or Wiener process with mean 0 and standard deviation t.

The solution to Equation (3) is

S ( t ) = S ( 0 ) e ( μ − 1 2 σ 2 ) t + σ W ( t ) (4)

In modelling the dynamics of the stock price in the jump diffusion setting the trajectory of the stock consist of two components. The first component is driven by the normal price changes due to the effect of economic factors such as disequilibrium in supply and demand on the market. This component is expressed by a standard Brownian motion with a constant drift, a constant volatility and almost continuous paths. The second component is described by changes of the stock price influenced by new available information. This jump part of the stock process is outlined as a Poisson process and by extension a compound Poisson process.

Consider the stock price S ( t ) , understood to be right-continuous function with left limits so that S ( t − ) represents the value of S ( t ) just before a possible jump at t. We write S ( t ) as the limit from the left, i.e.

S ( t − ) = lim S ↑ t S ( u ) (5)

Suppose that in the small-time interval Δ t the stock price jumps by Y ( t ) so that it jumps from S ( t ) to Y ( t ) S ( t ) as shown in

The percentage change in the stock price is thus given by

Δ S ( t ) S ( t ) = Y ( t ) S ( t ) − S ( t ) S ( t )

Δ S ( t ) S ( t ) = Y ( t ) S ( t ) S ( t ) − S ( t ) S ( t )

Δ S ( t ) S ( t ) = Y ( t ) − 1

In the infinitesimal limit

d S ( t ) S ( t ) = Y ( t ) − 1 (6)

It is realized that Y ( t ) ’s are non-negative random variables modeling the distribution of the jump sizes such that they are independently identically distributed and the trajectories of jumps sizes are piecewise constant, right-continuous with left limits. Equation (6) gives the relative jump amplitude or percentage change in stock price as (Y − 1).

In addition to the jump sizes the inter-arrival times of the jumps needed to be modeled. The arrival times of jumps t 1 , t 2 , ⋯ , t m is generated by a Poisson process N ( t ) independent of the jump sizes Y ( t ) with average arrival times or intensity

λ . By combining the jump times and the jump sizes the jump part of the stock price process is denoted by the compound Poisson process

S ( t ) = ∑ j = 1 N ( t ) Y i (7)

The consequence of Equation (7) is that in a small-time interval d t the likelihood of the Poisson event can be described as follows:

· Probability that the stock price jumps once

P r { the stock price jumps once } = P r { that the Poisson event d N ( t ) occurs } = P r { d N ( t ) = 1 } = λ d t

· Probability that the stock price does not jump

P r { the stock price does not jump } = P r { the Poisson event d N ( t ) does not occur } = P r { d N ( t ) = 0 } = 1 − λ d t

· Probability that the stock price jumps more than once

P r { the stock price jumps more than once } = P r { d N ( t ) ≥ 2 } = 0

· The random variables Y i defining the jump sizes are assumed to be normally distributed with mean m and variance δ 2 and has density f given by

f ( y ) = 1 2 π δ i exp { ( y − m ) 2 2 δ i 2 }

· The relative jump sizes ( Y j − 1 ) are also lognormally distributed with expected value and variance given by

E [ Y − 1 ] = e μ + 1 2 δ 2 − 1 = k

V a r [ Y ] = e 2 μ + δ 2 ( e δ 2 − 1 )

· The jumps occur at times t 1 , t 2 , ⋯ , t m and the intervals between jumps (the waiting times) are exponentially distributed. For t > 0 , N ( t ) has the Poisson distribution with parameter λ t , that is the probability of increment of jump times is given by the Poisson distribution

P r [ d N ( t ) = k ] = ( λ d t ) k e − λ d t k ! , k = 0 , 1 , ⋯ (8)

To model the dynamics of a stock price in the jump diffusion model, it is realized that the stock price path is driven by two stochastic processes: The first is the diffusion part driven by continuous Brownian motion and modeled by lognormal geometric Brownian motion. The second is the jump part driven by Poisson jumps and modelled by the compound Poisson process derived above. Hence the equation for the stock price path is given by

d S ( t ) = μ S ( t − ) d t + σ S ( t − ) d W ( t ) + S ( t − ) ∑ i = 1 N ( t ) Y i

where N ( t ) is a Poisson process independent of the Wiener process W ( t ) with constant arrival rate λ .

