International Journal of Systems Science and Applied Mathematics, 2017
This paper analyzes stock price behaviour on Ghana Stock Exchange (GSE) and develops a stochastic... more This paper analyzes stock price behaviour on Ghana Stock Exchange (GSE) and develops a stochastic model to predict the behaviour of stock prices on the exchange using Monte Carlo simulations. The first part looks at the various justifications and models that have been put forward to explain stock behaviour and its distribution elsewhere. It traces the foundations of the use of stochastic process as a means of predicting stock price behaviour from Louis Bachelier normality assumption to the works of Samuelson's lognormal supposition through to the doctoral thesis of Fama French in which he premised the behaviour of stock price to the idea of a random walk. We subsequently apply the Geometric Brownian Motion formulation to simulate stock price behaviour for all listed stocks on the GSE for the coming year (2015) using historical volatility and mean returns of the previous year (2014). The results find increasing evidence that the stochastic model consistently predict the stock price behaviour on the exchange in more than 80% of the listed stocks.
The behaviour of stocks on the Ghana stock exchange is examined to show that stock prices on the ... more 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.
Science Journal of Applied Mathematics and Statistics, 2018
We present a succinct new approach to derive the Black-Scholes partial differential equation and ... more We present a succinct new approach to derive the Black-Scholes partial differential equation and subsequently the Black-Scholes formula. We proceed to use the formula to price options using stocks listed on Ghana stock exchange as underlying assets. From one year historical stock prices we obtain volatilities of the listed stocks which are subsequently used to compute prices of three month European call option. The results indicate that it is possible to use the Black Scholes formula to price options on the stocks listed on exchange. However, it was realised that most call option prices tend to zero either due to very low volatilities or very low stock prices. On the other hand put options were found to give positive prices even for stocks with very low volatilities or low stock prices.
International Journal of Economics, Finance and Management Sciences, 2014
VaR is a potential loss. The VaR methodology gives the definition to risk-based capital, or econo... more VaR is a potential loss. The VaR methodology gives the definition to risk-based capital, or economic capital and confidence level reflects the risk appetite of the bank. This work is a delta-normal VaR application in the case of the Ghanaian economy. It assesses the exchange risk associated to the Ghana public debt portfolio. We used daily spot exchange rates of the Ghana cedi against the three main currencies, the dollar, the euro and the pound. We are interested in the period from 04/01/2000 to 31/12/2009. We demonstrated that the VaR result is very high and that there is a need for the government to also trade in a currency that can serve as a potential hedge against risk.
Mobile phone use by motorist is a known road accident risk factor. However, little known research... more Mobile phone use by motorist is a known road accident risk factor. However, little known research has been conducted in Ghana to determine the actual rate of mobile phone use by motorists. In this study, we try to estimate the rate of usage by motorists in Accra-Ghana. Drivers on six major roads in the city were observed in regards to their use of mobile phone while driving over four years. Gender of the driver and the type of vehicle used by the driver was also recorded. The results show that among 9,383 drivers that were observed over period, on average 9.4% (almost 1 in 10) motorists used their mobile phones everyday. In addition, mobile phone use increased over the period from 4.4% (1 in 25 drivers) in 2010 to 9.4% (1 in 10 drivers) in 2013. The results also show that there was a significant difference between men (9.0%) and women (12.2%) in the use of mobile phone by motorists in the Accra. In addition, average daily use of mobile phones among drivers of private vehicles (13.2%) was higher than among commercial vehicles (5.6%).
This research studied the trend of performance of Manchester United Football Club in the 1960-201... more This research studied the trend of performance of Manchester United Football Club in the 1960-2013 English premiership seasons. The three main variables involved in this study are the number of games won, games drawn and games lost by Manchester United for the study period. But this study concentrated on the number of games lost since the objective of every manager of a team is to minimize loss and maximize win or draw. Thus the objective of this study is to develop a model for predicting the number of games that would be lost by Manchester United in future seasons using games played in the previous seasons. The data used for this study are secondary data obtained from sportamok and English premiership websites. The statistical technique used for this study is time series analysis. Specifically, Autoregressive Integrated Moving Average (ARIMA) model was used to develop a model for the number of games lost by Manchester United. The model was used to forecast for the next fifteen seasons. The model predicted that Manchester United will lose six (6) games for 2013/2014 season.
