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.
ZHAO Pan, XIAO Qing-xian
. Pricing of European Options Based on Tsallis Distribution and Jump-diffusion Process[J]. Chinese Journal of Management Science, 2015
, 23(6)
: 41
-48
.
DOI: 10.16381/j.cnki.issn1003-207x.201.06.006
[1] Black F. Scholes M. The pricing of options and corporate liabilities[J]. Joumal of political Eco-nomy, 1973,81(3):133-155.
[2] Fama E F. The behavior of stock market prices[J]. Journal of Business, 1965, 38(1): 34-105.
[3] Mandelbrot B B. Fractional Brownian motions, fractional noises and applications[J]. SIAM review, 1968, 10(4): 422-437.
[4] Mandelbrot B B. Fractals and scaling in finance: Discontinuity, concentration, risk[M]. New York: Springer Verlag, 1997.
[5] Beben M, Ohowski A. Correlations in financial time series: Established versus emerging markets[J]. Eur. Phys. J.B, 2001, 20(4): 527-530.
[6] Lo A W. Long term memory in stock market prices[J]. Econometria, 1991, 59(5):1279-1313.
[7] Evertsz C J G. Fractal geometry of financial time series[J]. Fractals, 1995,3(3):609-616.
[8] Necula C. Option pricing in a fractional Brownian motion environment[R].Working Paper, Academy of Economic Studies, 2002.
[9] Xiao Weilin, Zhang Weiguo, Zhang Xili, et al. Pricing currency options in a fractional Brownian motion with jumps[J]. Economic Modelling, 2010, 27(5):935-942.
[10] Gu Hui, Liang Jinrong, Zhang Yunxiu. Time-changed geometric fractional Brownian motion and option pricing with transaction costs[J]. Physica A: Statistical Mechanics and its Applications, 2012, 391 (15):3971-3977.
[11] Merton R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Economics,1976, 3(1):125-144.
[12] 刘国买, 邹捷中,陈超.服从多种形式跳过程的期权定价模型[J].数量经济技术经济研究,2004,(4):110-114.
[13] 黄学军,吴冲锋.不确定环境下研发投资决策的期权博弈模型[J].中国管理科学,2006,14(5):33-37.
[14] Tsallis C.Possible generalization of Boltzmann-Gibbs statistics[J].Journal of Statistical Physics,1988,52(1):479-487.
[15] Rak R, Drozdz S, Kwapień J.Non-extensive statistical features of the Polish stock market fluctuations[J].Physica A, 2007, 374(1): 315-324.
[16] Kozaki M, Sato A H.Application of the Beck model to stock markets: Value-at-Risk and portfolio risk assessment[J]. Physica A, 2008, 387(5): 1225-1246.
[17] Queirós S M D, Moyano L G, de Souza J, et al.A non-extensive approach to the dynamics of financial observables[J].The European Physical,2007, 55(2): 161-167.
[18] Biró T S, Rosenfeld R, Journal B.Microscopic origin of non-Gaussian distributions of financial returns[J]. Physica A, 2008, 387(7): 1603-1612.
[19] Ishizaki R,Inoue M. Time-series analysis of foreign exchange rates using time-dependent pattern entropy[J]. Physica A, 2013, 392(16):3344-3350.
[20] Tapiero O J. A maximum (non-extensive) entropy approach to equity options bid-ask spread[J]. Physica A, 2013, 392(14): 3051-3060.
[21] Borland L. A theory of non-Gaussian option pricing[J].Quantitative Finance, 2002, 2(6):415-431.
[22] Katz Y A, Li Tian. q-Gaussian distributions of leverage returns, first stopping times, and default risk valuations[J]. Physica A, 2013, 392(20): 4989-4996.
