Volatility Estimation Using a Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process

Volatility is a measure of the level of uncertainty about the future of stock price.  Volatility estimation is a fundamental aspect of modern financial analysis and investment strategy formulation, security valuation, risk management and monetary policy making.  It determines the implementation and evaluation of asset and derivative pricing, which has a major input valuation of options in corporate and public liabilities.  The initial estimation of volatility of an asset, by Black-Scholes was assumed to be a constant throughout the duration of a derivative.  However, the discontinuous nature of asset pricing led to Merton developing a Merton Jump model as a better estimate of prices in a precise way.  Thereafter, models have been developed to include more information and better the Mertons’ Jump model; which included Verhults Logistic equation, the logistic Geometric Brownian Motion with jump diffusion, among others.  Existing models fall short of developing a model incorporating the dividend yields, Logistic aspect and jump diffusion at once.  This necessitated the development of a model that incorporated dividend yields; which is a reality in most companies do to its members, the jump diffusion, and the Logistic aspect. The model developed was used it to estimate volatility.  Analysis was done using Geometric Brownian Motion(GBM), Logistic Brownian Motion (LBM), Itôs process, Itôs lemma, Jump diffusion models, Fokker-Plancks’ equation and Dupire’s volatility equation.  Analysis of Variance was too used; graphs and statistical tables were used show volatility estimates between the model developed and other models. Using mathematical software STATA, data from Nairobi Security Exchange was analysed.  Comparative analysis between the developed model and other models; Black Scholes model, European Logistic Brownian Motion model, European Logistic Brownian Motion with jump diffusion process model, and found out that, it gave the lowest volatility meaning that it is the best when estimating volatility.