μ is the constant instantaneous expected rate of return for the stock;

σ is a constant volatility parameter of the stock.

The return distribution of the stock is given by

d S ( t ) S ( t − ) = μ d t + σ d W ( t ) + ∑ i = 1 N ( t ) Y i (9)

In a diffusion process the market model is complete and the existence and uniqueness of an equivalent martingale measure is guaranteed. In this framework the drift is determined by the condition that discounted stock price process is a martingale. The diffusion process under the equivalent martingale measure is given by

d S ( t ) = μ S ( t ) d t + σ S ( t ) d W ( t ) ℚ (10)

To ensure that S ( t ) is a martingale the jump part of the process is compensated by λ k d t such that the expected relative price change E [ d S S ] in the interval d t is E [ Y i − 1 ] = E [ Y ] − 1 = λ k . The term λ k compensates the jumps in the sense that the process Y ( t ) − λ k t is a martingale. Hence, Equation (6) can be written as

d S ( t ) S ( t − ) = μ d t + σ d W ( t ) + ∑ i = 1 N ( t ) Y i − λ k d t

d S ( t ) S ( t − ) = ( μ − λ k ) d t + σ d W ( t ) + ∑ i = 1 N ( t ) Y i (11)

Solution to the Jump Diffusion Model

Suppose S ( t ) is a stochastic process following the jump diffusion process in Equation (10) which can be rewritten as

d S ( t ) S ( t − ) = ( μ − λ k ) d t + σ d W ( t ) + h d N ( t )

Let f ( t ) = ln S ( t ) , then in between jumps, the log expansion of the Ito process is given by

d f ( t , S ( t ) ) = { ∂ f ∂ t ( t , S ( t ) ) + ( μ − λ k ) ∂ f ∂ x ( t , S ( t ) ) + 1 2 σ 2 ∂ 2 f ∂ x 2 ( t , S ( t ) ) } d t + σ ∂ f ∂ x ( t , S ( t ) ) d W ( t )

At the jump time t, N ( t ) has the jump size of Δ N ( t ) = N ( t ) − N ( t − ) = 1 and the induced jump is given by

Δ Z ( t ) = f ( t , S ( t ) ) − f ( t − , S ( t − ) ) .

The Ito expansion of JDM is now given by

d ln S ( t ) = ∂ ln S ( t ) ∂ t d t + ( μ − λ k ) ∂ ln S ( t ) ∂ S ( t ) d t + σ 2 ( t ) S ( t ) 2 ∂ 2 ln S ( t ) ∂ S ( t ) 2 d t + σ ( t ) S ( t ) ∂ ln S ( t ) ∂ S ( t ) d W ( t ) + [ ln Y ( t ) S ( t ) − ln S ( t ) ]

d ln S ( t ) = ( μ − λ k ) S ( t ) 1 S ( t ) d t + σ 2 ( t ) S ( t ) 2 2 ( 1 S ( t ) 2 ) d t + σ ( t ) S ( t ) 1 S ( t ) d W ( t ) + [ ln Y ( t ) + ln S ( t ) − ln S ( t ) ]

d ln S ( t ) = ( μ − λ k ) d t − σ 2 ( t ) 2 d t + σ ( t ) d W ( t ) + ln Y ( t )

ln S ( t ) = ( μ − σ 2 2 − λ k ) t + σ ( t ) [ W ( t ) − W ( t ) ] + ∑ i = 1 N ( t ) ln Y ( t )

S ( t ) = S ( 0 ) exp [ ( μ − σ 2 2 − λ k ) t + σ ( t ) W ( t ) ] + exp [ ∑ i = 1 N ( t ) ln Y ( t ) ]

S ( t ) = S ( 0 ) exp { ( μ − σ 2 2 − λ k ) t + σ ( t ) W ( t ) } ∏ i = 1 N ( t ) ln Y i ( t ) (12)

