一类交换期权和一类随机利率期权定价问题的研究
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摘要
期权理论是20世纪世界经济学领域最伟大的发现之一,与投资组合理论、资本资产定价理论、市场有效性理论以及代理问题一起,构成现代金融学的五大理论模块,对于传统的Black-Scholes模型,国内外学者已经做了大量研究工作,获得许多对金融实践有指导意义的结果。
本文主要得到如下结果:
(1)在风险中性的假设下,建立了跳过程为一类特殊的更新过程时的股票价格模型,利用鞅方法得到了此模型下的欧式广义交换期权的定价,并列出了此模型一些特例和推广,以及套期保值策略。
(2)建立了股票价格服从指数O-U过程的随机微分方程,在风险中性的假设下利用Girsanov定理找到了该模型的唯一等价鞅测度。利用期权定价的鞅方法,得到了随机利率情形下股票价格服从指数O-U过程,并且影响利率的因素与影响股票价格的因素相关时欧式期权的定价。
Option theory is one of the greatest findings in the area of the world'seconomics in the 20 century.Together with the portfolio selection theory,the capital asset pricing theory,the effectiveness theory of market and acting issue,it is regarded as one of the five theory modules in modern finance.Many scholars have done a great deal of researches on Black-Scholes model and obtained a lot of results which is instructive to financial practice.
In this paper it has made main conclusions as follows:
(1)Under the risk-neutral hypothesis , we construct the model of stock price which jump process is a kind of special renewal process and obtain European general exchange option pricing formula under this model by means of martingale method. At last, this paper list some special cases of this model and generalize the pricing model and hedging strategy.
(2)Construct stochastic different equation of stock price whose process is driven by exponential Ornstein-Uhlenbeck process .Under the risk-neutral hypothesis, then the equivalent martingale measure is found by means of Girsanov theorem, we obtain the European option pricing on stocks whose price is driven by exponential Ornstein-Uhlenbeck process under stochastic interest rate ,and the factors affect the interest rate and the price of the stocks are correlative.
本文主要得到如下结果:
(1)在风险中性的假设下,建立了跳过程为一类特殊的更新过程时的股票价格模型,利用鞅方法得到了此模型下的欧式广义交换期权的定价,并列出了此模型一些特例和推广,以及套期保值策略。
(2)建立了股票价格服从指数O-U过程的随机微分方程,在风险中性的假设下利用Girsanov定理找到了该模型的唯一等价鞅测度。利用期权定价的鞅方法,得到了随机利率情形下股票价格服从指数O-U过程,并且影响利率的因素与影响股票价格的因素相关时欧式期权的定价。
Option theory is one of the greatest findings in the area of the world'seconomics in the 20 century.Together with the portfolio selection theory,the capital asset pricing theory,the effectiveness theory of market and acting issue,it is regarded as one of the five theory modules in modern finance.Many scholars have done a great deal of researches on Black-Scholes model and obtained a lot of results which is instructive to financial practice.
In this paper it has made main conclusions as follows:
(1)Under the risk-neutral hypothesis , we construct the model of stock price which jump process is a kind of special renewal process and obtain European general exchange option pricing formula under this model by means of martingale method. At last, this paper list some special cases of this model and generalize the pricing model and hedging strategy.
(2)Construct stochastic different equation of stock price whose process is driven by exponential Ornstein-Uhlenbeck process .Under the risk-neutral hypothesis, then the equivalent martingale measure is found by means of Girsanov theorem, we obtain the European option pricing on stocks whose price is driven by exponential Ornstein-Uhlenbeck process under stochastic interest rate ,and the factors affect the interest rate and the price of the stocks are correlative.
引文
[1]Black F.Scholes M.The pricing of options and corporate liabilities[J].Journal of political Economy.1973,81:133-155.
[2]Hull,J.C.期权,期货和其他衍生产品(第三版).北京:华夏出版社,2003
[3]Markowitz.H,Portfolio selection.Journal of Financial,1952,7:77-91
[4]Sharp.W.F,Captial asset prices:a theory of market equilibrium under conditions of risk.Journal of Finance,1964,19:425-442
[5]Lintner.J,The valuation of risk assets and the selection of risky investments in stock protfolios and captiai budgets,Review of Economic and Statistics,1965,47:13-37
[6]Boness,A.J.Elements of a Theory of stock option value.Journal of Political Economy 1964,72:163-175
[7]Merton,R.C.,Influence of Mathematical Models in Finance On Practice:Past,Present and Futuer,Mathematial Model in Finance,Howison,Chapman&Hall,London,1995
[8]Cox.J.&Ross.S.The valuation of option for alternative stochastic processes.Journal of Financial Economics.1976,3,145-166
[9]Merton,M.C.,Option pricing when underlying stock returns are discontinuous.Journal of Financial Economics.1976,3,125-144
[10]Samuelson,P.A.Rational Theory of Warrant Pricing.Industrial Management Review.1965,6:13-31
[11]Bachelier,Louis,Theorie de la speculation,Annales de LEcole Normale Superieure1900,17:21-86
[12]Cox,J,S.Ross,and M.Rubinstein.Option pricing:A simplified Approach.Journal of Financial Economy 1979,7:229-263
[13]Cox,J.C.,and Ross,S.A.The valuation of options for processes.Journal of Financial Economics,1976,3:145-160
[14]Harrison M,Kreps D.Martingales and Arbitrage In Mufti-period Securities Markets.Economical Theory,1979,20:145-166
[15]Harrison,M.Pliska R.,Martingales and Stochastic Integrals in the theory of Continuous Trading.[J]Stochastic Process and their Application 1981,11(2):215-216
[16]Merton R C.Option pricing when underlying stock Returns are discontinous [J],Journal of Financial Ecnomical.May,1976,125-144.
