基于Vasicek随机利率下具有幂型支付的期权保险精算定价方法
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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。
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.
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。
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.
引文
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[10]Vasicek O. An equilibrium characterization of the term structure, Journ-al of financial economics,1977,5:177-188P
[11]Cox J, and Ingersoll J, Ross S. A theory of the term structure of interest rates. Econometrica,1985,53:385-408P
[12]Hull J, White A. Pricing interest rate derivative securities. Review of financial studies,1990,3:573-592P
[13]Health D, Jarrow R, Morton A. Bond pricing and the term structure of interest rates:a new methodology. Econometrica,1992,60,77-105P
[14]Brace A, Gatarek D, Musiela M. The market model of interest rate dynamics. Mathematical finance,1997,7(2):127-155P
[15]Cox J C, Ross S A, Rubinstein M. Option pricing:a simplified appr-oach. Journal of financial economic,1979,7(3):229-263P
[16]Harrison J M, Kreps D M. Martingales and arbitrage in multiperiod securiti-es markets. Journal of economic theory,1979,20:381-408P
[17]Harrison J M, Pliska S. Martingales and multiperiod securities marke-t. Journal of economic theory,1979,20(3):381-408P
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[3]Louis Bachelier. La theorie de la speculation. Ann sci ecole norm sup, 1900,17:21-86P
[4]Markowitz H. Portfolio selection. The Journal of finance,1952,7(1): 77-91P
[5]Sharp W F. Capital asset prices:a theory of market equilibrium under cond-itions of risk. Journal of Finance,1964,1(1):425-442P
[6]Linter J. The valuation of risk assets and the selection of risky investments in stock porfolios and capital budgets. Review of economics and statistic, 1965,47(2):13-37P
[7]Mossin J. Equilibrium in a capital asset market. Econcmetrica,1996,34: 768-783P
[8]Samuelson P A. Proof that properly anticipated prices fluctuate randoml-y. Industrial management review,1965,6:41-50P
[9]Black F, Scholes M. The pricing of options and corporate liabilitie-s. Journal of political economy,1973,81(2):637-659P
[10]Vasicek O. An equilibrium characterization of the term structure, Journ-al of financial economics,1977,5:177-188P
[11]Cox J, and Ingersoll J, Ross S. A theory of the term structure of interest rates. Econometrica,1985,53:385-408P
[12]Hull J, White A. Pricing interest rate derivative securities. Review of financial studies,1990,3:573-592P
[13]Health D, Jarrow R, Morton A. Bond pricing and the term structure of interest rates:a new methodology. Econometrica,1992,60,77-105P
[14]Brace A, Gatarek D, Musiela M. The market model of interest rate dynamics. Mathematical finance,1997,7(2):127-155P
[15]Cox J C, Ross S A, Rubinstein M. Option pricing:a simplified appr-oach. Journal of financial economic,1979,7(3):229-263P
[16]Harrison J M, Kreps D M. Martingales and arbitrage in multiperiod securiti-es markets. Journal of economic theory,1979,20:381-408P
[17]Harrison J M, Pliska S. Martingales and multiperiod securities marke-t. Journal of economic theory,1979,20(3):381-408P
[18]Harrison J M, Pliska S. Martingales and stochastic integrals in the theory of continuous trading. Stochastic processes and their applications,1981, 11(2):215-260P
[19]Harrison J M, Pliska S. A stochastic calculus model of continuous trading complete markets. Stochastic processes and their applications,1983,15: 313-316P
[20]Bladt M, Rydberg HT. An actuarial approach to option pricing under the physical measure and without market assumptions. Insurance:mathematics and economics,1998,22(1):65-73P
[21]Follmer H, Schweizer M. Hedging of contingent claims under incomplete information. Applied stochastic analysis, eds. M. H. A. Davis and R. J. Elliott, Stochastics monographs, Gordon and Breach, London/New York,1991,5:389-414P
[22]Ansel J P, Stricker C. Lois de martingale, densities et decomposition de Follmer-Schweizer. Ann. Inst. H. Poincare probed statist,1992,28, 375-392P
[23]Schweizer M. Martingale densities for general asset prices. Journal of mathematical economics,1992(a),21:363-378P
[24]Schweizer M. On the minimal martingale measure and the Follmer Schw-eizer decomposition. Stochastic analysis and applications,1995(b),13: 573-599P
[25]Schweizer M. Approximation pricing and the variance-optimal martingale measure. The annals of probability,1996,24(1):206-236P
[26]Schweizer M. A minimality property of the minimal martingale measure. S tatistics and probability letters,1999(a),42:27-31P
[27]Ping Li, Jian-ming Xia. Minimal martingale measures for discrete-time incomplete financial markets. Acta mathematics applicated sinica, English series,2002,18(2):349-352P
[28]Jean Paul Laurent, Huyen Pham. Dynamic programming and mean-varia-nce hedging. Finance stochastics,1999,3:83-110P
[29]Takuji Arai. The optimal martingale measure in continuous trading model-s. Statistics and probability letters,2001,54:93-99P
[30]Miyahara. Canonical martingale measures of incomplete assets market-s. Proceedings of the seventh Japan-Russian symposium Singapore:World scientific,1996:343-352P
[31]Tsukasa Fujiwara, Yoshio Miyahara. The minimal entropy martingale measure for geometric Levy processes. Finance and stochastics,2003,7: 509-531P
[32]Vicky Henderson. Analytical comparisons of option prices in stochastic volatility models, August 28,2002
[33]闫海峰,刘三阳.广义Black-Scholes模型期定价新方法-保险精算方法.应用数学和力学,2003,24(7)：730-738页
[34]刘倩,刘新平.外汇期权定价的新方法-保险精算方法.贵州大学学报(自然科学版),2004,21(1)：43-45页
[35]叶小青,蹇明,吴永红.亚式期权的保险精算定价.华中科技大学学报(自然科学版),2005,33(3)：91-121页
[36]张敏,王莺,何穗.股票价格遵循非时齐Poisson跳的亚式期权保险.湖北工业大学学报,2007,22(1)：97-100页
[37]毕学慧,杜雪樵.后定选择权的保险精算定价.合肥工业大学生学报(自然科学版),2704,30(5)：649-651页
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