From Equation (12) an explicit solution for the JDM is

S ( t ) = S ( 0 ) exp [ ( μ − σ 2 2 − λ k ) t + σ ( t ) W ( t ) ] ∏ i = 1 N ( t ) ln Y i ( t )

( S ( t ) S ( 0 ) ) = e σ W ( t ) + ( μ − λ k − σ 2 2 ) t ∏ i = 1 N ( t ) ln Y i ( t )

log ( S ( t ) S ( 0 ) ) = σ W ( t ) + ( μ − λ k − σ 2 2 ) t + ∑ i = 1 j V i

where V = ln Y ( t ) (this is Kou’s substitution).

Hence the log-return density ( log ( S ( t ) S ( 0 ) ) ) is normally distributed with mean ( σ W ( t ) + ( μ − λ k − σ 2 2 ) t ) and variance ( σ 2 t + j δ ) . That is

log ( S ( t ) S ( 0 ) ) ~ N ( σ W ( t ) + ( μ − λ k − σ 2 2 ) t , σ 2 t + j δ ) .

The related probability density at time t is

ψ t ( y ) = e − λ t ∑ j = 0 ∞ ( λ t ) j exp { − ( y − ( μ − λ k − σ 2 2 ) t − j m ) 2 2 ( σ 2 t + j δ 2 ) } j ! 2 π ( σ 2 t + j δ 2 ) (13)

· Merton’s Representation

Merton formally represented his model of the dynamics of the stock price as

d S ( t ) S ( t − ) = ( α − κ λ ) d t + σ d Z + d q ( t ) (14)

where

α is the instantaneous expected return on the stock;

σ 2 is the instantaneous variance of the return on the stock;

λ is the mean number of arrivals per unit time;

d Z is a standard Wiener process;

q ( t ) is the Poisson process independent Wiener process d Z such that in a time interval h,

P r { the jumps event does not occur in the interval ( t , t + h ) } = 1 − λ h + o ( h )

P r { the jump event occurs once in the interval ( t , t + h ) } = λ t + o ( h )

P r { the jumps event occurs more than once in the interval ( t , t + h ) } = o ( h )

where o ( h ) is the asymptotic order symbol defined by ψ ( h ) = o ( h ) if lim h → 0 [ ψ ( h ) / h ] = 0

κ ≡ E ( Y − 1 ) = E ( Y ) − 1 .

E is the expectation operator over the random variable Y.

Merton assumed that if α , λ , κ , σ are constants then Equation (14) has the solution

S ( t ) = S ( 0 ) exp { ( α − σ 2 2 − λ k ) t + σ Z ( t ) } Y ( n )

where

Y ( n ) = { 1 if n = 0 ∏ j = 1 n Y j if n ≥ 1

· Kou’s Double Exponential Model

In Kou’s model, the stochastic differential representing the return of the stock under the physical measure ℙ is given as

d S ( t ) S ( t − ) = μ d t + σ d W ( t ) + d ( ∑ i = 1 N ( t ) ( V i − 1 ) )

where W ( t ) is a standard Brownian motion, N ( t ) is a Poisson process with rate λ , and V i is a sequence of independent identically distributed (i.i.d.) nonnegative random variables such that Y = log V has an asymmetric double exponential distribution with the density

f ( y ) = p ⋅ η 1 e − η 1 y I ( y ≥ 0 ) + q ⋅ η 2 e − η 2 y I ( y < 0 )

η 1 > 1 , η 2 > 0

where p , q ≥ 0 , p + q = 1 , represent the probabilities of upward and downward jumps.

log ( V ) = Y = ︷ d { ξ + with probability p − ξ − with probability q

where ξ + and ξ − are exponential random variables with means 1 / η 1 and 1 / η 2 , respectively, and the notation = ︷ d means equal in distribution. The solution to Kou’s model gives the dynamics of the stock price as

S ( t ) = S ( 0 ) exp { ( μ − σ 2 2 ) t + σ ( t ) W ( t ) } ∏ i = 1 N ( t ) V i

Monte Carlo simulation of the jump diffusion model is carried out by simulating paths of a finite sample of the process S ( t ) ; t ∈ [ 0 , T ] . Sample paths of S ( t ) is obtained over fixed set of dates 0 = t 0 < t 1 < ⋯ < t n = T such that the observation times are equally spaced and Δ t = t i + 1 − t i = 1 day .

