由于经典的Black-Scholes期权定价模型的假设忽略了突发事件对资产价格的影响和"波动率微笑"对期权价值的影响而与实际情形往往存在偏差,因此学者们对Black-Scholes模型的改进则主要分别集中在带跳扩散过程的期权定价模型与具随机波动率的期权定价模型等两个方面,然而却少见将这两种模型结合起来的研究。本文首先在带跳扩散过程的期权定价模型与具随机波动率的期权定价模型的研究工作的基础上,建立了一种同时带跳扩散过程和具随机波动率的美式期权定价模型,并通过伊藤引理推导出了资产价格、随机波动率和期权满足的偏微分方程;然后,利用特征函数法和傅里叶变换导出了资产价格的随机分布,进而通过马尔科夫链方法给出了基于跳扩散过程和随机波动率的美式期权的数值解;最后,运用已建立的带跳扩散过程和随机波动率的美式期权定价模型对高新技术企业项目投资的专利权价值进行实物期权定价评估的案例研究,并对跳扩散强度参数和随机波动率参数进行敏感性分析,研究结果表明:将项目收益跳扩散过程和市场环境随机波动率加入到专利权实物期权定价模型中,可以有效避免专利权的期权价值被高估。
Since the hypotheses of the classical Black-Scholes option pricing model ignore the impact of sudden change on asset prices and the "volatility smile" on the option value, particularly, it deviates from the actual situations. Many improvements of the Black-Scholes model are based on two directions: one is the option pricing model with jump-diffusion, and the other is the option pricing model with stochastic volatility. But very few researches combine these two factors into a model. In this paper, based on the previous work of the option pricing model with jumps or stochastic volatility, an improved option pricing model is firstly constructed by taking into account jump diffusion and stochastic volatility at the same time. The corresponding partial differential equation of the asset pricing, stochastic volatility and option pricing is derived by Ito lemma. Then, the stochastic distribution of asset price is obtained by the characteristic function method and Fourier transform approach. The numerical solution of the American option with jump diffusion and stochastic volatility is obtained by employing the Markov Chain approach. Finally, by the empirical research, the real option price of the patent of the project investment in a high-tech company is evaluated by using our developed model, and the effects of value changes of jump diffusion strength parameters and stochastic volatility parameters on the value of the patent option price are reported. It is concluded that the overvaluation of the patent option price, due to the ignorance of the uncertainty of the financial market, can be avoided effectively by introducing the jump-diffusion process of project profit and the stochastic volatility of market environment into the patent real option pricing model.
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