[17]Merton R C.Theory of rational option pricing[J].Bell Journal of Economies and management Science 4,No.1,Springer,1997,141-183.
[18]Ball C A,Torous W N.On jumps in common stock prices and their impact on call option pricing[J]Journal of Finance,1985,40:155-173.
[19]Zhang Shuguang,Yuan Shuiyong,Wang Lijun.Prices of asian options under stochastic interest rates[J]Appl.Math.J.Chinese Univ.Ser.B 2006,21(2:135-142.
[20]Kwork,Y.K.Mathematical models of financial derivatives[M].Springer,1998:50-90.
[21]Hull,J.C.,and A.White.The pricing of options on assets with stochastic volatilities.Journal of Finance,1987,42:281-300.
[22]Peter G..Zhang.Exotic options,2nd Edtion.Singapore,World Scientific Publishing.Pte.Ltd,1998:329-340.
[23]Chan T.Pricing contingent claims on stocks driven by Levy process[J].Annals of Appl Prob,1999;9(2):504-528
[24]Margrabe W.The Value of Option to one Asset for Another Journal of Finance,Vol,ⅩⅩⅫⅠ:177-186.
[25]Knut K,AASE.Contingent claims valuation when the security price is combination of ito process and a random point process[J].Stoehastic process and their Application,1998:28(2):185-220.
[26]阎海峰,刘三阳,一类特殊模型的美式期权定价,西安电子科技大学学报,第30卷第1期,2003年2月,125-127
[27]郑小迎,陈金贤,关于美式期权定价方法的研究,陕西工学院学报,第15卷第3期,1999年9月,1-5
[28]孙国红,数学金融学中的期权定价问题,天津商学院学报,第23卷第3期,2003年5月,22-5
[29]周志超,随机利率下可转化债券定价一鞍方法,经济数学,2004,NO.1:102-109
[30]龚鲁光,随机微分方程引论[M],北京大学出版社,1987
[31]严加安.鞅与随机积分引论[M].上海:上海科技出版社,1981.
[32]金治明.随机分析基础及其应用[M].北京:国防工业出版社,2003:159-251.
[33]金治明.数学金融学基础[M].北京:国防工业科学出版社,2006.
[34]杜雪樵,惠军.随机过程.[M].合肥:合肥工业大学出版社,2006,65-72.
[35]张波,张景肖.应用随机过程[M],北京:清华大学出版社,2004,32-48
[36]闫海峰,刘三阳.带有跳的股票价格模型的期权定价[J].工程数学学报, 2003,5(3):35-40.
[37]薛红,彭玉成.鞅在未定权益定价中的应用[J].工程数学学报,2000,17(3):135-138.
[38]杨云峰,刘新平.一类具有随机利率的跳扩散模型的期权定价[J].纯粹数学与应用数学.2006,22(1):43-47.
[39]胡志锋,黄荣坦.纯生跳扩散型交换期权定价公式[J].数学研究,2005,38(3):333-338.
[40]陈超,邹捷中,王自后.股票价格服从纯生跳-扩散过程的期权定价模型[J].数量经济技术经济研究.
[41]宁丽娟,刘新平.股票价格服从跳-扩散过程的期权定价模型.[J].陕西师范大学学报:自然科学版,2003,31(4):16-19.
[42]魏正元.Black-Scholes期权定价公式推广[J].数学的实践与认识,2005,35(6):35-40.
[43]叶中行,林建忠,数理金融-资产定价与金融决策理论[M],北京:科学出版社。
[44]黄致远,随机分析学基础[M],北京:科学出版社,2001
[45]阎海峰,刘三阳,李文强.股票价格遵循指数O-U过程的最大值期权定价[J].工程数学学报:2004,21(3):397-402.
[46]王剑君.随机利率下欧式双向期权的定价[J].统计与决策:2006(2):30-32.
