欧式期权和美式期权定价的数值方法进一步研究
摘要
本文以无套利定价原则和风险中性定价原理为基础,对欧式期权和美式期权问题进行进一步研究.本文的主要工作包括:
(1)进一步讨论了欧式期权定价的相关理论和方法;
(2)分析研究了美式期权定价的数值方法,通过参数的改进,提高了模型的计算精度;
(3)基于G-融资策略通过套期保值方法进一步探讨了欧式期权公平价格和美式期权最佳交易时刻的问题.
对于欧式期权,本文采用Black-Scholes模型得出期权定价公式,并且讨论对比了非完备市场衍生资产定价的几种数值方法.而对于美式期权,主要探讨分析了两种数值方法:二叉树方法和有限差分法.由于普遍使用的二叉树参数模型带有缺陷,比如,在某种情况下将产生负的概率等,于是本文借助于随机误差校正思想,重新构造了二叉树参数模型,新模型永远不会产生负的概率且具有很高的计算精度,因而可实际应用于各种美式看跌期权的定价,而且这个新模型可推广到三叉树等更高阶的树图方法中去.在有限差分法中,为了保证计算结果更加精确,本文采用了控制变量技术.
最后,本文基于G-融资策略探讨了欧式期权和美式期权定价,并从理论上得出离散型美式期权的最佳套期交易时刻实际上是一个最优停时.
On the base of the non-arbitrage principle and the risk neutral valuation, we discuss European option and American option pricing. The main wok includes the following contents.
(1) Analyze relevant theories and methods of European option pricing;
(2) Investigate numerical methods of American option pricing, and improve precise valuation via new parameters;
(3) Study the fair price of European option and optimal time of American option through the hedging method based on G-hedging.
First, we get European option pricing formula by Black-Scholes model, and several numerical methods of derivative valuation in incomplete markets are discussed and compared. Then, we mainly deal with the binomial tree method and the finite difference method for American option pricing. The binomial tree parameter model has the flaw, for instance, it will have the negative probability in some kind of situation. So we construct new binomial tree parameter model, which will never have the negative probability and have the very high computation precision with the aid of the thought of the random error. Thus it might be applied to kinds of American put option pricing; moreover the new model might be extended to the trinomial tree and the higher order trees. In the finite difference method, we adopt control variable technique to obtain precise valuation.
At last, we discuss European option and American option pricing based on G- hedging, and prove optimal hedging-time of American option is a stopping-time.
(1)进一步讨论了欧式期权定价的相关理论和方法;
(2)分析研究了美式期权定价的数值方法,通过参数的改进,提高了模型的计算精度;
(3)基于G-融资策略通过套期保值方法进一步探讨了欧式期权公平价格和美式期权最佳交易时刻的问题.
对于欧式期权,本文采用Black-Scholes模型得出期权定价公式,并且讨论对比了非完备市场衍生资产定价的几种数值方法.而对于美式期权,主要探讨分析了两种数值方法:二叉树方法和有限差分法.由于普遍使用的二叉树参数模型带有缺陷,比如,在某种情况下将产生负的概率等,于是本文借助于随机误差校正思想,重新构造了二叉树参数模型,新模型永远不会产生负的概率且具有很高的计算精度,因而可实际应用于各种美式看跌期权的定价,而且这个新模型可推广到三叉树等更高阶的树图方法中去.在有限差分法中,为了保证计算结果更加精确,本文采用了控制变量技术.
最后,本文基于G-融资策略探讨了欧式期权和美式期权定价,并从理论上得出离散型美式期权的最佳套期交易时刻实际上是一个最优停时.
On the base of the non-arbitrage principle and the risk neutral valuation, we discuss European option and American option pricing. The main wok includes the following contents.
(1) Analyze relevant theories and methods of European option pricing;
(2) Investigate numerical methods of American option pricing, and improve precise valuation via new parameters;
(3) Study the fair price of European option and optimal time of American option through the hedging method based on G-hedging.
