A new mathematical model for stock price

Abstract

Geometric Brownian motion (GBM) has been commonly used for analyzing stock price movements. Fractal Brownian motion (fBm), introduced by Mandelbrot and Van Ness (1968) to capture long-term dependence, is also Gaussian. Motivated by fBM, Comte and Renault (1998) introduce fractional stochastic volatility models to explain the long-memory of the implied volatility. Further, fractional Black-Scholes formula is suggested by Mandelbrot 1997 and Shiryaev 1999. But these models provide for arbitrage opportunities. This is the focus of research by Oksendal (2003), Cheridito (2003) and Guasoni (2005). However, fBm is non-Markovian. That makes it very difficult to study and to implement. Cheridito (2001) introduces mixed fBM that is shown to be useful in modelling returns. Dhesi, Shakeel and Xiao (2015) investigate an extension of GBM by incorporating a weighting factor and a stochastic function that is a mixture of power and trigonometric functions. Simulation results show that their model is superior to GBM. Gajda and Wylomanska (2012) propose an alternative approach based on subordinated tempered stable GBM. Monte Carlo simulations show that their model is more successful. Ladde and Wu (2009) also develop modified linear models of GBM, under different data partitioning, with and without jumps. Monte Carlo results suggest that data partitioning improves the results, and the models with jumps are much better than those without them.