A Fractional Volatility Model : Estimation of the NASDAQ volatility paramater using Futures pricing

This paper explores a novel approach to modeling stochastic volatility using fractional Brownian motion, with a focus on long-memory characteristics often observed in financial time series. Building on the limitations of classical Markovian frameworks, we introduce a volatility process defined with a stochastic kernel. This formulation not only captures persistent dependencies in volatility but also offers tractability through Malliavin calculus and duality principles. We develop a pricing model for futures contracts in which the volatility component evolves according to this memory-inclusive structure. A calibration strategy is then proposed to estimate the underlying kernel function \( \hat{g}_\sigma \), leveraging futures prices as a bridge between theory and empirical dynamics. The proposed method enhances the representation of volatility in financial modeling, particularly for stock indices where long-term dependencies play a significant role, NASDAQ for instance was proved to follow a long memory path as its hurst parameter $H=0.59$, after we calibrated the theoritical volatility of NASDAQ with the FIGARCH volatility. JEL Classification. G13, C61, C63, C02.