The discretization representation

S ( t i + 1 ) = S ( t i ) e ( μ − σ 2 2 ) ( t i + 1 − t i ) + σ [ W ( t i + 1 ) − W ( t i ) ] ∏ j = N ( t i + 1 ) + 1 N ( t i + 1 ) Y j ( t ) (15)

is employed.

Direct simulation from the representation in Equation (15) is possible but in this case, it is appropriate to set X ( t ) = ln S ( t ) and write

X ( t i + 1 ) = X ( t i ) + ( μ − σ 2 2 ) + ( t i + 1 − t i ) + σ ( t ) [ W ( t i + 1 ) − W ( t i ) ] + ∑ j = N ( t i + 1 ) + 1 N ( t i + 1 ) ln Y j (16)

where the product over j is equal to 1 if N ( t i + 1 ) = N ( t i ) . This recursion replaces products with sums and it is preferable.

The procedure can be summarized into the following steps:

1) Generate Z ∼ N ( 0 , 1 ) ;

2) Generate Z ∼ Poisson ( λ ( t i + 1 − t i ) ) if n = 0 , set M = 0 and go to Step 4;

3) Generate ln Y 1 , ⋯ , ln Y N from the common distribution and set M = ln Y 1 + ⋯ + ln Y N ;

4) Set X ( t i + 1 ) = X ( t i ) + ( μ − 1 2 σ 2 ) Δ t + σ Δ t Z + M .

The parameters for the simulation are estimated from GSE historical market data, specifically the daily log-returns of closing stock prices from January 3, 2017 to December 29, 2017. There are 247 daily closings for stock prices in the period and Δ t is set as

Δ t = 1 TradingDays = 1 247 = 0.004

The JDM is a five-set parameter model admitting the mean return rate μ , volatility of the diffusion process σ 2 , the log-return mean of the jump sizes m, variance of the jump sizes δ 2 and the jump intensity of the Poisson process λ . In the simulation procedure the return data is divided into two groups D and J . Group D includes log-returns with absolute value less than ε . For this group there are no jumps and μ and σ are estimated from the historical data. Group J represents the jumps. In determining whether a jump has occurred, the decision rule is that a jump occurs if the absolute value of the log-return is larger than some positive value ε , In this study we set ε = 0.1 . This means that if there is a daily price change of 10% or more it is considered as a jump from the previous price. The mean jump height m is determined from the average of the jump sizes over the year. δ is the standard deviation from the mean jump sizes. The parameters for the simulation for all stocks are given in TableA1 in Appendix. For example, the estimated parameters for the Standard Chartered Bank are: μ = 0.00295 , σ 2 = 0.0217 , λ = 4 , m = 0.12 , δ 2 = 0.01 and for SIC Insurance, μ = − 0.0074 , σ 2 = 0.037589 , λ = 12 , m = 0.14 , δ 2 = 0.03 .

As stated earlier, if the stock price follows GBM, the empirical returns should be at least moderately close to a normal density; otherwise the returns will have jumps or will follow some other appropriate distribution. Consequently, the properties of the first four moments, including skewness and excess kurtosis are employed to determine whether a simulated stock follows JDM or GBM. The moments and properties of skewness and kurtosis of the simulated stock prices are thus compared to the observed market data. The parameters of the simulated stock paths and the realized stock prices are presented in TableA2 in the Appendix. Stocks listed 1 - 9 in the table have means and variances that match the parameters of the actual stocks. In addition, the simulated paths also match the high skewness and excess kurtosis shown by the actual stock paths. This shows that the JDM model is more suited to modelling these stocks. Stocks listed 10 - 25 on the other hand, have means and variances that match the parameters of the actual stocks. In addition, the simulated paths also match the low skewness and low kurtosis shown by the actual stock paths. This shows that the GBM is more suited to modelling these stocks. These results are further established by the graphs of the simulated stocks.

