Research for fractal market influenced by stochastic volatility

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In the financial model derived by Black, F. and Scholes, M. (1973), the volatility, which represents the degree of randomness of the asset price, is quantified by a constant parameter. In this research, to obtain a more realistic model, the volatilities are formulated by fractional stochastic processes which have statistic self-similarity and dependence on the memory to the past. Moreover, in the case when the volatility-driving processes have the ‘long and short term-memories’ and the ‘fast and slow mean-reversions’, we solved the problem for option pricing in financial derivatives.