股价波动源模型的期权定价
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摘要
股票价格波动模型是用于描述股票价格波动的数学模型，一直是金融学者们长期研究的问题。目前存在的模型主要有随机游走模型、对数正态模型等，鉴于股价波动的随机游走模型和对数正态模型均经过实证分析，表明不完全符合现实的股票市场，目前理论研究者提出一种更符合实际股票市场的股价模型-股价波动源模型(文[5]的作者将股价异常变化带来的短期收益率函数附加在几何Brown运动上，推广了对数正态模型)及研究出了另一种混合形式下(见文[15])的期权定价方程。
本文基于股价符合波动源模型的假设，综合运用随机微分理论等数学原理和无套利理论等金融理论，依此对短期收益率函数为分段阶梯函数和Possion跳跃过程的股价波动源模型分别在无风险利率是常数和随机过程的条件下作了期权定价，推导出了相应的期权定价偏微分方程，结果表明：
1、由异常波动源带来的短期收益率函数是分段阶梯函数时，这种对股价对数正态分布模型的修正不能改善期权价格，因为基于这种模型的期权定价偏微分方程与基于股价对数正态分布模型的期权定价偏微分方程完全相同(见方程2.14)。
2、假设无风险利率是随机利率，我们在详细考察基础变量-利率和股票价格行为特征的基础上，运用无套利原理推导出波动源模型的双因素期权定价方程(见方程3.19)，其特例便是Black-Scholes模型。
3、在非有效市场中，投资者将面临数量可观、不容忽视的交易成本。本文在界定交易成本的基础上，建立了离散交易时间条件下的非线性期权定价模型(见方程3.33)，并分别讨论了有交易成本的欧式期权多头和空头的定价方法。
4、将短期收益率函数由确定函数修改为Possion跳跃过程后，文[15]推导出的期权定价偏微分方程(见方程4.2)虽然推广了Black-Scholes期权定价偏微分方程，但此时依旧假设利率是常数，这与实际生活中的不符，我们研究了一个随机利率下短期收益率函数是Possion跳跃过程的期权定价模型(见5.13)，该模型既改变了股票价格波动源模型中短期收益率函数的形式，避免了异常波动源带来的收益率函数的简单化。又在放松了推导Black-Scholes期权定价偏微分方程的6个严格假设条件之一的前提下，使得期权定价偏微分方程的建立更接近实际股票市场，使推导出的期权定价偏微分方程更具实际意义。
The models of the stock price fluctuation is a mathematics model discribing the fluctuation of the stock price,It is all along the question financial scholars research over a long period of time, The models existing at present are mainly the model of Randonm Walk and the model of lognormal distribution etc.Economists analyse the two models by authentic proof,which indicates that this two models do not fully qualify the actual stock market.In view of the above- mentioned facts,at the time some scholar have studied a new model of the stock price that even conforms to the actual stock market- that is the model of lognormal distribution.
Underlying the assumption that the stock price accords with the model of the stock price fluctuating sources,By comprehensivily applying the stochasitic differential theory and no-arbitriagc thcory,this paper,under the conditions that the risk-free rate r is constant or Ito stochasitic process, successively works out the option pricing about the stock price model with that the short-term profit function is piecewise lecture function arid that one with that the short-term profit function is Possion Jump Process,derivats counterpart partial differential equation of option pricing.The outcome states:
1.When the short-term profit function is unusual flunctuating sources bring out a piecewise lecture function,this amendment on the lognormal distribution model does not improve the option price,because this partial differential equation of option pricing is the same one underlying the lognormal distribution model(see equation 2.14).
2.When the risk-free rate r is stochastie interest rate,the partial differential equation of option pricing,underlying the above-mentioned stock price model,popularizes the Black-Scholes partial differential equation(see equation 3.19).
3. In no-effective market,investors will face trading costs which cann't be neglected.Based on defining the trading costs,this paper set up a no-linear
option pricing model with discrete trading time, and discuss European option long position and option short position with trading cost(3.33).
4.After changing the short-term profit function to Possion Jump Process, in the view of that the derivated partial differential equation of the option pricing which different from Black-Scholes partial differential equation still is that interest rate is constant(4.2), the model which does not accord with the real market under the assumption. At last,we derivat a new model of option pricing whoso profit rate is Possion Jump Process under stochastic interest rate(5.13),this model not only changes the form of the short-term profit function of the stock price model and avaids the simplization of the profit rate function the unusual flunction sources bring about,but also relaxes the basis assumption of Black-Scholes option pricing model and makes that the partial differential equation builds the foundation which even approaches the actual market.