Modelling and forecasting the exchange rate volatility is a crucial area, as it has implications ... more Modelling and forecasting the exchange rate volatility is a crucial area, as it has implications for many issues in the arena of finance and economics. Generalised Autoregressive Conditional Heteroskedasticity (GARCH) models with their modifications, is used in capturing the volatility of the exchange rates. Simple rate of returns is employed to model the currency exchange rate volatility of Ghana Cedi-United States Dollar. The daily closing exchange rates were used as the daily observations. The parameters of these models are estimated using the maximum likelihood method. The results indicate that the volatility of the GHC_USD exchange rate is persistent. The asymmetry terms for TARCH are not statistically significant. Also in TARCH case, the coefficient estimate is negative, suggesting that positive shocks imply a higher next period conditional variance than negative shocks of the same sign. This is the opposite to what would have been expected in the case of the application of a GARCH model to a set of stock returns. But arguably, neither the leverage effect or volatility feedback explanations for asymmetries in the context of stocks apply here.
Science Journal of Applied Mathematics and Statistics, 2015
We outline the ingredients necessary to compute the Joint Default Probability from which we obtai... more We outline the ingredients necessary to compute the Joint Default Probability from which we obtain Default Correlation, an important risk quantity in the determination of Internal Rating Based Approach in Basel II and III documents on banking supervision and regulations. We discuss Merton's structural approach of which one key drawback is the difficulty in tracking and calibrating asset value processes and the limitations of variant models which tend to be analytically too complex and computationally intensive. We address these issues by simulating all the possible asset value processes of a firm whose asset paths we assume to be Gaussian. By generating random values that simulate all the possible asset value processes, we are able to capture all the possible default horizons within a certain macroeconomic framework. Drawing standardised normally distributed assets values of obligors we obtain a range of values of Joint Default Probabilities at a specified asset correlation from which the corresponding range of default correlations are obtained. The results is a simplified approach to the determination of default correlation, easily implementable in Excel and less analytically complicated than existing procedures.
Science Journal of Applied Mathematics and Statistics, 2014
One of the approaches to determining and quantifying the credit risk of a loan portfolio is by ob... more One of the approaches to determining and quantifying the credit risk of a loan portfolio is by obtaining the distribution of losses of the portfolio and determining the risk quantities from such distributions. In this paper, we describe the challenges to using this approach and illustrate a practical solution where simulation methods are used to obtain loss distribution for a two obligor portfolio. This is then extended to ten and hundred obligor portfolios. Existing probability distributions with specified parameters are then used to approximate the loss distributions obtained. Using such parameters of the existing probability distributions, we obtain the risk quantities associated with the loan portfolio including Expected and Unexpected losses. We realized that depending on the confidence interval for which we measure the Unexpected Loss, Stress Losses are needed to account for the total loss of the portfolio.
International Journal of Economics, Finance and Management Sciences, 2013
This paper investigates the complexity involved in the quantitative measurement of Economic Capit... more This paper investigates the complexity involved in the quantitative measurement of Economic Capital and proposes simulation methods as a practical solution for obtaining the loss distribution of a portfolio of obligors. The paper examines a one factor model to generate loss distribution which establishes the necessary ingredients to measure the credit risk quantities in a loan portfolio. The general elements of credit risk modeling are outlined and then a specific model that employs a Monte Carlo simulation is developed. An example is provided that calculates the risk quantities in a loan portfolio from which the Economic Capital in a credit risk portfolio is obtained.
We propose a stochastic process modelling of covid-19 deaths in Ghana. The objective is to accura... more We propose a stochastic process modelling of covid-19 deaths in Ghana. The objective is to accurately capture the death processes resulting from the pandemic and to predict future deaths resulting from Covid-19 infections in Ghana. The mathematical derivation is based strictly on the compound Poisson process, a class of a Levy process. The model is verified by using empirical data of deaths resulting from Covid-19 from the onset of the pandemic up to the time of writing this report. That is, Covid-19 deaths in Ghana from March to August 2020. The method departs slightly from the usual differential equations used in modeling pandemics due to the unique occurrence of deaths from the disease in Ghana. As the methods are basically compound Poisson process, we delve into Levy processes as it allows us to effectively simulate the future behaviour of the death process. To test the effectiveness of the model, we compared the simulated results to the actual reported number of deaths from Covid-19 cases in Ghana from March to August 2020. The results show that at a 95% confidence interval there is no significant difference between the actual deaths and the simulated results. The results of the simulation, when extended to February 2021 (one year after the advent of the pandemic) shows that if the current conditions remain same, that is, if there is no immediate intervention by the discovery of an effective drug or a vaccine, then the number of deaths could reach four hundred and forty six (446) by February 28, 2020.