In this paper, significant progress has been made towards simplifying the mathematics of developing the jump diffusion model. The results from theoretical and empirical studies of stock behaviour have also given further insights into the understanding of the stock market in Ghana. When the theoretical model is used to simulate the log returns of stock prices and compared to the actual stock data in 2018, it is determined that out of the thirty-six (36) listed stocks, nine (9) follow paths that can be modeled by jump diffusion models. Nineteen stocks were found to follow continuous paths and can be modeled as by diffusion models. Thus, the theoretical models simulating the path of the stocks show that the Jump Diffusion Model and Geometric Brownian Motion are possible models for examining stocks on the exchange. The results confirm our assertion that no single model can be used to predict stock behaviour on the exchange.

Ten stocks exhibited paths that are neither purely continuous nor continuous with jumps. These stocks have prices that remain constant for long periods and only change in price at intermittent periods. They therefore have zero or values close to zero in their return distributions and thus cannot be modelled by any of the two models. This shows that the present state of understanding of the stock behaviour on the exchange is far from being conclusive. Further studies are required to completely model the behaviour of the stocks. To pursue for further research, a new model which can track the trajectories of such stocks and can be used to price derivatives on the simulated prices as well as to the real data in the Ghanaian market is required.

The authors declare no conflicts of interest regarding the publication of this paper.

Antwi, O., Bright, K. and Wereko, K.A. (2020) Jump Diffusion Modeling of Stock Prices on Ghana Stock Exchange. Journal of Applied Mathematics and Physics, 8, 1736-1754. https://doi.org/10.4236/jamp.2020.89131

Stock | μ | σ | λ | m | δ | Stock | μ | σ | λ | m | δ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | Access Bank Ghana PLC | 0.0000 | 0.029841 | 0.00 | 0.00 | 0.00 | 19 | Golden Star Resources Limited | −0.00011 | 0.001723 | 0.00 | 0.00 | 0.00 |

2 | African Champion Limited | 0.00 | 0.00 | 7.00 | 0.45 | 0.23 | 20 | Golden Web Limited | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

3 | Agricultural Development Bank | 0.001715 | 0.012896 | 1.00 | 0.20 | 0.01 | 21 | HFC Bank (Ghana) Limited | 0.002508 | 0.021791 | 0.00 | 0.00 | 0.00 |

4 | AngloGold Ashanti Limited | 0.00000 | 0.0000 | 0.00 | 0.00 | 0.00 | 22 | Mega African Capital Limited | −0.00145 | 0.00015 | 0.00 | 0.00 | 0.00 |

5 | Aluworks Limited | 0.000543 | 0.016869 | 2.00 | 0.13 | 0.004 | 23 | Mechanical Lloyd Co. Limited | −0.00372 | 0.020868 | 0.00 | 0.00 | 0.00 |

6 | Ayrton Drug Manufacturing Limited | −0.00074 | 0.011624 | 1.00 | 0.19 | 0.003 | 24 | Pioneer Kitchenware Limited | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

7 | Benso Oil Palm Plantation | 0.004387 | 0.018568 | 4.00 | 0.12 | 0.01 | 25 | PBC Limited | 0.01 | 0.04573 | 12.00 | 0.20 | 0.03 |

8 | Cal Bank Limited | 0.001428 | 0.031976 | 9.00 | 0.14 | 0.02 | 26 | PZ Cussons Ghana Limited | −0.00039 | 0.006077 | 0.00 | 0.00 | 0.00 |

9 | Clydestone (Ghana) Limited | 0.0000 | 0.0000 | 0.00 | 0.00 | 0.00 | 27 | Standard Chartered Bank Ghana Limited | 0.002954 | 0.021696 | 4.00 | 0.12 | 0.01 |

10 | Camelot Ghana Limited | −0.00035 | 0.005548 | 0.00 | 0.00 | 0.00 | 28 | SIC Insurance Company Limited | −0.00074 | 0.037589 | 12.00 | 0.14 | 0.03 |

