套利定价理论及保险精算方法在期权定价中的应用
|
摘要
套利是一种常见的，以获利为目的的交易行为，它是基于无风险的超额利润产生的。如果某个资产(组合)存在无风险的超额利润，就会产生套利行为。套利是市场有效的内在力量，所有的市场价格都完全及时地反映了所有可能得到的信息，任何暂时的价格偏离都将因套利而迅速消失，当市场达到均衡(供给=需求)状态时，套利将不复存在。因此套利定价理论是以市场有效性为基础的一种均衡价格方法论，是金融产品及其衍生产品定价的理论基础。
期权定价问题是金融数学的核心问题之一。在市场有效，均衡的条件下，对期权进行定价可以采用套利定价方法。根据套利理论的思想，当不存在套利机会时，一个无风险的组合只能获得正常的无风险收益，或者说它的价值只能以无风险利率增长，我们可以通过构造无风险的证券组合，或者将证券拆分与复合来为期权定价。
然而，对期权进行定价的传统的套利定价方法，只适用于金融市场完备时的期权定价问题。在套利定价理论中，市场的完备性是对于任一随机现金流均能进行无套利定价的一个很重要的条件。如果市场是不完备的，则可能对任一随机现金流就不能进行无套利定价，此时可以将期权定价问题转化为等价的公平保费问题，利用公平保费原则和价格过程的实际概率测度为期权定价，这种定价方法我们称之为保险精算方法。
本文的研究目的就是要对金融学中若干期权定价问题，通过考虑各种因素，运用套利定价理论建立偏微分方程、构造等价鞅测度以及利用保险精算方法这些数学工具建立期权定价的数学模型，推导出期权价值方程及合理的期权价值，试图得到一些对金融实践具有指导意义并且易于操作的结果。
本文研究并主要解决了下面几个问题：
1)指出套利定价理论的基本思想和意义以及它在金融产品和金融衍生产品定价中的应用问题。
2)将套利定价理论应用到期权定价中，首先改变Black-Scholes模型的基本假设条件，通过构造无风险的证券组合，建立偏微分方程，得到有交易成本的股票价格服从混合过程的欧式期权定价模型并推导出期权的定价公式；其次利用鞅论的有关知识，通过无套利机会等价于存在一个风险中性概率测度(等价概率鞅测度)的资产定价基本定理，构造等价鞅测度，推导出欧式未定权益的一般定价公式；
最后指出了套利定价和风险中性定价的一致性.
3)引入了一种期权定价的新方法，即保险精算方法，可以解决当市场是有套
利、非均衡、不完备时的期权定价问题，将期权定价问题转化为等价的公平保费
的确定问题，给出了股票的连续复利预期收益率为产(S(t))和波动率为口(S(t))的广
义B一S模型的期权定价的一般方法.利用保险精算方法给出了股票价格遵循广义
0一U过程模型的欧式期权的精确定价公式和买权与卖权之间的平价关系以及股票
价格过程遵循指数Levy过程的定价模型，最后推导出利率确定情形下外汇期权的
定价公式，并且给出了保险精算方法在可转换债券定价中的应用.
Arbitrage is a usual transaction behavior in order to make profits. It is the possibility of riskless profits. If an asset (portfolio) has riskless super profits, there is arbitrage opportunity in the market. Arbitrage is an internal power to make the market efficient, all the prices in the market reflect all the information entirely and immediately. Any price temporarily deviating from its nature will disappear by the arbitrage, that is to say, when capital market reaches balance (supply = demand), there is no arbitrage. Therefore, arbitrage pricing theory, which is based on the efficiency of the market, uses equilibrium methodology. The theory can be applied to the pricing of financial products and their derivative products widely.
The problem of option pricing is one of key problems in financial mathematics. Under the condition of efficient, equilibrium markets, options can be priced using arbitrage pricing method. According to arbitrage theory, when there is no arbitrage, a riskless portfolio can only obtain normal riskless profits. Therefore, we can make use of constructing a risk free portfolio or separating and compounding securities for option pricing.
However, traditional arbitrage pricing method can only be applied to complete financial market substantially. Completeness is an important condition in no-arbitrage pricing. So incompleteness implies that there exist securities which can not be priced using this method. But we can calculate the price of option as the fair premium needed to insure the potential loss from the issuers' point of view, using the principle of fair premium and physical probabilistic measure of price process. We call this method insurance actuary pricing.