[2]Hull,J.C.期权,期货和其他衍生产品(第三版).北京:华夏出版社,2003
[3]Markowitz.H,Portfolio selection.Journal of Financial,1952,7:77-91
[4]Sharp.W.F,Captial asset prices:a theory of market equilibrium under conditions of risk.Journal of Finance,1964,19:425-442
[5]Lintner.J,The valuation of risk assets and the selection of risky investments in stock protfolios and captiai budgets,Review of Economic and Statistics,1965,47:13-37
[6]Boness,A.J.Elements of a Theory of stock option value.Journal of Political Economy 1964,72:163-175
[7]Merton,R.C.,Influence of Mathematical Models in Finance On Practice:Past,Present and Futuer,Mathematial Model in Finance,Howison,Chapman&Hall,London,1995
[8]Cox.J.&Ross.S.The valuation of option for alternative stochastic processes.Journal of Financial Economics.1976,3,145-166
[9]Merton,M.C.,Option pricing when underlying stock returns are discontinuous.Journal of Financial Economics.1976,3,125-144
[10]Samuelson,P.A.Rational Theory of Warrant Pricing.Industrial Management Review.1965,6:13-31
[11]Bachelier,Louis,Theorie de la speculation,Annales de LEcole Normale Superieure1900,17:21-86
[12]Cox,J,S.Ross,and M.Rubinstein.Option pricing:A simplified Approach.Journal of Financial Economy 1979,7:229-263
[13]Cox,J.C.,and Ross,S.A.The valuation of options for processes.Journal of Financial Economics,1976,3:145-160
[14]Harrison M,Kreps D.Martingales and Arbitrage In Mufti-period Securities Markets.Economical Theory,1979,20:145-166
[15]Harrison,M.Pliska R.,Martingales and Stochastic Integrals in the theory of Continuous Trading.[J]Stochastic Process and their Application 1981,11(2):215-216
[16]Merton R C.Option pricing when underlying stock Returns are discontinous [J],Journal of Financial Ecnomical.May,1976,125-144.
[17]Merton R C.Theory of rational option pricing[J].Bell Journal of Economies and management Science 4,No.1,Springer,1997,141-183.
[18]Ball C A,Torous W N.On jumps in common stock prices and their impact on call option pricing[J]Journal of Finance,1985,40:155-173.
[19]Zhang Shuguang,Yuan Shuiyong,Wang Lijun.Prices of asian options under stochastic interest rates[J]Appl.Math.J.Chinese Univ.Ser.B 2006,21(2:135-142.
[20]Kwork,Y.K.Mathematical models of financial derivatives[M].Springer,1998:50-90.
[21]Hull,J.C.,and A.White.The pricing of options on assets with stochastic volatilities.Journal of Finance,1987,42:281-300.
[22]Peter G..Zhang.Exotic options,2nd Edtion.Singapore,World Scientific Publishing.Pte.Ltd,1998:329-340.
[23]Chan T.Pricing contingent claims on stocks driven by Levy process[J].Annals of Appl Prob,1999;9(2):504-528
[24]Margrabe W.The Value of Option to one Asset for Another Journal of Finance,Vol,ⅩⅩⅫⅠ:177-186.
[25]Knut K,AASE.Contingent claims valuation when the security price is combination of ito process and a random point process[J].Stoehastic process and their Application,1998:28(2):185-220.
[26]阎海峰,刘三阳,一类特殊模型的美式期权定价,西安电子科技大学学报,第30卷第1期,2003年2月,125-127
[27]郑小迎,陈金贤,关于美式期权定价方法的研究,陕西工学院学报,第15卷第3期,1999年9月,1-5
[28]孙国红,数学金融学中的期权定价问题,天津商学院学报,第23卷第3期,2003年5月,22-5
[29]周志超,随机利率下可转化债券定价一鞍方法,经济数学,2004,NO.1:102-109
[30]龚鲁光,随机微分方程引论[M],北京大学出版社,1987
[31]严加安.鞅与随机积分引论[M].上海:上海科技出版社,1981.
[32]金治明.随机分析基础及其应用[M].北京:国防工业出版社,2003:159-251.
[33]金治明.数学金融学基础[M].北京:国防工业科学出版社,2006.
[34]杜雪樵,惠军.随机过程.[M].合肥:合肥工业大学出版社,2006,65-72.
[35]张波,张景肖.应用随机过程[M],北京:清华大学出版社,2004,32-48
[36]闫海峰,刘三阳.带有跳的股票价格模型的期权定价[J].工程数学学报, 2003,5(3):35-40.
[37]薛红,彭玉成.鞅在未定权益定价中的应用[J].工程数学学报,2000,17(3):135-138.
[38]杨云峰,刘新平.一类具有随机利率的跳扩散模型的期权定价[J].纯粹数学与应用数学.2006,22(1):43-47.
[39]胡志锋,黄荣坦.纯生跳扩散型交换期权定价公式[J].数学研究,2005,38(3):333-338.
[40]陈超,邹捷中,王自后.股票价格服从纯生跳-扩散过程的期权定价模型[J].数量经济技术经济研究.
[41]宁丽娟,刘新平.股票价格服从跳-扩散过程的期权定价模型.[J].陕西师范大学学报:自然科学版,2003,31(4):16-19.
[42]魏正元.Black-Scholes期权定价公式推广[J].数学的实践与认识,2005,35(6):35-40.
[43]叶中行,林建忠,数理金融-资产定价与金融决策理论[M],北京:科学出版社。
[44]黄致远,随机分析学基础[M],北京:科学出版社,2001
[45]阎海峰,刘三阳,李文强.股票价格遵循指数O-U过程的最大值期权定价[J].工程数学学报:2004,21(3):397-402.
[46]王剑君.随机利率下欧式双向期权的定价[J].统计与决策:2006(2):30-32.