Vasicek利率模型下几种欧式未定权益的定价
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摘要
对未定权益进行定价是金融数学领域内一个既具有理论意义又有实际应用价值的重要问题。关于欧式未定权益定价的研究，在利率为常数或者是时间的确定性函数时，国内外学者已经做了大量地研究工作，得到了许多结果。然而，当利率是随机变量时，目前的研究成果并不多见。但是，这种情形在实际中又是存在的现象，因此必须考虑到利率的不确定性对衍生资产定价的影响。本文考虑的是一个完备的，连续的市场模型，其中资产价格的运动过程假设服从对数正态分布，利率运动过程假设为Vasicek利率模型。在这种情形下，利用Black-Scholes风险中性定价原则，推导出了几种欧式未定权益的定价公式。
主要得到了如下结果：
(1)在资产运动过程服从对数正态分布，利率服从Vasicek模型的假设下，利用风险中性定价原则，讨论了二元期权并得到了欧式买权的定价公式，以及买权和卖权的平价关系。
(2)利用(1)得到的平价关系和条件数学期望的性质，推导了任选期权的定价公式，并得到了显示解析式。
(3)采用几何平均计算资产的平均价格，在连续平均和离散平均两种情形下，分别给出了平均价格型和平均执行价格型亚式期权的定价公式及相应的买权和卖权的平价关系。
It is an important essay that studying the pricing of contingent claims is of theoretical significance and of practical value in financial mathematic. In the fields of the study of pricing of European contingent claims with interst rate being constant or deterministic functions, many authors have done many researchs and acquired a lot of achievements. However,when the interest rate being stochastic,these are not more.But the case about interest rate being stochastic exists in fact.So we must consider the random of interest rate affects the asset pricing. A complete and continuous market model which the process of asset price is assumed to be lognormal distribution arid the process of interest rate is assumed to be Vasicek are studied in this paper. By using Black-Scholes' risk-neutral valuation principal , pricing formulae of some European contingent claims are deduced .
main results as follows:
(1) The pricing formula of European call option and the put-call parity relation are deduced;
(2)By using the above put-call parity relationship and the properties of conditional expectation, we also get the formula of as-you-like-it option under Vasicek model.
(3)By applying the geometric average to compute the average price of assets which consists of two cases of continuous and discrete,the pricing formula for the Asian option of averag asset and average strike price and the parity relationship are obtained.
主要得到了如下结果：
(1)在资产运动过程服从对数正态分布，利率服从Vasicek模型的假设下，利用风险中性定价原则，讨论了二元期权并得到了欧式买权的定价公式，以及买权和卖权的平价关系。
(2)利用(1)得到的平价关系和条件数学期望的性质，推导了任选期权的定价公式，并得到了显示解析式。
(3)采用几何平均计算资产的平均价格，在连续平均和离散平均两种情形下，分别给出了平均价格型和平均执行价格型亚式期权的定价公式及相应的买权和卖权的平价关系。
It is an important essay that studying the pricing of contingent claims is of theoretical significance and of practical value in financial mathematic. In the fields of the study of pricing of European contingent claims with interst rate being constant or deterministic functions, many authors have done many researchs and acquired a lot of achievements. However,when the interest rate being stochastic,these are not more.But the case about interest rate being stochastic exists in fact.So we must consider the random of interest rate affects the asset pricing. A complete and continuous market model which the process of asset price is assumed to be lognormal distribution arid the process of interest rate is assumed to be Vasicek are studied in this paper. By using Black-Scholes' risk-neutral valuation principal , pricing formulae of some European contingent claims are deduced .
main results as follows:
(1) The pricing formula of European call option and the put-call parity relation are deduced;
(2)By using the above put-call parity relationship and the properties of conditional expectation, we also get the formula of as-you-like-it option under Vasicek model.
(3)By applying the geometric average to compute the average price of assets which consists of two cases of continuous and discrete,the pricing formula for the Asian option of averag asset and average strike price and the parity relationship are obtained.
引文
[1] A.Dravid,M.Richardson,and T.Sun, " Pricing Foreign Index Contingent Claims: An Application to Nikkei Index Warrants ", The Journal of Derivatives, Fall 1993.
[2] 章珂，周文彪，沈荣芳，“几何亚式期权的定价方法”，同济大学学报第29卷第8期，2001．
[3] 王莉军，张曙光，“随机利率下重置期权的定价问题’，高校应用数学学报A辑2002,17(4)：471-478
[4] 王莉军，张曙光，“Vasiek利率模型下的亚式期权定价问题和数值分析”，应用数学学报第26卷第3期，2003年7月。
[5] 林建忠，“亚洲期权定价的一个逼近解”，上海交通大学学报，第35卷第12期，2001年12月．
[6] 陈松男，“金融工程学”，复旦大学出版社，2002年11月，p 106-111，130-133．
[7] Damien Lamberton, Bernard Lapeyre, "Introduction To Stochastic Calculus Applied To Finance", Chapman & Hall,1996,p 137-138.
[8] Steven Shreve," Stochastic Calculus and Finance", 1997年10月, p 286-291.
[9] Yue-Kuen KwoK,"Mathematical Models of Financial Derivatives", Springer,1999, p. 81-82,282-291.
[10] 雍炯敏，刘道百，“数学金融学”，上海人民出版社，2003，p 237-239．
[11] 洛伦兹．格利茨著，唐旭等译，“金融工程学”，(修订版，)经济科学出版社，1999年11月，p
[2] 章珂，周文彪，沈荣芳，“几何亚式期权的定价方法”，同济大学学报第29卷第8期，2001．
[3] 王莉军，张曙光，“随机利率下重置期权的定价问题’，高校应用数学学报A辑2002,17(4)：471-478
[4] 王莉军，张曙光，“Vasiek利率模型下的亚式期权定价问题和数值分析”，应用数学学报第26卷第3期，2003年7月。
[5] 林建忠，“亚洲期权定价的一个逼近解”，上海交通大学学报，第35卷第12期，2001年12月．
[6] 陈松男，“金融工程学”，复旦大学出版社，2002年11月，p 106-111，130-133．
[7] Damien Lamberton, Bernard Lapeyre, "Introduction To Stochastic Calculus Applied To Finance", Chapman & Hall,1996,p 137-138.
[8] Steven Shreve," Stochastic Calculus and Finance", 1997年10月, p 286-291.
[9] Yue-Kuen KwoK,"Mathematical Models of Financial Derivatives", Springer,1999, p. 81-82,282-291.
[10] 雍炯敏，刘道百，“数学金融学”，上海人民出版社，2003，p 237-239．
[11] 洛伦兹．格利茨著，唐旭等译，“金融工程学”，(修订版，)经济科学出版社，1999年11月，p