Modeling Options Prices Considering Mandelbrotian Movement of Prices

Classical Black Scholes model, however still widely used in the financial circles, is known for its inability to generate volatility satisfying the market observations. Besides the assumption of the normal distribution of the underlying assets and perfect liquidity of the market, classical Black-Scholes provides poor modeling of the in-the-money call option price, and does not account for the term structure of the options. Current work concentrates on the fractional integration of a diffusion process, relaxes classical assumptions, and provides theoretically solid modeling of volatility smile and the term structure of options. Inspired by the work of Comte, Coutin, and Renault(2003), we present a modification of the classical Heston model, where the volatility process is defined through a fractional integration of a diffusion process. Our construction allows us to compute a martingale representation for the volatility process efficiently. From this representation, and using calibrated parameters, we provide a numerical result of the model. Results give evidence that the fractional version of the Heston volatility model has better accuracy than the classical Black-Scholes model in terms of prediction error. Among other advantages, the fractional Heston model is a robust heuristical approach to the model parameters, which allows us to interpret them to understand the market behavior