Volatility Estimation Using European-Logistic Brownian Motion with Jump Diffusion Process

Abstract

Volatility is the measure of how we are uncertain about the future of stock or asset prices. Black-Scholes model formed the foundation of stock or asset pricing. However, some of its assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. The measure of volatility and good forecasts of future volatility are crucial for implementation, evaluation of asset and derivative pricing of asset. In particular, volatility has been used in financial markets in assessment of risk associated with short-term fluctuations in financial time-series. Constant volatility is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the ``volatility smile". To address the above problem the Jump-Diffusion Model and the Kou Double-Exponential Jump-Diffusion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to flatten out for longer maturities; the two models abilities to produce accurate results are reduced. This study ventures into a research that will involve volatility estimation using European logistic-type option pricing with jump diffusion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump diffusion model for price process.