Incorporating stochastic volatility and long memory into geometric Brownian motion model to forecast performance of Standard and Poor's 500 index

It is known in the financial world that the index price reveals the performance of economic progress and financial stability. Therefore, the future direction of index prices is a priority of investors. This empirical study investigated the effect of incorporating memory and stochastic volatility into geometric Brownian motion (GBM) by forecasting the future index price of S&P 500. To conduct this investigation, a comparison study was implemented between twelve models; six models without memory (GBM) and six models with memory (GFBM) under two different assumptions of volatility; constant, which were computed by three methods, and stochastic volatility, obeying three deterministic functions. The results showed that the best performance model was for GFBM under a stochastic volatility assumption using the identity deterministic function $\sigma \left(Y_t\right)=Y_t$, according to the smallest values of mean square error (MSE) and mean average percentage error (MAPE). This revealed the direct positive effect of incorporating memory and stochastic volatility into GBM to forecast index prices, and thus can be applied in a real financial environment. Furthermore, the findings showed invalidity of the models with exponential deterministic function $\sigma \left(Y_t\right)=e^{Y_t}$ in forecasting index prices according to huge values of MAPE and MSE.