给出了标的资产服从混合分数跳-扩散过程的几何平均亚式期权定价的解析解.运用广义Itô引理和自融资交易策略得到混合分数布朗运动下带跳的几何平均亚式期权定价的偏微分方程模型.结合边值条件，通过求解该偏微分方程得到亚式期权定价的解析解.通过数值试验，讨论各定价参数对期权价值的影响.本文推广了一些已有的结论，所得结果更贴近实际金融市场.
This paper mainly studied the geometric average Asian option pricing on the condition that the underlying asset followed mixed jump-fraction process. The general Itô's lemma and the self-financing dynamic strategy were obtained by using the partial differential equation of such option pricing in the mixed fractional environment with jump. With the combination of boundary condition, an analytic formula for the geometric average Asian option was derived by solving the partial differential equation. The numerical experiments were showed to discuss the influence of different parameters on the option value. The results were the generalization of some existing results which was closer to the real financial market.
[1] KEMNA A G Z, VORST A C F. A pricing method for options based on average asset values [J]. Journal of Banking and Finance, 1990, 14(1): 113-129.
[2] WONG H Y, CHEUNG H L. Geometric Asian options: Valuation and calibration with stochastic volatility [J]. Quantitative Finance, 2004, 4(3): 301-314.
[3] CHOU C S, LIN H J. Asian options with jumps [J]. Statistics & Probability, 2006, 6(14): 1983-1993.
[4] CHERIDITO P. Regularizing fractional Brownian motion with a view towards stock price modelling [D]. Zürich: Swiss Federal Institute of Technology, 2001.
[5] CHERIDITO P. Arbirage in fractional Brownian motion models [J]. Finance and Stochastics, 2003, 7(4): 533-553.
[6] KUZNETSOV Y A. The absence of arbitrage in a model with fractal Brownian motion [J]. Russian Mathematical Surveys, 1999, 54(4): 847-848.
[7] ZÄHLE M. Long range dependence, no arbitrage and the Black–Scholes formula [J]. Stochastics and Dynamics, 2002, 2(2): 265-280.
[8] MISHURA Y S. Stochastic Calculus for Fractional Brownian Motion and Related Processes [M]. Berlin: Springer, 2008.
[9] WANG X T. Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model [J]. Physica A, 2010, 389(3): 438-444.
[10] WANG X T. Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black–Scholes model with transaction costs [J]. Physica A, 2011, 390(9): 1623-1634.
[11] MEHRDOUST F, SABER N. Pricing arithmetic Asian option under a two-factor stochastic volatility model with jumps [J]. Journal of Statistical Computation and Simulation, 2015, 85(18): 3811-3819.
[12] RAMBEERICH N. A high order finite element scheme for pricing options under regime switching jump diffusion processes [J]. Journal of Computational and Applied Mathematics, 2016, 300(2): 83-96.
[13] XIAO W L. Pricing currency options in a fractional Brownian motion with jumps[J]. Economic Modelling, 2010, 27(5): 935-942.
[14] PENG B. Pricing Asian power options under jump-fraction process [J]. Journal of Economics, Finance and Administrative Science, 2012, 17(33): 2-9.
[15] 丰月姣. 带跳混合分数布朗运动下利差期权定价[J]. 佳木斯大学学报,(自然科学版), 2013, 30(6): 922-925.
[16] 孙玉东. 带跳混合分数布朗运动下利差期权定价[J]. 系统科学与数学, 2013, 32(11): 1377-1385.
[17] SHOKROLLAHI F. Actuarial approach in a mixed fractional Brownian motion with jumps environment for pricing currency option [J]. Advances in Difference Equations, 2015, 257(1): 1-8.
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