【作者】 刘凤波；

【导师】 郑晓阳；

【作者基本信息】 哈尔滨工程大学 ， 应用数学， 2009， 硕士

【摘要】 金融数学经历了近百年的发展,主要研究风险资产的定价、利率衍生证券定价和最优投资消费策略,其中风险资产定价是金融数学研究的核心问题。由于衍生证券一般都是长期的,所以利率变化对金融衍生证券定价影响也比较大。1977年,Vasicek提出了一个受市场不确定因素影响而呈现随机波动现象的短期利率模型,使得利率衍生证券定价成为投资者关注的又一个焦点,因此对随机利率下的衍生证券定价研究是具有重要的理论意义和实际应用价值的。本文主要以随机分析、鞅理论和随机过程为核心来构造金融市场的数学模型,研究了在随机利率下,尤其是函数Vasicek模型下欧式期权定价公式,并且给出了几何平均亚式期权的定价公式；同时还用保险精算法给出了随机利率Vasicek模型下具有幂型支付的欧式期权定价公式。本文的主要成果及创新如下：(1)运用Ito积分和随机微分方程的方法,讨论了奇异期权中具有代表性的几何平均亚式期权在函数Vasicek模型下的定价问题,并且得到了函数Vasicek模型下推广了的Black-Scholes期权定价模型。(2)运用Mogens Bladt和Tina Hviid Rydberg(1998)提出的期权保险精算法,得了函数Vasicek模型下具有幂型支付的欧式看涨(看跌)期权定价公式,并予以改进和推广。更多还原

【Abstract】 Finance calculus which has experienced nearly one hundred years mainly researched on the pricing of risky assets, the pricing of interest rate derivative securities and optimal investment and consumption strategy, morever the pricing of interest rate derivative securities is one of the kernel problems. Due to the long duration of the derivative securities, the moment of interest rates becomes more important in pricing such long-dated derivative options. In 1977, Vasicek proposed a model of short-term interest rate, which made more investors focus on the pricing of interest rate derivative securities. Thus the study of the pricing of derivative securities with stochastic interest rate not only has important theoretic significance but also the practical value.This paper mainly applies stochastic analysis, martingale theory and stochastic processes to simulate the mathematic model of financial market, studies the pricing of European option with stochastic interest rate, and gets the pricing of geometric-average Asian options. Meanwhile, this paper give the pricing of European option in the case of power-option pricing formula under the model of Vasicek stochastic interest rate with the actuarial approach. The following are this paper’s main results and innovations:(1)Using the method of Ito integral and stochastic differential equation, the problems of exotic options’pricing are taken into consideration, of which the most representative geometric average Asian options with the function of Vasicek stochastic interest rates. Forthermorever, getting the promoted option pricing model of Black-Scholes with the function of Vasicek stochastic interest rates.(2)Using the method of the actuarial approach proposed by Mogens Bladt and Tina Hviid Rydberg in 1998. The call(put) European options with power payoffs pricing formula are gained with the function of Vasicek model, and then improved and promoted.更多还原