First, we get European option pricing formula by Black-Scholes model, and several numerical methods of derivative valuation in incomplete markets are discussed and compared. Then, we mainly deal with the binomial tree method and the finite difference method for American option pricing. The binomial tree parameter model has the flaw, for instance, it will have the negative probability in some kind of situation. So we construct new binomial tree parameter model, which will never have the negative probability and have the very high computation precision with the aid of the thought of the random error. Thus it might be applied to kinds of American put option pricing; moreover the new model might be extended to the trinomial tree and the higher order trees. In the finite difference method, we adopt control variable technique to obtain precise valuation.
At last, we discuss European option and American option pricing based on G- hedging, and prove optimal hedging-time of American option is a stopping-time.
引文
[1]John C.Hull.Options,Future,And other Derivatives[MJ.NY:Prentice Hall,2006
[2]Gemmill.G.Options Pricing[M].New York:McGraw Hill,1992,203-245
[3]Myneni,R.The pricing of the American option[J].Annals of Applied Probability,1992,2(1):1-23
[4]Bunch,David S,and Herb Johnson.The American put option and its critical stock price[J].Journal of Finance,2000,5:2333-2356
[5]Alfredo Ibanez.Factorization of European and American option prices under complete and incomplete markets[J].Journal of Banking & Finance,2008,32(2):311-325
[6]R.Merton.The theory of rational option pricing[J].Bell Journal of Economics and Management Science,1973,4:141-183
[7]郑承利.美式期权的几种蒙特卡罗仿真定价方法比较[J].系统仿真学报,2006,18(10):2929-2935
[8]D.Duffie,Dynamic Asset Pricing Theory(third ed)[J].Princeton University Press,Princeton,New Jersey,2001
[9]M.Broadie and J.B.Detemple,Option pricing:Valuation models and applications[J].Management Science,2004,50:l145-1177
[10]F.Black and M.Scholes.The pricing of options and corporate liabilities[J].Journal of Political Economy,1973,81:637-654
[11]M.Broadie and J.Detemple.Option pricing:valuation models and applications[J].Management Science,2004,50(9):1145-1177
[12]M.J.Brennan and E.S.Schwartz.The valuation of American put options[J].Journal of Finance,1977,32(2):449-462
[13]姜礼尚.期权定价的数学模型和方法[M].北京:高等教育出版社,2003
[14]刘海龙,吴冲锋.期权定价方法综述[J].管理科学学报,2002,5(2):67-73
[15]张铁.美式期权定价问题的数值方法[J].应用数学学报,2002,1:113-122
[16]张铁.一个新型的期权定价二叉树参数模型[J].系统工程理论与实践,2000,11:90-94
[17]陈万义.非风险中性定价意义下的欧式期权定价公式[J].数学的实践与认识,2004,34(1):76-79
[18]Cox J C,Rubinstein M.option Markets[M].Prentice Hall,1985
[19]吴恒煜.布莱克-斯科尔斯期权定价公式的推导及推广[J].商业研究,2006,348:42-45
[20]Vicky Henderson.Valuing the option to invest in an incomplete market[J].Springer-Verlag,2007,1:103-128
[21]YOU Su-rong.Option Pricing Method in a Market Involving Interval Number Factors[J].Journal of Donghua University,22(4):47-51
[22]Alefeld,G.and Herzberger.J.Introduction to Interval Computations[M].New York Academic Press,New York,1983
[23]吴金美,金治明,刘旭.支付持续红利的欧式和美式期权定价问题的研究[J].经济数学,2007,24(2):147-152
[24]Jaemin Ahn,Minsu Song.Convergence of the trinomial tree method for pricing European/American options[J].Applied Mathematics and Computation,2007,189:575-582
[25]Stylianos Perrakis,Jean Lefoll.Option pricing and replication with transaction costs and dividends[J].Journal of Economic Dynamics & Control,2000,24:1527-1561
[26]姚建平.离散型美式期权的最优套期交易时刻的选择[J].重庆建筑大学学报,2003,25(2):99-102