This paper deals with some option pricing problems in finance with all kinds of factors, by establishing partial differential equations, constructing equivalent martingale measure and using insurance actuary pricing method. It calculates equations of the price of options and rational price of options, trying to obtain some outcomes to guide financial practice and make it easy to be operated.
The essay mainly answers the following problems:
1) We point out the basic ideas and significance of arbitrage pricing theory and its uses in the pricing of financial products and their derivative products.
2) We apply the theory of arbitrage pricing to option pricing. Firstly, by changing basic assumption of Black-Scholes option pricing model to the assumption that stock pricing process is a mixed process, give basic option pricing equations with transaction costs whose stock pricing process is a mixed process replicating the payoff of option by portfolio and solving partial differential equations. Secondly, deal with general pricing formulas of European contingent claim by means of martingale method of constituting equivalent martingale measure on the basis of the basic theorem of asset pricing that no-arbitrage opportunity is equivalent to the existence of neutral-risk probabilistic measure (equivalent martingale measure). Thirdly, argue about the consistency between arbitrage pricing and neutral-risk pricing.
3) We introduce a new method to option pricing-an actuarial approach. It turns option pricing into an equivalent insurance or a fair premium determination. The approach is valid even when arbitrage exists and the market is incompleteness and un-equilibrium. Under the assumption that the expected rate μ(S(t)) , volatility
σ(S(t)) are functions of risk asset S(t), and stock pricing process respectively driven by a general O-U process and an exponential of a Levy process, we obtain the accurate pricing formulas and put-call parity of European option. In the end, assume that riskless rate is given, we deal with pricing formulas of European option on foreign currency and apply the approach to the pricing of the convertible bond.
期权定价问题是金融数学的核心问题之一。在市场有效，均衡的条件下，对期权进行定价可以采用套利定价方法。根据套利理论的思想，当不存在套利机会时，一个无风险的组合只能获得正常的无风险收益，或者说它的价值只能以无风险利率增长，我们可以通过构造无风险的证券组合，或者将证券拆分与复合来为期权定价。
然而，对期权进行定价的传统的套利定价方法，只适用于金融市场完备时的期权定价问题。在套利定价理论中，市场的完备性是对于任一随机现金流均能进行无套利定价的一个很重要的条件。如果市场是不完备的，则可能对任一随机现金流就不能进行无套利定价，此时可以将期权定价问题转化为等价的公平保费问题，利用公平保费原则和价格过程的实际概率测度为期权定价，这种定价方法我们称之为保险精算方法。
本文的研究目的就是要对金融学中若干期权定价问题，通过考虑各种因素，运用套利定价理论建立偏微分方程、构造等价鞅测度以及利用保险精算方法这些数学工具建立期权定价的数学模型，推导出期权价值方程及合理的期权价值，试图得到一些对金融实践具有指导意义并且易于操作的结果。
本文研究并主要解决了下面几个问题：
1)指出套利定价理论的基本思想和意义以及它在金融产品和金融衍生产品定价中的应用问题。
2)将套利定价理论应用到期权定价中，首先改变Black-Scholes模型的基本假设条件，通过构造无风险的证券组合，建立偏微分方程，得到有交易成本的股票价格服从混合过程的欧式期权定价模型并推导出期权的定价公式；其次利用鞅论的有关知识，通过无套利机会等价于存在一个风险中性概率测度(等价概率鞅测度)的资产定价基本定理，构造等价鞅测度，推导出欧式未定权益的一般定价公式；
最后指出了套利定价和风险中性定价的一致性.
3)引入了一种期权定价的新方法，即保险精算方法，可以解决当市场是有套
利、非均衡、不完备时的期权定价问题，将期权定价问题转化为等价的公平保费
的确定问题，给出了股票的连续复利预期收益率为产(S(t))和波动率为口(S(t))的广
义B一S模型的期权定价的一般方法.利用保险精算方法给出了股票价格遵循广义
0一U过程模型的欧式期权的精确定价公式和买权与卖权之间的平价关系以及股票
价格过程遵循指数Levy过程的定价模型，最后推导出利率确定情形下外汇期权的
定价公式，并且给出了保险精算方法在可转换债券定价中的应用.
Arbitrage is a usual transaction behavior in order to make profits. It is the possibility of riskless profits. If an asset (portfolio) has riskless super profits, there is arbitrage opportunity in the market. Arbitrage is an internal power to make the market efficient, all the prices in the market reflect all the information entirely and immediately. Any price temporarily deviating from its nature will disappear by the arbitrage, that is to say, when capital market reaches balance (supply = demand), there is no arbitrage. Therefore, arbitrage pricing theory, which is based on the efficiency of the market, uses equilibrium methodology. The theory can be applied to the pricing of financial products and their derivative products widely.