本文基于股价符合波动源模型的假设，综合运用随机微分理论等数学原理和无套利理论等金融理论，依此对短期收益率函数为分段阶梯函数和Possion跳跃过程的股价波动源模型分别在无风险利率是常数和随机过程的条件下作了期权定价，推导出了相应的期权定价偏微分方程，结果表明：
1、由异常波动源带来的短期收益率函数是分段阶梯函数时，这种对股价对数正态分布模型的修正不能改善期权价格，因为基于这种模型的期权定价偏微分方程与基于股价对数正态分布模型的期权定价偏微分方程完全相同(见方程2.14)。
2、假设无风险利率是随机利率，我们在详细考察基础变量-利率和股票价格行为特征的基础上，运用无套利原理推导出波动源模型的双因素期权定价方程(见方程3.19)，其特例便是Black-Scholes模型。
3、在非有效市场中，投资者将面临数量可观、不容忽视的交易成本。本文在界定交易成本的基础上，建立了离散交易时间条件下的非线性期权定价模型(见方程3.33)，并分别讨论了有交易成本的欧式期权多头和空头的定价方法。
4、将短期收益率函数由确定函数修改为Possion跳跃过程后，文[15]推导出的期权定价偏微分方程(见方程4.2)虽然推广了Black-Scholes期权定价偏微分方程，但此时依旧假设利率是常数，这与实际生活中的不符，我们研究了一个随机利率下短期收益率函数是Possion跳跃过程的期权定价模型(见5.13)，该模型既改变了股票价格波动源模型中短期收益率函数的形式，避免了异常波动源带来的收益率函数的简单化。又在放松了推导Black-Scholes期权定价偏微分方程的6个严格假设条件之一的前提下，使得期权定价偏微分方程的建立更接近实际股票市场，使推导出的期权定价偏微分方程更具实际意义。
The models of the stock price fluctuation is a mathematics model discribing the fluctuation of the stock price,It is all along the question financial scholars research over a long period of time, The models existing at present are mainly the model of Randonm Walk and the model of lognormal distribution etc.Economists analyse the two models by authentic proof,which indicates that this two models do not fully qualify the actual stock market.In view of the above- mentioned facts,at the time some scholar have studied a new model of the stock price that even conforms to the actual stock market- that is the model of lognormal distribution.
Underlying the assumption that the stock price accords with the model of the stock price fluctuating sources,By comprehensivily applying the stochasitic differential theory and no-arbitriagc thcory,this paper,under the conditions that the risk-free rate r is constant or Ito stochasitic process, successively works out the option pricing about the stock price model with that the short-term profit function is piecewise lecture function arid that one with that the short-term profit function is Possion Jump Process,derivats counterpart partial differential equation of option pricing.The outcome states:
1.When the short-term profit function is unusual flunctuating sources bring out a piecewise lecture function,this amendment on the lognormal distribution model does not improve the option price,because this partial differential equation of option pricing is the same one underlying the lognormal distribution model(see equation 2.14).
2.When the risk-free rate r is stochastie interest rate,the partial differential equation of option pricing,underlying the above-mentioned stock price model,popularizes the Black-Scholes partial differential equation(see equation 3.19).
3. In no-effective market,investors will face trading costs which cann't be neglected.Based on defining the trading costs,this paper set up a no-linear
option pricing model with discrete trading time, and discuss European option long position and option short position with trading cost(3.33).
4.After changing the short-term profit function to Possion Jump Process, in the view of that the derivated partial differential equation of the option pricing which different from Black-Scholes partial differential equation still is that interest rate is constant(4.2), the model which does not accord with the real market under the assumption. At last,we derivat a new model of option pricing whoso profit rate is Possion Jump Process under stochastic interest rate(5.13),this model not only changes the form of the short-term profit function of the stock price model and avaids the simplization of the profit rate function the unusual flunction sources bring about,but also relaxes the basis assumption of Black-Scholes option pricing model and makes that the partial differential equation builds the foundation which even approaches the actual market.
引文
[1] 叶中行，林建忠，数理经融-资产定价与决策理论，科学技术出版社，北京，1998，31-33，141-144。
[2] 罗开位，候振挺，李致中，期权定价理论的产生与发展，系统工程，2000，1，11-5．
[3] Black, F., and Scholes, M. J. 1973, The pricing of option and corporate liabilites. Journal of Political Economy 81(3), May, 637-654.
[4] 吴冲锋，王海成，吴文锋，金融工程研究，上海交通大学出版社，2000，第一版，387-418。
[5] 吴文锋，吴冲锋，股票价格波动源模型探讨，系统工程理论与实践，2000，4，63-69．
[6] John．Hull[美] 著，张陶伟，期权期货和其他衍生产品(第三版)，华夏出版社，2000，1，188-200。
[6] 林建华，王福昌，冯敬海，股价波动的指数O-U过程模型，经济数学，2000，4(17)，29-32。
[7] 郑小迎，陈军，陈金贤，可转换债券定价模型探讨，系统工程理论与实践，2000，8，24-28．
[8] 郑立辉，连续时间资产定价理论进展评述，系统工程理论与实践，1998，5，29-33．
[9] 郑小迎，陈金贤，杨屹，关于利率衍生工具定价的研究，系统工程学报，2001，6。
[10] Vasicek O. Anequilibrium Characterization of the term structure [J] . Journal of Financial Economics. 1977,20(5):177-188.
[11] Cox J C, Ingersoll J, Ross S A. A theory of term structure of interest rate[J] . Econometerica, 1985,53(2):385-407.