[27]Whaley,R.E.On the valuation of American call options on stocks with known dividends[J].Journal of Financial Economics,1981,9:207-212
[28]Jichao Zhao,Matt Davison,Robert M.Corless.Compact finite difference method for American option pricing[J].Journal of Computational and Applied Mathematics,2007,206:306-321
[29]M.Ahmed.An exploration of compact finite difference methods for the numerical solution of PDE[M].Ph.D.Thesis,the University of Western Ontario,1997
[30]S.D.Jacka.Optimal stopping and the American put option[J].Math.Financel,1991,2:1-14
[31]A.Bensoussan,On the theory of option pricing.Acta Appl[J].Math.1984,2(2):139-158
[32]郑振龙.衍生产品[M].武汉:武汉大学出版社,2005,220-232
[33]廖作鸿,李晓昭.控制变量技术在美式期权定价中的应用[J].石家庄经济学院学报,2005,28(6):784-786
[34]Brennan,M.& Schwartz,E.The valuation of American put options[J].Journal of Finance,1977,32:449-462
[35]Whaley,R.On the valuation of American call options on stocks with known dividends[J].Journal of Financial Economics,June,1981,207-211
[36]A.Sattauz,M.Pratelli.Optimal stopping and American options with discrete dividends and exogenous risk[J].Insurance:Mathematics and Economics,2004,35:255-265
[37]Protter,P.Stochastic Integration and Differential Equations[M].A New Approach.Springer,1990
[38]Marcozzi,M.D.On the approximation of optimal stopping problems with applications to financial mathematics[J].SIAM Journal on Scientific Computing,2001,22(5):1865-1884
[39]Ross S A.The arbitrage theory of capital asset pricing[]].Journal of Economic theory,1976,13(3):341-360
[40]雍炯敏,刘道百.数学金融学[M].上海:上海人民出版社,2003,96-99
[41]Jiang,L.Analysis of pricing American options on the maximum(minimum) of two risk assets[M].Interface & Free Boundaries,2002,4:27-46
[42]Cheng,W.& Zhang,S.The analysis of reset options[J].The Journal of Derivatives.2000,59-71
[43]Jiang,L&Dai,M.Convergence of explicit difference scheme and binomial tree method for American options[M].Journal of Computational Mathematics,2004,22:371-380
[44]许海峰,王凤荣,王德强.股票期权[M].北京:人民法院出版社,2005
[46]Merton R.Theory of rational option pricing[J].Bell Journal of Economics and Management Science.1973,4(1):141-183
[47]Cox J,Ross S.The valuation of options for alternation stochastic processes[J].Journal of Financial Economics,1976,3(3):145-166
[48]Rubinstein M.The valuation of uncertain income streams and the pricing of options[J].Bell Journal of Economics and Management Science,1976,7(2):407-425
[49]Cox J.Ross S.A survey of some new results in financial option pricing theory[]].Journal of Financial,1976,31(2):383-402
[50]Tolt K.On the mean-variance tradeoff in option replication with transactions costs[J].Journal of Financial and Quantitative Analysis,1996,31:233-263
[51]罗开位,侯振挺,李致中.期权定价理论的产生与发展[J].系统工程,2000,18(6):1-5
[52]Kassouf S T.An econometric model for option price with implications for investors expectations and audacity[J].E-conometrica.1969,37(4):685-694
[53]Boness A J.Elements of a theory of stock option value[]].Journal of Political Economy,1964,72(2):163-175
[54]Samuelon P A.Rational theory of warrant pricing[]].Industrial Management Review,1965,6(2):13-32
[55]Musiela M,Kutkowski M.Martingale methods in financial modeling theory and application[M].Beling Heidel-berg New York:Springer-Verlay,1997,128-180
[56]Phclim P B,Vorst T.Option replication in discrete time with transaction costs[J].Journal of Finance,1992,47(1):271-293
[57]David S B,Johnson H.The American put option and its critical stock price[J].Journal of Finance,2000,55(5):2333-2356
[58]蒲兴成,郑继明,游晓黔.基于Black-Scholes公式下美式期权价格的计算[J].重庆大学学报,2004,27(7):102-104
