Abstract: 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.

Key words: Tsallis distribution, option pricing, jump, anomalous diffusion