International Journal of Systems Science and Applied Mathematics, 2017
This paper analyzes stock price behaviour on Ghana Stock Exchange (GSE) and develops a stochastic... more This paper analyzes stock price behaviour on Ghana Stock Exchange (GSE) and develops a stochastic model to predict the behaviour of stock prices on the exchange using Monte Carlo simulations. The first part looks at the various justifications and models that have been put forward to explain stock behaviour and its distribution elsewhere. It traces the foundations of the use of stochastic process as a means of predicting stock price behaviour from Louis Bachelier normality assumption to the works of Samuelson's lognormal supposition through to the doctoral thesis of Fama French in which he premised the behaviour of stock price to the idea of a random walk. We subsequently apply the Geometric Brownian Motion formulation to simulate stock price behaviour for all listed stocks on the GSE for the coming year (2015) using historical volatility and mean returns of the previous year (2014). The results find increasing evidence that the stochastic model consistently predict the stock price behaviour on the exchange in more than 80% of the listed stocks.
The behaviour of stocks on the Ghana stock exchange is examined to show that stock prices on the ... more 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.
Science Journal of Applied Mathematics and Statistics, 2018
We present a succinct new approach to derive the Black-Scholes partial differential equation and ... more We present a succinct new approach to derive the Black-Scholes partial differential equation and subsequently the Black-Scholes formula. We proceed to use the formula to price options using stocks listed on Ghana stock exchange as underlying assets. From one year historical stock prices we obtain volatilities of the listed stocks which are subsequently used to compute prices of three month European call option. The results indicate that it is possible to use the Black Scholes formula to price options on the stocks listed on exchange. However, it was realised that most call option prices tend to zero either due to very low volatilities or very low stock prices. On the other hand put options were found to give positive prices even for stocks with very low volatilities or low stock prices.
International Journal of Economics, Finance and Management Sciences, 2014
VaR is a potential loss. The VaR methodology gives the definition to risk-based capital, or econo... more VaR is a potential loss. The VaR methodology gives the definition to risk-based capital, or economic capital and confidence level reflects the risk appetite of the bank. This work is a delta-normal VaR application in the case of the Ghanaian economy. It assesses the exchange risk associated to the Ghana public debt portfolio. We used daily spot exchange rates of the Ghana cedi against the three main currencies, the dollar, the euro and the pound. We are interested in the period from 04/01/2000 to 31/12/2009. We demonstrated that the VaR result is very high and that there is a need for the government to also trade in a currency that can serve as a potential hedge against risk.
Mobile phone use by motorist is a known road accident risk factor. However, little known research... more Mobile phone use by motorist is a known road accident risk factor. However, little known research has been conducted in Ghana to determine the actual rate of mobile phone use by motorists. In this study, we try to estimate the rate of usage by motorists in Accra-Ghana. Drivers on six major roads in the city were observed in regards to their use of mobile phone while driving over four years. Gender of the driver and the type of vehicle used by the driver was also recorded. The results show that among 9,383 drivers that were observed over period, on average 9.4% (almost 1 in 10) motorists used their mobile phones everyday. In addition, mobile phone use increased over the period from 4.4% (1 in 25 drivers) in 2010 to 9.4% (1 in 10 drivers) in 2013. The results also show that there was a significant difference between men (9.0%) and women (12.2%) in the use of mobile phone by motorists in the Accra. In addition, average daily use of mobile phones among drivers of private vehicles (13.2%) was higher than among commercial vehicles (5.6%).
This research studied the trend of performance of Manchester United Football Club in the 1960-201... more This research studied the trend of performance of Manchester United Football Club in the 1960-2013 English premiership seasons. The three main variables involved in this study are the number of games won, games drawn and games lost by Manchester United for the study period. But this study concentrated on the number of games lost since the objective of every manager of a team is to minimize loss and maximize win or draw. Thus the objective of this study is to develop a model for predicting the number of games that would be lost by Manchester United in future seasons using games played in the previous seasons. The data used for this study are secondary data obtained from sportamok and English premiership websites. The statistical technique used for this study is time series analysis. Specifically, Autoregressive Integrated Moving Average (ARIMA) model was used to develop a model for the number of games lost by Manchester United. The model was used to forecast for the next fifteen seasons. The model predicted that Manchester United will lose six (6) games for 2013/2014 season.