[12] Gurdip Baskhi, Charles cao and Zhiwu Chen. Enpirital Performance of Alternative Option Pricing Mldels. The Journal of Finance,1997,11(5):55-64.
[13] 薛兴恒，数学物理偏微分方程，中国科技大学出版社，1995。
[14] 龚光鲁，随机微分方程引论，北京大学出版社，1995。
[15] 马超群，陈牡妙，标的资产服从混合过程的期权定价模型，系统工程理论与实践，99，4。
[16] 吴志刚，金朝嵩，标的股价服从混合过程的期权定价公式及有限元算法，经济数学，2002，6，28-31．
[17] 冯广波，刘再明，蔡海涛，标的资产价格服从跳——扩散过程的具有随机寿命的未定权益定价，经济数学，2002，6，21-27．
[18] 南京工学院数学教研组，数学物理方程与特殊函数，高教出版社，1982，58-65。
[19] Guy Barles, Halil Mete, Option Pricing with transaction costs and a nonlinear Black-Scholes equation. Jonuaral Stockast, 1998,72(2):369-397.
[20] 朱玉旭，黄洁纲，吴冲锋，系统工程理论与实践，1998，5：12-16。
[21] 潭俊德，胡宗义，收益率周期波动的股票欧式期权定价，经济数学，2002，3。
[22] R. C. Merton, "Theory of Rational Option Pricing", Bell Journal of Economical and Management Science, 4(Spring 1973),141-83.
[23] Ho T, Lee S. Term structure movements and Pricing interest-rate contingent claims [J] , Journal of Finance, 1986,41(2):1011-1029.
[24] 马超群，陈牡妙，袁敢，一类期权定价方法，
[25] 董太亨，金融资产定价理论的方法比较，数量经济技术经济研究，2002，1。
[26] 钱敏平，龚光鲁，随机过程论(第二版)，北京大学出版社，2000。
[2] 罗开位，候振挺，李致中，期权定价理论的产生与发展，系统工程，2000，1，11-5．
[3] Black, F., and Scholes, M. J. 1973, The pricing of option and corporate liabilites. Journal of Political Economy 81(3), May, 637-654.
[4] 吴冲锋，王海成，吴文锋，金融工程研究，上海交通大学出版社，2000，第一版，387-418。
[5] 吴文锋，吴冲锋，股票价格波动源模型探讨，系统工程理论与实践，2000，4，63-69．
[6] John．Hull[美] 著，张陶伟，期权期货和其他衍生产品(第三版)，华夏出版社，2000，1，188-200。
[6] 林建华，王福昌，冯敬海，股价波动的指数O-U过程模型，经济数学，2000，4(17)，29-32。
[7] 郑小迎，陈军，陈金贤，可转换债券定价模型探讨，系统工程理论与实践，2000，8，24-28．
[8] 郑立辉，连续时间资产定价理论进展评述，系统工程理论与实践，1998，5，29-33．
[9] 郑小迎，陈金贤，杨屹，关于利率衍生工具定价的研究，系统工程学报，2001，6。
[10] Vasicek O. Anequilibrium Characterization of the term structure [J] . Journal of Financial Economics. 1977,20(5):177-188.
[11] Cox J C, Ingersoll J, Ross S A. A theory of term structure of interest rate[J] . Econometerica, 1985,53(2):385-407.
[12] Gurdip Baskhi, Charles cao and Zhiwu Chen. Enpirital Performance of Alternative Option Pricing Mldels. The Journal of Finance,1997,11(5):55-64.
[13] 薛兴恒，数学物理偏微分方程，中国科技大学出版社，1995。
[14] 龚光鲁，随机微分方程引论，北京大学出版社，1995。
[15] 马超群，陈牡妙，标的资产服从混合过程的期权定价模型，系统工程理论与实践，99，4。
[16] 吴志刚，金朝嵩，标的股价服从混合过程的期权定价公式及有限元算法，经济数学，2002，6，28-31．
[17] 冯广波，刘再明，蔡海涛，标的资产价格服从跳——扩散过程的具有随机寿命的未定权益定价，经济数学，2002，6，21-27．
[18] 南京工学院数学教研组，数学物理方程与特殊函数，高教出版社，1982，58-65。
[19] Guy Barles, Halil Mete, Option Pricing with transaction costs and a nonlinear Black-Scholes equation. Jonuaral Stockast, 1998,72(2):369-397.
[20] 朱玉旭，黄洁纲，吴冲锋，系统工程理论与实践，1998，5：12-16。
[21] 潭俊德，胡宗义，收益率周期波动的股票欧式期权定价，经济数学，2002，3。
[22] R. C. Merton, "Theory of Rational Option Pricing", Bell Journal of Economical and Management Science, 4(Spring 1973),141-83.
[23] Ho T, Lee S. Term structure movements and Pricing interest-rate contingent claims [J] , Journal of Finance, 1986,41(2):1011-1029.
[24] 马超群，陈牡妙，袁敢，一类期权定价方法，
[25] 董太亨，金融资产定价理论的方法比较，数量经济技术经济研究，2002，1。
[26] 钱敏平，龚光鲁，随机过程论(第二版)，北京大学出版社，2000。