978-1-5386-5541-2/18/$31.00 ©2018 IEEE Ito calculus-machine learning projection of forward US dollar-Ghana cedi rates Paul A. Agbodza Department of Mathematical Sciences University of Mines and Technology Tarkwa, Ghana https://orcid.org/0000-0003-0069-5298 NB. This is a preprint published in 2019 International Conference on Computing, Computational Modelling and Applications. IEEE Explore 20 June 2019. DOI: 10.1109/ICCMA.2019.00023 Abstract—In this paper the fundamental solution to CMSVJD is implemented to simulate USD/GHS forward exchange rates under the condition of jumps. CMSVJD was generalized in this study to include conditions of ‘no jump’ and ‘reduced parameters’. The model captures both stochastic volatility and jumps. Stochastic calculus modeling and machine learning interface is the answer to speculators and chartists driving prices in the Ghana interbank market (a frontier market). Codes written in R were used for simulation and plot of results. A 5-fold cross validation was used to create the test sample data, as proxy for forward rates required to validate the performance of the model. The resultant machine yields a predictive result of forward USD/GHS rates with a mean squared error of 0.169% and 0.5%. Keywords—USD/GHS, volatility, foreign exchange, jumps, frontier markets, cross validation I. INTRODUCTION A new equation to study the dynamics of the USD/GHS foreign exchange returns is the Correlated Multifactor Stochastic Variance Jump Diffusion (CMSVJD) model [1]. Since the early approach of [2] to apply learning networks to model derivative securities, the application of machine learning tools to financial engineering is an area of current research. Projecting foreign exchange rates is the most difficult endeavor. In a frontier market, where there is no derivative securities trading, forward rates are non-existent. Annual reports [3] and [4] show that speculative activity is brisk and drives prices in the Ghana interbank market. CMSVJD was implemented to resolve these difficulties. This is a two-state simultaneous equation of the evolution of rates and variance. The fundamental solution to this equation gives the discounting factor for determining forward rates of the USD/GHS exchange rate. In the next four sections, the model and the results from implementing the model were presented. Finally, the results were discussed and the implications of the study given in the conclusion. II. METHOD The method used is stochastic calculus modelled as Brownian motion and Poisson jumps. The multifactor jump- diffusion equation, CMSVJD of [1] is the main tool for implementation in this paper. Cross-validation of machine learning was also applied [5]. Five-fold cross validation was used. Computer-intensive methods (monte carlo or random simulation) were employed to train and test the machine. Bootstrap samples were created as initial FX values to be fed in the machine for projection. The data used for implementation of the model was obtained from [6] and [7]. The data spans July 2, 2007 to February 18, 2019 totaling 3,075 trading days or data points. Common abbreviations used are FX for foreign exchange; and USD/GHS for US Dollar-Ghana Cedi exchange rate. Mean squared error is abbreviated as mse. III. FORWARD RATE PROJECTION MODEL In this paper, the model is implemented on the data split with a 5-fold cross validation to determine the forward exchange rates. There are jumps in the data which is modelled as a compound Poisson process and the volatility is itself modelled as a stochastic process. Implementing the model requires a 5-fold cross validation to generate training and test samples in the data. This method was adopted to solve the problem of lack of forward rates in a frontier financial market without derivative trading. A. Econometric Characteristics of Data The distribution of the returns data is shown in the log rate of returns graph (Fig. 1) which exhibits features of random walk. They exhibit random volatility. Fig 1: US Dollar-Ghana Cedi Log Returns (02/07/2007 – 18/02/2019) From Fig. 1, the jumps are conspicuously seen. These are up and down movements as a result of unexpected events or crashes during trading days. These jumps can be considered outliers. The kernel density (Fig. 2) shows that the data is unimodal. Different kernels were chosen but the kernel choice did not alter the nature of the modality. This suggests that the USD/GHS data has homogenous features.