分形金融市场中的若干问题研究

This project is concerned in the modelling of the stock return time series and their distributions as well as the option pricing from the methods and the views of the fractal geometry. First, we will present the new multifractal stock return models.Those models can capture the thick tails, volatility persistence, moment scaling, and discontinuity exhibited by many financial time series. In the no-arbitrage pricing regime, the European option pricing problem associated with those new models is considered. Through the asymptotic expansion of the multifractal measure of the stock return series, we will obtain the asymptotic closed-form expansion for the option pricing formula with respect to the fractal time scaling . We will discover that the fractal scaling and fractal dimension play an important role in the precise pricing option and in the explanation for the phenomenon of the implied volatility.smiles . We will make an empirical analysis to show that the effect of the fractal scaling and fractal dimension on the option pricing and the implied volatility smiles .Second, we will propose the new stock price models to make the corresponding European option pricing and get their closed-form solutions .Those new models have the following features: the long-range dependence is present in the absolute and squared returns, and return distributions have regularly varying tails ,i.e. , higher-peaked and heavier-tailed than Gaussian distribution.Third ，we will study the classical extensions of the Black-Scholes model and the Merton model for option pricing,often known as the long memory stochastic volatility models .Assume that the drived noise of the stochastic volatility is the fractional Brownian motion. We will price option in the following two cases : a. There exist the whitening representations for the fractional Brownian motion.In this case, we will discover that the historical volatility has a important influence on the current option value and investors ,with the different information structures of the historical volatility ,will obtain the different values for the same option by themselves. The historical volatility may be another reason for the implied volatility smiles; in addition, we will study how European option is priced while Hurst exponent in the long memory stochastic volatility model is time-varying.b. There does not exist the whitening representation for the fractional Brownian motion. In this case, in the no-arbitrage pricing regime, European option value is a random variable .In order to obtain the exact value of the option ,we propose that the value of the option would be determined in one way: so that the variance of the pricing error is minimized ,and so that a VAR-type inequality holds for the volatility.

本项目是分形几何在金融中的应用研究。首先，我们构造具有重分形结构的股票价格模型，这些模型可刻画收益分布的高峰厚尾、波动率的集聚性、收益序列的矩标度特征及样本路径的不连续性；相应的期权定价公式可研究分形标度及分形维数对期权定价及隐含波动率微笑现象的影响。其次，我们将构造收益分布尾部是正则变化的股票价格模型并进行相应的期权定价，得出其闭形解；这些模型可刻画收益序列函数的自相关性和正则变化尾部对期权定价及隐含波动率微笑现象的影响。最后，我们研究股票波动率的长期记忆性对期权定价的影响；分两种情况讨论：a.当分数Brown运动有白化滤波表示时，我们发现股票价格过去的波动率对期权定价有影响，不同信息结构的投资者，对相同的期权有不同的定价；b.当分数Brown运动没有白化滤波表示时，我们极小化定价误差并用VAR型不等式确定期权的价格。我们将对某些结果进行实证分析，检验它们对隐含波动率微笑现象的解释能力。

首先，在不完全市场条件下，我们研究了分形标度、残留风险及不完美规避策略对期权定价的影响。在不完美规避条件下，欧式期权价格满足双曲型偏微分方程，我们得到了该方程的闭式解，并且当时间标度趋于零时，期权价格收敛到Black-Scholes公式，数值方法显示我们得到的期权定价公式可很好地拟合实际数据。其次，在离散时间不完全市场我们考虑了分形标度、残留风险及混合规避策略对期权定价的影响。实证分析显示这三个指标对准确定价期权至关重要。再者，在离散时间不完全市场条件下，我们研究了存在成比例交易费时，分形标度与投资者风险偏好对欧式期权定价的影响。我们的结果显示分形标度及投资者风险偏好对隐含波动率的微笑特征有重要的影响。最后，我们研究了当股票价格是一个Student跳跃过程并且波动率未知时的欧式期权定价问题。在不完全信息场合，我们提出使用平均自融资策略定价期权，并得到期权价格的闭形解；我们还提出用Value-at-Risk型不等式估计期权价格的隐含波动率，并得到具体的估值步骤。实证分析显示，在不完全信息场合，期权并非总是可以定价的，这具有非常重要的现实意义。