准确描述资产价格的运行规律是进行衍生产品定价及风险控制的基础。受金融市场外部环境的影响,资产收益率常常具有尖峰厚尾和偏尾的现象,为了准确地描述资产价格的运动规律,本文利用具有长程记忆及统计反馈性质的Tsallis熵分布和一类更新过程,建立了跳-反常扩散的股票价格运动模型。利用随机微分和鞅方法,在风险中性的条件下,得到了欧式期权的定价公式,该公式推广了文献11和21的相应结论。最后,利用上证指数数据分别计算出了各模型的参数以及对资产收益率拟合的平均绝对误差,数据分析结果表明本文模型与文献11和21相比其平均绝对误差分别减小了10.4%和25.1%。说明了本文模型对资产收益率尖峰厚尾及偏尾等现象的捕捉更为准确。
The accurate description of the motion law of asset prices is the foundation of pricing and controlling risk of derivatives. The distribution of yields often has a peak, fat or skewed tail, because of influence of the external environment of financial market. Tsallis distribution has the characteristics of long-term memory and statistical feedback. So, the peak or fat tail of yields can be captured, through fitting non-extensive parameter qof Tsallis distribution. In addition, asymmetric jump processes can fit the skewed tail of returns. Tsallis distribution and renewal jump process are employed in this paper, then, an abnormal jump diffusion model of share price movements is established. In the risk-neutral condition, the pricing formulas of European options were obtained by using the stochastic differential and martingale method. But, in the literature of Merton[11], the model employed the Poisson jump process and normal distribution. The literature of Borland[21] only used Tsallis distribution without considering the skewed tail of yields. Therefore, they were included in our model as special cases. Using the actual data of China's shanghai index, the parameters of the models and the mean absolute error of yields are calculated,respectively. The results showed that the mean absolute error of our model was reduced respectively by 10.4% and 25.1% compared with ones of the literature 11 and 21.It explained that our model can fit accurately the motion law of asset prices. In addition, our model can also be used to price or measure and control risk of other derivatives, such as warrants and other types of options.
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