Mathematical model for non-linear stock price adjustment

Abstract

The field of financial mathematics has drawn a lot of interest from both practitioners and academicians since the derivation of Black- Scholes model of 1970s. :Ihe celebrated option pricing formula, the so-called Black- Scholes -Merton option pricing model was developed by the use of geometric Brownian motion. But the model has a shortcoming because it assumes that the volatility is constant, when in reality it is not. To overcome this, Hull and White developed a stochastic volatility model in 1987. Recently in 2003, Onyango relaxed the geometric Brownian motion assumptions by applying the Walrasian price adjustment mechanisms, taking the supply and demand functions to respond to random fluctuation in asset trading. From the available literature, the work so far done on logistic Brownian motion does not include those assets that pay continuous dividends during the life of the option. This justified the need to develop a model that will take care of continuous dividend payments. In this study we have developed mathematical model for non-linear stock price adjustment that will be used to fit the prices of assets that pay continuous dividends and follow a non-linear trend. To achieve this, we have applied the knowledge of logistic Brownian motion and analysed the geometric Brownian motion model analytically. To verify this model, secondary data from London Stock Exchange has been used. Since dividend payments in financial markets are critical in securing investors, the result obtained from this study will help decision-makers in determining the prices of assets that pay continuous dividends that would attract more investors. We believe that this study has also contributed more knowledge to the field of mathematics of finance.