The problem of option pricing is one of key problems in financial mathematics. Under the condition of efficient, equilibrium markets, options can be priced using arbitrage pricing method. According to arbitrage theory, when there is no arbitrage, a riskless portfolio can only obtain normal riskless profits. Therefore, we can make use of constructing a risk free portfolio or separating and compounding securities for option pricing.
However, traditional arbitrage pricing method can only be applied to complete financial market substantially. Completeness is an important condition in no-arbitrage pricing. So incompleteness implies that there exist securities which can not be priced using this method. But we can calculate the price of option as the fair premium needed to insure the potential loss from the issuers' point of view, using the principle of fair premium and physical probabilistic measure of price process. We call this method insurance actuary pricing.
This paper deals with some option pricing problems in finance with all kinds of factors, by establishing partial differential equations, constructing equivalent martingale measure and using insurance actuary pricing method. It calculates equations of the price of options and rational price of options, trying to obtain some outcomes to guide financial practice and make it easy to be operated.
The essay mainly answers the following problems:
1) We point out the basic ideas and significance of arbitrage pricing theory and its uses in the pricing of financial products and their derivative products.
2) We apply the theory of arbitrage pricing to option pricing. Firstly, by changing basic assumption of Black-Scholes option pricing model to the assumption that stock pricing process is a mixed process, give basic option pricing equations with transaction costs whose stock pricing process is a mixed process replicating the payoff of option by portfolio and solving partial differential equations. Secondly, deal with general pricing formulas of European contingent claim by means of martingale method of constituting equivalent martingale measure on the basis of the basic theorem of asset pricing that no-arbitrage opportunity is equivalent to the existence of neutral-risk probabilistic measure (equivalent martingale measure). Thirdly, argue about the consistency between arbitrage pricing and neutral-risk pricing.
3) We introduce a new method to option pricing-an actuarial approach. It turns option pricing into an equivalent insurance or a fair premium determination. The approach is valid even when arbitrage exists and the market is incompleteness and un-equilibrium. Under the assumption that the expected rate μ(S(t)) , volatility
σ(S(t)) are functions of risk asset S(t), and stock pricing process respectively driven by a general O-U process and an exponential of a Levy process, we obtain the accurate pricing formulas and put-call parity of European option. In the end, assume that riskless rate is given, we deal with pricing formulas of European option on foreign currency and apply the approach to the pricing of the convertible bond.
引文
[1] Black F., Scholes, M., The pricing of options and corporate liabilities[J], Journal of Political Economy, 1973, 8(7), 637-655.
[2] Merton, R.C., Theory of rational option pricing[J], Bell J of Econ and Management Sci, 1973, (4), 141-183.
[3] Harrison, J. M., Brownian motion and stochastic flow systems[J], John Wiley and Sons, New York, 1985.
[4] Harrison, J. M., and D. M. Kreps, Martingales and arbitrage in multiperiod markets [J], Journal of Economic Theory, 1979, 20, 381-408.
[5] Cox, J., Ross, S. and Rubinstein, M., Option pricing: A simplified approach[J], J. Finan. Econ., 3,145-166.
[6] Gerber, H.U., Shiu, S.W., Option pricing by Esscher transform[J], Transaction of the Society of Actuaries, 1994, 46, 99-140.
[7] 程希骏，张伟，非完整市场模型下的期权定价研究[J]，预测，2000，5，60-61．
[8] Mogens Bladt, Tina Hviid Rydberg, An actuarial approach to option pricing under the physical measure and without market assumptions[J], Insurance: Mathematics and Economics 1998; 22(1), 65-73.
[9] 约翰·赫尔，期权，期贷和衍生证券[M]，华夏出版社，1996．
[10] 弗兰克·J·法博齐，弗朗哥·莫迪利亚尼，资本市场：机构与工具[M]，经济科学出版社，1998．
[11] William, F. S., Gordon, J. A., Jeffrey V B, Investment[M], New York, Prentice Hall Inc, 1995.
[12] 约翰·马歇尔，维普尔·班赛尔，金融工程[M]，清华大学出版社，1998．
[13] Lamberton ,D., Lapeyre, B., Introduction to stochastic calculus applied to finance [M], Chapman and Hall, 1996.