Modelling and forecasting the exchange rate volatility is a crucial area, as it has implications ... more Modelling and forecasting the exchange rate volatility is a crucial area, as it has implications for many issues in the arena of finance and economics. Generalised Autoregressive Conditional Heteroskedasticity (GARCH) models with their modifications, is used in capturing the volatility of the exchange rates. Simple rate of returns is employed to model the currency exchange rate volatility of Ghana Cedi-United States Dollar. The daily closing exchange rates were used as the daily observations. The parameters of these models are estimated using the maximum likelihood method. The results indicate that the volatility of the GHC_USD exchange rate is persistent. The asymmetry terms for TARCH are not statistically significant. Also in TARCH case, the coefficient estimate is negative, suggesting that positive shocks imply a higher next period conditional variance than negative shocks of the same sign. This is the opposite to what would have been expected in the case of the application of a GARCH model to a set of stock returns. But arguably, neither the leverage effect or volatility feedback explanations for asymmetries in the context of stocks apply here.
Science Journal of Applied Mathematics and Statistics, 2015
We outline the ingredients necessary to compute the Joint Default Probability from which we obtai... more We outline the ingredients necessary to compute the Joint Default Probability from which we obtain Default Correlation, an important risk quantity in the determination of Internal Rating Based Approach in Basel II and III documents on banking supervision and regulations. We discuss Merton's structural approach of which one key drawback is the difficulty in tracking and calibrating asset value processes and the limitations of variant models which tend to be analytically too complex and computationally intensive. We address these issues by simulating all the possible asset value processes of a firm whose asset paths we assume to be Gaussian. By generating random values that simulate all the possible asset value processes, we are able to capture all the possible default horizons within a certain macroeconomic framework. Drawing standardised normally distributed assets values of obligors we obtain a range of values of Joint Default Probabilities at a specified asset correlation from which the corresponding range of default correlations are obtained. The results is a simplified approach to the determination of default correlation, easily implementable in Excel and less analytically complicated than existing procedures.
Science Journal of Applied Mathematics and Statistics, 2014
One of the approaches to determining and quantifying the credit risk of a loan portfolio is by ob... more One of the approaches to determining and quantifying the credit risk of a loan portfolio is by obtaining the distribution of losses of the portfolio and determining the risk quantities from such distributions. In this paper, we describe the challenges to using this approach and illustrate a practical solution where simulation methods are used to obtain loss distribution for a two obligor portfolio. This is then extended to ten and hundred obligor portfolios. Existing probability distributions with specified parameters are then used to approximate the loss distributions obtained. Using such parameters of the existing probability distributions, we obtain the risk quantities associated with the loan portfolio including Expected and Unexpected losses. We realized that depending on the confidence interval for which we measure the Unexpected Loss, Stress Losses are needed to account for the total loss of the portfolio.
International Journal of Economics, Finance and Management Sciences, 2013
This paper investigates the complexity involved in the quantitative measurement of Economic Capit... more This paper investigates the complexity involved in the quantitative measurement of Economic Capital and proposes simulation methods as a practical solution for obtaining the loss distribution of a portfolio of obligors. The paper examines a one factor model to generate loss distribution which establishes the necessary ingredients to measure the credit risk quantities in a loan portfolio. The general elements of credit risk modeling are outlined and then a specific model that employs a Monte Carlo simulation is developed. An example is provided that calculates the risk quantities in a loan portfolio from which the Economic Capital in a credit risk portfolio is obtained.
We propose a stochastic process modelling of covid-19 deaths in Ghana. The objective is to accura... more We propose a stochastic process modelling of covid-19 deaths in Ghana. The objective is to accurately capture the death processes resulting from the pandemic and to predict future deaths resulting from Covid-19 infections in Ghana. The mathematical derivation is based strictly on the compound Poisson process, a class of a Levy process. The model is verified by using empirical data of deaths resulting from Covid-19 from the onset of the pandemic up to the time of writing this report. That is, Covid-19 deaths in Ghana from March to August 2020. The method departs slightly from the usual differential equations used in modeling pandemics due to the unique occurrence of deaths from the disease in Ghana. As the methods are basically compound Poisson process, we delve into Levy processes as it allows us to effectively simulate the future behaviour of the death process. To test the effectiveness of the model, we compared the simulated results to the actual reported number of deaths from Covid-19 cases in Ghana from March to August 2020. The results show that at a 95% confidence interval there is no significant difference between the actual deaths and the simulated results. The results of the simulation, when extended to February 2021 (one year after the advent of the pandemic) shows that if the current conditions remain same, that is, if there is no immediate intervention by the discovery of an effective drug or a vaccine, then the number of deaths could reach four hundred and forty six (446) by February 28, 2020.
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