11 | Cocoa Processing Co. Limited | 0.0000 | 0.072554 | 0.00 | 0.00 | 0.00 | 29 | Starwin Products Limited | 0.0100 | 0.084415 | 14.00 | 0.35 | 0.57 |

12 | Ecobank Ghana Ltd. | 0.000636 | 0.000176 | 0.00 | 0.00 | 0.00 | 30 | Societe Generale Ghana Limited | 0.001137 | 0.010978 | 0.00 | 0.00 | 0.00 |

13 | Enterprise Group Limited | 0.00176 | 0.018091 | 1.00 | 0.11 | 0.00 | 31 | Sam Woode Limited | 0.000907 | 0.014227 | 0.00 | 0.00 | 0.00 |

14 | Ecobank Transnational Inc. | 0.001911 | 0.034102 | 2.00 | 0.57 | 0.61 | 32 | Trust Bank Ghana Limited | 0.001208 | 0.023732 | 6.00 | 0.14 | 0.01 |

15 | Fan Milk Limited | 0.00186 | 0.01192 | 1.00 | 0.10 | 0.00 | 33 | Total Petroleum Ghana Limited | 0.00231 | 0.013432 | 1.00 | 0.12 | 0.00 |

16 | GCB Bank Limited | 0.001376 | 0.011604 | 0.00 | 0.00 | 0.00 | 34 | Tullow Oil Plc | −0.00183 | 0.015226 | 2.00 | 0.16 | 0.01 |

17 | Guinness Ghana Breweries Limited | 0.000952 | 0.000115 | 0.00 | 0.00 | 0.00 | 35 | Unilever Ghana Limited | 0.001672 | 0.01281 | 0.00 | 0.00 | 0.00 |

18 | Ghana Oil Company Limited | 0.003635 | 0.011702 | 0.00 | 0.00 | 0.00 | 36 | UT Bank Limited | 0.003984 | 0.066734 | 0.00 | 0.00 | 0.00 |

Stock | Stock Path | Mean | Variance | Skew | Kurtosis | Stock | Stock Path | Mean | Variance | Skew | Kurtosis | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | Aluworks Limited | Actuals | −0.0028 | 0.0007 | 13.9989 | 13.4400 | 19 | Golden Web Limited | Actuals | 0.0278 | 0.1217 | 1.1106 | 0.5173 |

Predicted | −0.0005 | 0.0001 | 13.6600 | 20.3400 | Predicted | 0.0007 | 0.0000 | −0.2600 | −0.8500 | ||||

2 | Benso Oil Palm Plantation | Actuals | −0.0008 | 0.0002 | 9.5710 | 10.1330 | 20 | HFC Bank (Ghana) Ltd. | Actuals | 0.00591 | 0.00994 | 1.96957 | 1.5447 |

Predicted | −0.0005 | 0.0003 | 15.8500 | 9.7300 | Predicted | 0.00641 | 0.00007 | 1.0600 | −1.1100 | ||||

3 | Ecobank Transnational Inc. | Actuals | 0.0020 | 0.0014 | 10.4612 | 12.9413 | 21 | Mechanical Lloyd Co. Ltd. | Actuals | −0.0007 | 0.00401 | −1.6125 | 1.6063 |

Predicted | 0.0018 | 0.0001 | 10.5538 | 21.2631 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

4 | Fan Milk Limited | Actuals | −0.0032 | 0.0003 | 4.6480 | 33.2835 | 22 | PZ Cussons Ghana Ltd. | Actuals | 0.00501 | 0.17822 | 0.03927 | 15.1281 |

Predicted | −0.0013 | 0.0001 | 7.7800 | 24.3400 | Predicted | 0.00005 | 0.00002 | 0.4000 | −0.5900 |

5 | Produce Buying Company Ltd. | Actuals | 0.00778 | 0.02473 | 13.5271 | 20.5417 | 23 | SIC Insurance Company Ltd. | Actuals | 0.0026 | 0.00162 | 0.96564 | 7.0041 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Predicted | 0.00314 | 0.00205 | 13.9200 | 16.4300 | Predicted | −0.0692 | 0.00208 | 0.9400 | 3.0050 | ||||