[14] 南京工学院数学教研组，数学物理方程与特殊函数[M]，高等教育出版社，1982，56-65．
[15] Merton, M.C., Option pricing when underlying stock returns are discontinuous[J], Journal of Financial Economics, 1976(3), 125-144.
[16] Barles, G., Mete, H., Option pricing with transaction costs and a nonlinear Black-Scholes equation[J], Journal of Stochastic, 1998,72(2),69-397.
[17] 郑小迎，陈金贤，有交易成本的期权定价研究[J]，管理工程学报，2001，15(3)，35-37．
[18] Poul Wilmott, San Howynne, The mathematics of financial derivatives[M], London,
Cambridge University Press, 1995, 58-67.
[19] 马超群，陈牡妙，标的资产服从混合过程的期权定价模型[J]，系统工程理论与实践，1999，19(4)，41-46．
[20] 金治明，随机分析基础及其应用[J]，国防工业出版社，2003．
[21] ZHANG Shui-ming, General Black-Scholes model of security valuation[M], Acta Mathematica Scientia, 1999, 19(3), 279-288.
[22] 薛红，姚慧君，欧式未定权益的一般定价公式及其应用[J]，长安大学学报(自然科学版)，2002，22(1)，85-88．
[23] Karatzas, I., S.E.Shreve, Brown motion and stochastic calculus[J], Springer- Verlag, 1988.
[24] 姜礼尚，期权定价的数学模型和方法[M]，高等教育出版社，2002．
[25] 黄兴旺，朱楚珠，金融套利定价理论及应用分析[J]，武汉汽车工业大学学报，2000， 22(5)，102-105．
[26] Kouritzin, M. A., De Li, On explicit solution to stochastic differential equation[J], 2000, 18(4), 571-580.
[27] 武宝亭，李庆士，杨跃武，随机过程与随机微分方[J]，电子科技大学出版社，1994．
[28] 闫海峰，刘三阳，广义Black-Scholes模型期权定价新方法—保险精算方法[M]，应用数学和力学，2003，24(7)，730-739
[29] 闫海峰，刘三阳，带有Poisson跳的股票价格模型的期权定价[J]，工程数学学报，2003，20(2)，35-40．
[30] Eberlein, E., Keller, U., Hyperbolic distributions in finance[J], Bernoulli 1,281-299.
[31] Barndorff-Nielsen, O.E., Normal inverse Gaussian distributions and stochastic volatility modeling [J], Scandinavian Journal of Statistics 24, 1-13.
[32] Rydberg, T. H., The normal inverse Gaussian Levy process[J], Simulation and Approximation, Stochastic Models 13(4).
[33] Garman and Kohlhagen, Foreign currency option values[J], Journal of International Money and Finance, 1983, 23, 1-7.
[34] 薛红，聂赞坎，随机利率情形下外汇未定权益定价[J]，系统工程学报，2000，15(3)， 287-289．
[35] Musiela, M., Rutkowski, M., Martingale methods in financial modeling[M], Berlin, Heidelberg, Newyork: Springer-Verlag, 1997, 109-181
[36] 严加安，鞅与随机分析引论[M]，上海科技出版社，1981．
[37] 严加安，彭实戈，方诗赞，吴黎明等，随机分析选讲[M]，科学出版社，2000．
[38] Jia-An Yan, Introduction to martingale methods in option pricing[M], Lecture Notes in Mathematics, Liu Bie Ju Centre for Mathematical Sciences.
[39] Joseph Stampfli, Victor Goodman, The mathematics of finance: Modeling and hedging [M], China Machine Press, 2003.
[40] 何声武，随机过程引论[M]，高等教育出版社，1999．
[2] Merton, R.C., Theory of rational option pricing[J], Bell J of Econ and Management Sci, 1973, (4), 141-183.
[3] Harrison, J. M., Brownian motion and stochastic flow systems[J], John Wiley and Sons, New York, 1985.
[4] Harrison, J. M., and D. M. Kreps, Martingales and arbitrage in multiperiod markets [J], Journal of Economic Theory, 1979, 20, 381-408.
[5] Cox, J., Ross, S. and Rubinstein, M., Option pricing: A simplified approach[J], J. Finan. Econ., 3,145-166.
[6] Gerber, H.U., Shiu, S.W., Option pricing by Esscher transform[J], Transaction of the Society of Actuaries, 1994, 46, 99-140.