[59]S.L.Heston.A closed-form solution for options with stochastic volatility with application to bond and currency options[J].Review of Financial Studies,1993,6:327-343
[60]K.Borovkov,and A.Novikov.A new approach to calculating expectations for option pricing[J].Journal of Applied Probability,2002,39:889-895
[61]P.Carr,and D.B.Madan.Option valuation using the fast Fourier transform[J].Journal of Computational Finance,1999,2:61-73
[62]Y.K.Kwok.Mathematical Models of Financial Derivatives[M].Singapore:Springer-Verlag,1998,212-219
[2]Gemmill.G.Options Pricing[M].New York:McGraw Hill,1992,203-245
[3]Myneni,R.The pricing of the American option[J].Annals of Applied Probability,1992,2(1):1-23
[4]Bunch,David S,and Herb Johnson.The American put option and its critical stock price[J].Journal of Finance,2000,5:2333-2356
[5]Alfredo Ibanez.Factorization of European and American option prices under complete and incomplete markets[J].Journal of Banking & Finance,2008,32(2):311-325
[6]R.Merton.The theory of rational option pricing[J].Bell Journal of Economics and Management Science,1973,4:141-183
[7]郑承利.美式期权的几种蒙特卡罗仿真定价方法比较[J].系统仿真学报,2006,18(10):2929-2935
[8]D.Duffie,Dynamic Asset Pricing Theory(third ed)[J].Princeton University Press,Princeton,New Jersey,2001
[9]M.Broadie and J.B.Detemple,Option pricing:Valuation models and applications[J].Management Science,2004,50:l145-1177
[10]F.Black and M.Scholes.The pricing of options and corporate liabilities[J].Journal of Political Economy,1973,81:637-654
[11]M.Broadie and J.Detemple.Option pricing:valuation models and applications[J].Management Science,2004,50(9):1145-1177
[12]M.J.Brennan and E.S.Schwartz.The valuation of American put options[J].Journal of Finance,1977,32(2):449-462
[13]姜礼尚.期权定价的数学模型和方法[M].北京:高等教育出版社,2003
[14]刘海龙,吴冲锋.期权定价方法综述[J].管理科学学报,2002,5(2):67-73
[15]张铁.美式期权定价问题的数值方法[J].应用数学学报,2002,1:113-122
[16]张铁.一个新型的期权定价二叉树参数模型[J].系统工程理论与实践,2000,11:90-94
[17]陈万义.非风险中性定价意义下的欧式期权定价公式[J].数学的实践与认识,2004,34(1):76-79
[18]Cox J C,Rubinstein M.option Markets[M].Prentice Hall,1985
[19]吴恒煜.布莱克-斯科尔斯期权定价公式的推导及推广[J].商业研究,2006,348:42-45
[20]Vicky Henderson.Valuing the option to invest in an incomplete market[J].Springer-Verlag,2007,1:103-128
[21]YOU Su-rong.Option Pricing Method in a Market Involving Interval Number Factors[J].Journal of Donghua University,22(4):47-51
[22]Alefeld,G.and Herzberger.J.Introduction to Interval Computations[M].New York Academic Press,New York,1983
[23]吴金美,金治明,刘旭.支付持续红利的欧式和美式期权定价问题的研究[J].经济数学,2007,24(2):147-152
[24]Jaemin Ahn,Minsu Song.Convergence of the trinomial tree method for pricing European/American options[J].Applied Mathematics and Computation,2007,189:575-582
[25]Stylianos Perrakis,Jean Lefoll.Option pricing and replication with transaction costs and dividends[J].Journal of Economic Dynamics & Control,2000,24:1527-1561
[26]姚建平.离散型美式期权的最优套期交易时刻的选择[J].重庆建筑大学学报,2003,25(2):99-102
[27]Whaley,R.E.On the valuation of American call options on stocks with known dividends[J].Journal of Financial Economics,1981,9:207-212
[28]Jichao Zhao,Matt Davison,Robert M.Corless.Compact finite difference method for American option pricing[J].Journal of Computational and Applied Mathematics,2007,206:306-321
[29]M.Ahmed.An exploration of compact finite difference methods for the numerical solution of PDE[M].Ph.D.Thesis,the University of Western Ontario,1997
[30]S.D.Jacka.Optimal stopping and the American put option[J].Math.Financel,1991,2:1-14
[31]A.Bensoussan,On the theory of option pricing.Acta Appl[J].Math.1984,2(2):139-158