6 | Standard Chartered Bank Ghana. Ltd. | Actuals | 0.00075 | 0.00045 | 10.9790 | 28.5262 | 24 | Societe Generale Ghana Ltd. | Actuals | −0.0004 | 0.00054 | −0.58737 | 1.2515 |

Predicted | 0.01804 | 0.00033 | 6.3900 | 12.600 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

7 | Starwin Products Ltd. | Actuals | −0.00164 | 0.01773 | −10.0784 | 8.53299 | 25 | Total Petroleum Ghana Ltd. | Actuals | −0.0002 | 0.00071 | −1.6952 | 2.04099 |

Predicted | 0.10701 | 0.03703 | 12.4100 | 21.2100 | Predicted | 0.0082 | 0.00049 | 0.1800 | 1.5700 | ||||

8 | Trust Bank Ghana Ltd. | Actuals | 0.0017 | 0.0007 | 13.2535 | 31.9042 | 26 | Unilever Ghana Ltd. | Actuals | 0.00132 | 0.00008 | 0.72769 | 1.01634 |

Predicted | 0.03945 | 0.00020 | 7.2800 | 28.9600 | Predicted | 0.01101 | 0.00231 | 0.7500 | 1.0002 | ||||

9 | Tullow Oil Plc | Actuals | 0.00147 | 0.00022 | 10.7760 | 16.5308 | 27 | Access Bank | Actuals | −0.0005 | 0.00241 | −0.30962 | 8.1141 |

Predicted | 0.00628 | 0.00155 | 10.2800 | 14.8900 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

10 | African Champion | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 28 | AngloGold Ashanti Limited | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | 0.0005 | 0.0020 | 0.9800 | 1.1300 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

11 | Ayrton Drug Manufacturing Limited | Actuals | −0.0014 | 0.0005 | −0.2615 | 0.0723 | 29 | Clydestone (Ghana) Limited | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | −0.0080 | 0.0000 | 0.4900 | −0.2400 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

12 | Cal Bank Limited | Actuals | −0.0004 | 0.0007 | −0.8781 | 1.9381 | 30 | Cocoa Processing Co. Limited | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | −0.0004 | 0.0012 | 3.0500 | 0.5500 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.00000 | ||||

13 | Camelot Ghana Limited | Actuals | −0.0004 | 0.0005 | −1.5120 | 3.4298 | 31 | Golden Star Resources Limited | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | 0.0000 | 0.0000 | −0.0300 | −0.8700 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

14 | Ecobank Ghana Limited | Actuals | −0.0001 | 0.0006 | 0.6055 | 1.9457 | 32 | Mega African Capital Ltd. | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | 0.0009 | 0.0001 | −0.1900 | −1.2600 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

15 | Enterprise Group Limited | Actuals | −0.0225 | 0.0002 | 1.6500 | 0.2000 | 33 | Pioneer Kitchenware Ltd. | Actuals | −0.0009 | 0.00263 | −0.2852 | 2.2805 |

Predicted | −0.0020 | 0.0003 | −2.2000 | 2.8255 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

16 | Ghana Commercial Bank Limited | Actuals | −0.0004 | 0.0004 | −0.4943 | 1.7835 | 34 | Sam Woode Ltd. | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | 0.0086 | 0.0000 | −0.3600 | −1.3400 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

17 | Guinness Ghana Breweries Limited | Actuals | 0.0002 | 0.0001 | 0.7534 | 1.2409 | 35 | Transaction Solutions (Ghana) Ltd. | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | 0.0004 | 0.0000 | −0.1300 | −1.3400 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||||

18 | Ghana Oil Company Limited | Actuals | 0.0006 | 0.0004 | 0.5471 | 2.0007 | 36 | UT Bank Ltd. | Actuals | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Predicted | 0.0019 | 0.0000 | 2.7494 | 2.0808 | Predicted | 0.0000 | 0.0000 | 0.0000 | 0.0000 |