Jump Diffusion Modeling of Stock Prices on Ghana Stock Exchange

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.


Introduction
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      (2) where σ is the standard deviation of a stock's return.
For the standard normal density, 0 S = and 3 K = 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 Table 1.
The results in Table 1 show that several stocks exhibited skewness and kurtosis far in excess of 0 and 3 respectively. This confirms the presence of jumps in some stock price returns and as such there is the need to critically review approaches of stock modeling in regards to the exchange. One problem for investors on Ghana stock exchange is the absence of models to predict stock behaviour. Even in cases where models exist closed form expressions for the stock density are generally are not available. Antwi (2017) [1] used GBM to model stock behaviour on the exchange but in the light of current evidence it is essential to consider the use of other models for some stocks. Thus, the aim of this paper is to examine the use of Jump Diffusion Model (JDM) to predict stock price behaviour on the exchange. The use of JDM will ensure that the jump behaviour exhibited by some stocks can be incorporated into stock price modeling on the exchange.

Literature Review
The literature on stock price modeling is extensive. The foundations were laid by Bachelier [2] who assumed that the stock price dynamics follows a normal distribution. Samuelson [3]  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 [5] was the first to consider the use of jump diffusion in modeling stock prices, assuming that the jumps in stocks follows a compound Poisson process allowing jump times to follow Poisson distribution and jump sizes to be normally distributed. Accordingly, several extensions and variants of jump diffusion models have been developed. Kou [6], assumed that jump sizes follow a double-exponential distribution. In Madan and Seneta [7], the distribution of the uncertainty in the stock price is Gaussian, conditional on a variance that is distributed as a gamma variate. The advantages of Madan and Seneta include the ability to track long tail distributions, continuous-time specification, finite moments, elliptical multivariate unit period distributions and being a good empirical fit. Matsuda [8], introduced an approach to jump diffusions and derived the moments, skewness and kurtosis of the model. Amin [9], developed a discrete time model which converges weakly to the diffusion component of the jump diffusion process by superimposing jumps on the existing local price changes. Multivariate jumps were superimposed on the binomial model of Cox, Ross, and Rubinstein [10] to obtain a model which converges to a limiting jump diffusion process. Cont and Tankov [11] gave a complete treatment of jump diffusion models.
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 [12], in which the instantaneous volatility is allowed to follow a stochastic process. There are several other stochastic volatility models that allow arbitrage free prices in volatile markets as in Heston [13]. Duffie, Pan and Singleton [14] examined the impact of stochastic volatility on jumps diffusion for both jump amplitude as well as jump timing in an option pricing setting. These ideas are extended in Chernov et al. [15]. Barndorff-Nielson and Shephard [16] showed that the difference between realized variance and realized bipower variation estimates the quadratic variation of the jump component when volatility is stochastic. The follow up to jump diffusion and stochastic volatility models are the Affine Stochastic-Volatility and Affine Jump-Diffusion models which combine stochastic volatility and jump-diffusions models as in Duffie [17]. The basic affine jump diffusion is a stochastic process which consists of a Geometric Brownian motion with a stochastic volatility, a compound Poisson process and an independent exponentially distributed jumps with specified mean and variance.
The Constant Elasticity of Variance (CEV) model was first proposed by Cox and Ross [18] and extended by Davydov and Linetsky [19]. CEV is a one-dimensional Journal of Applied Mathematics and Physics diffusion model with instantaneous volatility specified by a power function of the underlying stock price. Although introduced as an alternative to GBM the model is more related to the Bessel processes and it is analytically tractable, allowing for closed-form pricing formulas. Other models include chaos theory and fractal Brownian motions which were considered by Mandelbrot [20]. There are the Generalized hyperbolic models and log-hyperbolic model of Barndorff-Nielsen and Shephard [21], Samorodnitsky and Taqqu [22], Blattberg and Gonedes [23] which replaces the log-normal distribution assumed in GBM by some other dis-

tributions.
To implement simulation of the jump diffusion models, Glasserman [24], discussed Monte Carlo approaches of simulating the process at fixed dates and at specified jump times. Hanson [25], gives practical examples for jump-diffusions in continuous time, including jumps driven by compound Poisson process that allow randomly distributed jump-amplitudes, state-dependent jump-diffusions and multidimensional jump-diffusions. Hanson and Westman [26], compared the performance of three jump-diffusion models using normal, uniform and double-exponential jump-amplitude distributions. The parameters of all three models were fit to Standard and Poor's 500 log-return market data, given the same first moment and second central moments. Beskos and Roberts [27], introduced a simple algorithm that simulates exact sample paths of stochastic differential equations. This was extended by Casella and Roberts [28], who introduced an algorithm to simulate from a class of one-dimensional jump-diffusion processes with state-dependent intensity. Advances in simulation procedures has been carried out by Pollock, et al. [29], by introducing a framework for simulating finite dimensional representations of jump diffusion sample paths over finite intervals without discretization error.

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

Developing the Jump Diffusion Model
In 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.
( ) ( ) Suppose that in the small-time interval t ∆ the stock price jumps by ( ) Figure 4.
The percentage change in the stock price is thus given by It is realized that ( ) The consequence of Equation (7) is that in a small-time interval dt the likelihood of the Poisson event can be described as follows: • The relative jump sizes ( ) 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

JDM under Equivalent Martingale Measures
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 To ensure that ( ) S t is a martingale the jump part of the process is compen- The term k λ compensates the jumps in the sense that the process ( ) Y t kt λ − is a martingale. Hence, Equation (6) can be written as 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

Density of Jump Distribution
From Equation (12) an explicit solution for the JDM is Hence the log-return density The related probability density at time t is

Merton and Kou's Models • Merton's Representation
Merton formally represented his model of the dynamics of the stock price as where α is the instantaneous expected return on the stock;  is the expectation operator over the random variable Y.
Merton assumed that if , , , α λ κ σ are constants then Equation (14) has the In Kou's model, the stochastic differential representing the return of the stock under the physical measure  is given as

Monte Carlo Simulation
Monte The discretization representation is employed.
Direct simulation from the representation in Equation (15) where the product over j is equal to 1 if This recursion replaces products with sums and it is preferable.
The procedure can be summarized into the following steps: and go to Step 4; 3) Generate 1 ln , , ln N Y Y  from the common distribution and set In the simulation procedure the return data is divided into two groups  and  . Group  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  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 0.1 ε = . 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 Table A1 in Appendix. For example, the estimated parameters for the Standard Chartered

Results
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 Table A2 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. Figure 5(a) and Figure 5

Conclusions
In this paper, significant progress has been made towards simplifying the ma-