[32]郑振龙.衍生产品[M].武汉:武汉大学出版社,2005,220-232
[33]廖作鸿,李晓昭.控制变量技术在美式期权定价中的应用[J].石家庄经济学院学报,2005,28(6):784-786
[34]Brennan,M.& Schwartz,E.The valuation of American put options[J].Journal of Finance,1977,32:449-462
[35]Whaley,R.On the valuation of American call options on stocks with known dividends[J].Journal of Financial Economics,June,1981,207-211
[36]A.Sattauz,M.Pratelli.Optimal stopping and American options with discrete dividends and exogenous risk[J].Insurance:Mathematics and Economics,2004,35:255-265
[37]Protter,P.Stochastic Integration and Differential Equations[M].A New Approach.Springer,1990
[38]Marcozzi,M.D.On the approximation of optimal stopping problems with applications to financial mathematics[J].SIAM Journal on Scientific Computing,2001,22(5):1865-1884
[39]Ross S A.The arbitrage theory of capital asset pricing[]].Journal of Economic theory,1976,13(3):341-360
[40]雍炯敏,刘道百.数学金融学[M].上海:上海人民出版社,2003,96-99
[41]Jiang,L.Analysis of pricing American options on the maximum(minimum) of two risk assets[M].Interface & Free Boundaries,2002,4:27-46
[42]Cheng,W.& Zhang,S.The analysis of reset options[J].The Journal of Derivatives.2000,59-71
[43]Jiang,L&Dai,M.Convergence of explicit difference scheme and binomial tree method for American options[M].Journal of Computational Mathematics,2004,22:371-380
[44]许海峰,王凤荣,王德强.股票期权[M].北京:人民法院出版社,2005
[46]Merton R.Theory of rational option pricing[J].Bell Journal of Economics and Management Science.1973,4(1):141-183
[47]Cox J,Ross S.The valuation of options for alternation stochastic processes[J].Journal of Financial Economics,1976,3(3):145-166
[48]Rubinstein M.The valuation of uncertain income streams and the pricing of options[J].Bell Journal of Economics and Management Science,1976,7(2):407-425
[49]Cox J.Ross S.A survey of some new results in financial option pricing theory[]].Journal of Financial,1976,31(2):383-402
[50]Tolt K.On the mean-variance tradeoff in option replication with transactions costs[J].Journal of Financial and Quantitative Analysis,1996,31:233-263
[51]罗开位,侯振挺,李致中.期权定价理论的产生与发展[J].系统工程,2000,18(6):1-5
[52]Kassouf S T.An econometric model for option price with implications for investors expectations and audacity[J].E-conometrica.1969,37(4):685-694
[53]Boness A J.Elements of a theory of stock option value[]].Journal of Political Economy,1964,72(2):163-175
[54]Samuelon P A.Rational theory of warrant pricing[]].Industrial Management Review,1965,6(2):13-32
[55]Musiela M,Kutkowski M.Martingale methods in financial modeling theory and application[M].Beling Heidel-berg New York:Springer-Verlay,1997,128-180
[56]Phclim P B,Vorst T.Option replication in discrete time with transaction costs[J].Journal of Finance,1992,47(1):271-293
[57]David S B,Johnson H.The American put option and its critical stock price[J].Journal of Finance,2000,55(5):2333-2356
[58]蒲兴成,郑继明,游晓黔.基于Black-Scholes公式下美式期权价格的计算[J].重庆大学学报,2004,27(7):102-104
[59]S.L.Heston.A closed-form solution for options with stochastic volatility with application to bond and currency options[J].Review of Financial Studies,1993,6:327-343
[60]K.Borovkov,and A.Novikov.A new approach to calculating expectations for option pricing[J].Journal of Applied Probability,2002,39:889-895
[61]P.Carr,and D.B.Madan.Option valuation using the fast Fourier transform[J].Journal of Computational Finance,1999,2:61-73
[62]Y.K.Kwok.Mathematical Models of Financial Derivatives[M].Singapore:Springer-Verlag,1998,212-219
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