Research for fractal market influenced by stochastic volatility

在Black， F.和Scholes， M.（1973）推导的金融模型中，波动性用一个常数参数来量化，波动性代表资产价格的随机程度。在本研究中，为了得到一个更真实的模型，波动性是由具有统计自相似性和依赖于过去记忆的分数阶随机过程来表示的。此外，在波动驱动过程具有“长期和短期记忆”以及“快和慢均值回归”的情况下，我们解决了金融衍生品期权定价问题。

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.