[7] 程希骏，张伟，非完整市场模型下的期权定价研究[J]，预测，2000，5，60-61．
[8] Mogens Bladt, Tina Hviid Rydberg, An actuarial approach to option pricing under the physical measure and without market assumptions[J], Insurance: Mathematics and Economics 1998; 22(1), 65-73.
[9] 约翰·赫尔，期权，期贷和衍生证券[M]，华夏出版社，1996．
[10] 弗兰克·J·法博齐，弗朗哥·莫迪利亚尼，资本市场：机构与工具[M]，经济科学出版社，1998．
[11] William, F. S., Gordon, J. A., Jeffrey V B, Investment[M], New York, Prentice Hall Inc, 1995.
[12] 约翰·马歇尔，维普尔·班赛尔，金融工程[M]，清华大学出版社，1998．
[13] Lamberton ,D., Lapeyre, B., Introduction to stochastic calculus applied to finance [M], Chapman and Hall, 1996.
[14] 南京工学院数学教研组，数学物理方程与特殊函数[M]，高等教育出版社，1982，56-65．
[15] Merton, M.C., Option pricing when underlying stock returns are discontinuous[J], Journal of Financial Economics, 1976(3), 125-144.
[16] Barles, G., Mete, H., Option pricing with transaction costs and a nonlinear Black-Scholes equation[J], Journal of Stochastic, 1998,72(2),69-397.
[17] 郑小迎，陈金贤，有交易成本的期权定价研究[J]，管理工程学报，2001，15(3)，35-37．
[18] Poul Wilmott, San Howynne, The mathematics of financial derivatives[M], London,
Cambridge University Press, 1995, 58-67.
[19] 马超群，陈牡妙，标的资产服从混合过程的期权定价模型[J]，系统工程理论与实践，1999，19(4)，41-46．
[20] 金治明，随机分析基础及其应用[J]，国防工业出版社，2003．
[21] ZHANG Shui-ming, General Black-Scholes model of security valuation[M], Acta Mathematica Scientia, 1999, 19(3), 279-288.
[22] 薛红，姚慧君，欧式未定权益的一般定价公式及其应用[J]，长安大学学报(自然科学版)，2002，22(1)，85-88．
[23] Karatzas, I., S.E.Shreve, Brown motion and stochastic calculus[J], Springer- Verlag, 1988.
[24] 姜礼尚，期权定价的数学模型和方法[M]，高等教育出版社，2002．
[25] 黄兴旺，朱楚珠，金融套利定价理论及应用分析[J]，武汉汽车工业大学学报，2000， 22(5)，102-105．
[26] Kouritzin, M. A., De Li, On explicit solution to stochastic differential equation[J], 2000, 18(4), 571-580.
[27] 武宝亭，李庆士，杨跃武，随机过程与随机微分方[J]，电子科技大学出版社，1994．
[28] 闫海峰，刘三阳，广义Black-Scholes模型期权定价新方法—保险精算方法[M]，应用数学和力学，2003，24(7)，730-739
[29] 闫海峰，刘三阳，带有Poisson跳的股票价格模型的期权定价[J]，工程数学学报，2003，20(2)，35-40．
[30] Eberlein, E., Keller, U., Hyperbolic distributions in finance[J], Bernoulli 1,281-299.
[31] Barndorff-Nielsen, O.E., Normal inverse Gaussian distributions and stochastic volatility modeling [J], Scandinavian Journal of Statistics 24, 1-13.
[32] Rydberg, T. H., The normal inverse Gaussian Levy process[J], Simulation and Approximation, Stochastic Models 13(4).
[33] Garman and Kohlhagen, Foreign currency option values[J], Journal of International Money and Finance, 1983, 23, 1-7.
[34] 薛红，聂赞坎，随机利率情形下外汇未定权益定价[J]，系统工程学报，2000，15(3)， 287-289．
[35] Musiela, M., Rutkowski, M., Martingale methods in financial modeling[M], Berlin, Heidelberg, Newyork: Springer-Verlag, 1997, 109-181
[36] 严加安，鞅与随机分析引论[M]，上海科技出版社，1981．
[37] 严加安，彭实戈，方诗赞，吴黎明等，随机分析选讲[M]，科学出版社，2000．
[38] Jia-An Yan, Introduction to martingale methods in option pricing[M], Lecture Notes in Mathematics, Liu Bie Ju Centre for Mathematical Sciences.
[39] Joseph Stampfli, Victor Goodman, The mathematics of finance: Modeling and hedging [M], China Machine Press, 2003.
[40] 何声武，随机过程引论[M]，高等教育出版社，1999．
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |