期权定价模型及其改进推广
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摘要
在国际衍生金融市场的形成和发展过程中，期权的合理定价是困扰投资者的一大难题。1997年10月14日，瑞典皇家科学院在斯德哥尔摩宣布，将1997年度诺贝尔经济学奖授予美国哈拂大学教授罗伯特．默顿(Robert Merton)、迈伦．斯科尔斯(Myron．Scholes)，以表彰他们和已故的费希尔．布莱克(Fisher．Black)在期权和其它衍生金融产品定价方面所做的开创性工作。他们创立和发展的布莱克—斯科尔斯期权定价模型(Black—Scholes Option Pricing Model)为包括股票、债券、货币、商品等在内的新兴衍生金融产品的合理定价奠定了基础。
布莱克和斯科尔斯在建立期权定价模型时作了以下前提假设：①原生资产的价格演化适合几何布朗运动；②在期权有效期内，无风险利率和金融资产收益的波动率是恒定不变的；③市场是无摩擦，即不存在税收和交易费用；④金融资产在期权有效期内无红利及其它所得；⑤该期权是欧式期权。
然而在现实证券市场中，投资者将面临数量可观和不容忽视的交易费用，而且股票投资回报的波动率和无风险利率随着时间的变化也在不断变化，这与布莱克—斯科尔斯模型的假设有所出入。正是基于这一情况，本文对其进行了某些修正和推广，主要做了以下几方面的工作：
(1)对于现在常用的Black—Scholes模型和二叉树模型进行了详细的介绍和分析：期权定价的离散模型——二叉树方法，是基于无套利原理，利用△-对冲技巧，定义了一个风险中性世界，使证券组合的预期收益率是无风险利率，从而给出了一个独立于每个投资人风险偏好的公平价格；期权定价的连续模型—Black—Scholes模型，是基于原生资产价格演化遵循几何Brown运动，利用△-对冲技巧和It(?)公式，给出了期权价格满足的Black—Scholes方程，通过求解Black—Scholes方程的终值问题，给出了一个独立于每个投资人风险偏好的欧式期权的公平价格Black—Scholes公式。由此指明建立期权定价模型的正确方法及这一方法的合理性。
(2)用证券组合模拟期权收益来构造有交易费用的欧式期权定价的基本方程，也就是建立一个包含股票头寸和基于该股票的衍生证券头寸在内的证券组合Π，并利用该组合Π的收益等于无风险收益的瞬间(即△-对冲)为期权定价。并用二叉树模型予以验证。
(3)在Black甲Scholes模型中，假设了在期权有效期内股票投资回报的波动
率。和无风险利率r都是固定的.而在现实中，它们很难保持不变.本文考虑。和r
在期权有效期内是时间t的已知函数，建立了。和:是时间t的函数的期权定价模
型，在此模型下推出了欧式期权的定价公式.对二叉树模型(C ox.Ross and
Rubinstein)进行了改进，使结果更加令人满意.
(4)用期权作为金融工具对已暴露的风险进行管理时，几个用希腊字母表示
的比率是非常重要的.本文介绍了Delta(△)、Gamma(r)、Theta(0)三个重要比
率及其应用，并定义了两个新的比率Phi(。)和Psi(甲)，分析讨论了期权价值变
化相对于红利率和执行价格变化的比率Phi值和Psi值，在一定程度上提高了我们
正确运用期权定价模型的准确性和制定期权交易策略抗风险的能力.
The Black--Scholes Option Pricing Model has been a useful tool
in proper pricing of option .It is on the following conditions :
(1)The price of underlying assets derives in geometric Brown motion way. (2)In the term of validity, risk-free rate and financial capital payoff's volatility are constant. (3)There exists no frictions on markets i.e. there are no taxes and transaction costs. (4) Financial assets have no dividend or other income during the term of validity.(5)The options are European Options.
However, there are some differences between these conditions and the real society. According to this, the writer makes some adaptation and generalization to that model mainly on the following aspects:
(1) The writer made a detailed introduction and analysis to the Black-Scholes model and the Binomial Tree Option pricing model. On the basis of arbitrage-free principle.the binomial tree methods define a risk-neutral world by making use of
-hedging skills. As a result , the expected yield of all security are risk-free .So a
fair price independent on every investor's risk aversion is given. The continuous model of option pricing -Black-Scholes model, which is on the basis of the consumption that the price of the underlying assets derivates in the geometric Brown motion,the
writer uses the A-hedging skills and ltd formula and gives the Black-Schcoles
equation which the option price satisfies. By computing its terminal value, the model gives a fair price of European option - Black-Scholes formula independent on every investor' risk aversion. By introducing the two models, the paper show the right way for option price modeling and its reasonability.
(2) This paper tries to build the basic equation of the European option pricing with transaction costs by simulating the option payoff using the portfolio i.e. build a portfolio including the option positions and the derivative security position on the basis of this option, and give the option a price when the payoff of this portfolio position is equal to the risk-free payoff (the A- hedging).The paper has tested it by binomial tree model.
(3) In Black-Scholes pricing model, the payoff volatility of stock investment a and risk-free rate r are constant during the term of validity. In fact it is hard for them to remain the same in real society. The writer consideres cr and r as functions of time t, then the writer gets a pricing formula for European option. The paper also makes improvements to binomial tree model to made the results more satisfying.
(4) When people are performing risk management to the appeared risk by using the option as a financial tool, the following ratios denoted by some Greek letters are very important. This paper introduces three important ratios Delta, Gamma, Theta, and their applications, and discussed the Phi value and Psi value of the changes of the option value relative to the changes of the interest rate and that of the exercises price . To some extent, it improves our abilities to use the option pricing model more exactly and make option transaction strategies against risks.
布莱克和斯科尔斯在建立期权定价模型时作了以下前提假设：①原生资产的价格演化适合几何布朗运动；②在期权有效期内，无风险利率和金融资产收益的波动率是恒定不变的；③市场是无摩擦，即不存在税收和交易费用；④金融资产在期权有效期内无红利及其它所得；⑤该期权是欧式期权。
然而在现实证券市场中，投资者将面临数量可观和不容忽视的交易费用，而且股票投资回报的波动率和无风险利率随着时间的变化也在不断变化，这与布莱克—斯科尔斯模型的假设有所出入。正是基于这一情况，本文对其进行了某些修正和推广，主要做了以下几方面的工作：
(1)对于现在常用的Black—Scholes模型和二叉树模型进行了详细的介绍和分析：期权定价的离散模型——二叉树方法，是基于无套利原理，利用△-对冲技巧，定义了一个风险中性世界，使证券组合的预期收益率是无风险利率，从而给出了一个独立于每个投资人风险偏好的公平价格；期权定价的连续模型—Black—Scholes模型，是基于原生资产价格演化遵循几何Brown运动，利用△-对冲技巧和It(?)公式，给出了期权价格满足的Black—Scholes方程，通过求解Black—Scholes方程的终值问题，给出了一个独立于每个投资人风险偏好的欧式期权的公平价格Black—Scholes公式。由此指明建立期权定价模型的正确方法及这一方法的合理性。
(2)用证券组合模拟期权收益来构造有交易费用的欧式期权定价的基本方程，也就是建立一个包含股票头寸和基于该股票的衍生证券头寸在内的证券组合Π，并利用该组合Π的收益等于无风险收益的瞬间(即△-对冲)为期权定价。并用二叉树模型予以验证。
(3)在Black甲Scholes模型中，假设了在期权有效期内股票投资回报的波动
率。和无风险利率r都是固定的.而在现实中，它们很难保持不变.本文考虑。和r
在期权有效期内是时间t的已知函数，建立了。和:是时间t的函数的期权定价模
型，在此模型下推出了欧式期权的定价公式.对二叉树模型(C ox.Ross and
Rubinstein)进行了改进，使结果更加令人满意.
(4)用期权作为金融工具对已暴露的风险进行管理时，几个用希腊字母表示
的比率是非常重要的.本文介绍了Delta(△)、Gamma(r)、Theta(0)三个重要比
率及其应用，并定义了两个新的比率Phi(。)和Psi(甲)，分析讨论了期权价值变
化相对于红利率和执行价格变化的比率Phi值和Psi值，在一定程度上提高了我们
正确运用期权定价模型的准确性和制定期权交易策略抗风险的能力.
The Black--Scholes Option Pricing Model has been a useful tool
in proper pricing of option .It is on the following conditions :
(1)The price of underlying assets derives in geometric Brown motion way. (2)In the term of validity, risk-free rate and financial capital payoff's volatility are constant. (3)There exists no frictions on markets i.e. there are no taxes and transaction costs. (4) Financial assets have no dividend or other income during the term of validity.(5)The options are European Options.
However, there are some differences between these conditions and the real society. According to this, the writer makes some adaptation and generalization to that model mainly on the following aspects:
(1) The writer made a detailed introduction and analysis to the Black-Scholes model and the Binomial Tree Option pricing model. On the basis of arbitrage-free principle.the binomial tree methods define a risk-neutral world by making use of
-hedging skills. As a result , the expected yield of all security are risk-free .So a
fair price independent on every investor's risk aversion is given. The continuous model of option pricing -Black-Scholes model, which is on the basis of the consumption that the price of the underlying assets derivates in the geometric Brown motion,the
writer uses the A-hedging skills and ltd formula and gives the Black-Schcoles
equation which the option price satisfies. By computing its terminal value, the model gives a fair price of European option - Black-Scholes formula independent on every investor' risk aversion. By introducing the two models, the paper show the right way for option price modeling and its reasonability.
(2) This paper tries to build the basic equation of the European option pricing with transaction costs by simulating the option payoff using the portfolio i.e. build a portfolio including the option positions and the derivative security position on the basis of this option, and give the option a price when the payoff of this portfolio position is equal to the risk-free payoff (the A- hedging).The paper has tested it by binomial tree model.
(3) In Black-Scholes pricing model, the payoff volatility of stock investment a and risk-free rate r are constant during the term of validity. In fact it is hard for them to remain the same in real society. The writer consideres cr and r as functions of time t, then the writer gets a pricing formula for European option. The paper also makes improvements to binomial tree model to made the results more satisfying.
(4) When people are performing risk management to the appeared risk by using the option as a financial tool, the following ratios denoted by some Greek letters are very important. This paper introduces three important ratios Delta, Gamma, Theta, and their applications, and discussed the Phi value and Psi value of the changes of the option value relative to the changes of the interest rate and that of the exercises price . To some extent, it improves our abilities to use the option pricing model more exactly and make option transaction strategies against risks.
引文
[1] Black F ,Scholes M. The pricing of options and corporate liabilities[J].Journal of Political Economy,1973, 8, 1:637--655
[2] COX J .Ross S, Rubinstein M .option pricing :a simplified approach[J] Journal of Financial Economics. 1979,7,3:145--166,229---263
[3] Barles G ,Mete H ,option pricing with transaction costs and a nonlinear Black--scholes equation[J],Journal of stochast, 1998,72(2):369--397
[4] Poul Wil mott ,Sam Howynne. The Mathematics of Financial Derivatives [M]London:Cambridge University Press, 1995,58--67
[5] 约翰．赫尔，张陶伟译．期权，期货，与衍生证券[M]．北京：华夏出版社，1997．399—447．
[6] Merton R c .Theory of Rational option Pricing [J] .Bell Journal of economics and Management Science, 1973,46:141--183
[7] Duffle D .An Extention of the Black--scholse Model of Security Voluation[J]Journal of Economics Theory, 1988,46:194--204
[8] Duffle D Security Markets :Stochastic Models[M],New York :Academic Press, 1998
[9] Black F, 1976. The Pricing of commodity Contracts. Journal of Financial Economics 3 (1--2), January--March L: 167--179
[10] Black, F. and Scholes,M,M .1972.The Valuation of option contracts and a test of market efficiency. Journal of Finance 27 (2). May 399--417
[11] Boness ,A.J.1964 a Elements of a theory of Stock-option value.Journal of Political Economy72(2),April: 163--175
[12] Fama, E,F.1970.Efficient capital markets: a review of theory and empirical work .Journal of Finance,May:383---417.
[13] Gould ,J,P. and Galai ,D. 1994,transactions costs and the relationship between Put and Call Prices .Journal of financial Economics 1,105---129
[14] klemkosky, R and Resnick ,B. 1979.Put--Call Parity and market efficiency.Journal of Finance(5),December: 1141--1155.
[15] MecBeth ,J. and Merville, L. 1980.Tests of the Black--Scholes and Cox Call option valuation models,Journal of Finance,(2),May:285--300
[16] Smith, c.w, Jr. 1976.Option Pricing: a review. Journal of Financial Economics 3(1-2),Jcinury-March:3-51
[17] Hull J. Option, Futures and other Derivative Securities [M], Second Edition.PrenticeHall, 1993
[18] Abrahamazks. Arnuities under Random rate of interest[J] Insurancel :Mathematics & Economics,2001,1:1-11.
[19] Westcott. DA. Moment of compound interest function under fluctuating interest rates [J]. Scandinavian Actuarial Journal, 1981,4:237-249
[20] Boylepp.Rates of return as of risk and insurance, 1976,4 693-713
[21] 张铁．一个新型的期权定价二差数参数模型[J]，系统工程理论与实践，2000．
[22] 关莉，李耀堂．修正的Black-scholes期权定价模型[J]．云南大学学报，2001，23(2)：84-86
[23] (美)Sheldon Natenbeg．option volatilitg & pricing．中译书《期权价格波动率与定价理论》．刘鸿儒主编，经济科学出版社．2000．
[24] 姜礼尚．期权定价的数学模型和方法．高等教育出版社．2003．
[25] 林元烈．应用随机过程．清华大学出版社．2002．
[26] 孔爱国．现代投资学．上海人民出版社．2003．
[27] 范龙振，胡畏．金融工程学．上海人民出版社．2003．
[28] 雍炯敏，刘道百．数学金融学．上海人民出版社．2003．
[29] 严士健，王隽骧，刘秀芳．概率论基础．科学出版社．1982．
[30] 黄志远．随机分析学基础(第二版)．科学出版社．2001．
[31] 姜礼尚，陈亚浙．数学物理方程讲义．高等教育出版社．1986．
[2] COX J .Ross S, Rubinstein M .option pricing :a simplified approach[J] Journal of Financial Economics. 1979,7,3:145--166,229---263
[3] Barles G ,Mete H ,option pricing with transaction costs and a nonlinear Black--scholes equation[J],Journal of stochast, 1998,72(2):369--397
[4] Poul Wil mott ,Sam Howynne. The Mathematics of Financial Derivatives [M]London:Cambridge University Press, 1995,58--67
[5] 约翰．赫尔，张陶伟译．期权，期货，与衍生证券[M]．北京：华夏出版社，1997．399—447．
[6] Merton R c .Theory of Rational option Pricing [J] .Bell Journal of economics and Management Science, 1973,46:141--183
[7] Duffle D .An Extention of the Black--scholse Model of Security Voluation[J]Journal of Economics Theory, 1988,46:194--204
[8] Duffle D Security Markets :Stochastic Models[M],New York :Academic Press, 1998
[9] Black F, 1976. The Pricing of commodity Contracts. Journal of Financial Economics 3 (1--2), January--March L: 167--179
[10] Black, F. and Scholes,M,M .1972.The Valuation of option contracts and a test of market efficiency. Journal of Finance 27 (2). May 399--417
[11] Boness ,A.J.1964 a Elements of a theory of Stock-option value.Journal of Political Economy72(2),April: 163--175
[12] Fama, E,F.1970.Efficient capital markets: a review of theory and empirical work .Journal of Finance,May:383---417.
[13] Gould ,J,P. and Galai ,D. 1994,transactions costs and the relationship between Put and Call Prices .Journal of financial Economics 1,105---129
[14] klemkosky, R and Resnick ,B. 1979.Put--Call Parity and market efficiency.Journal of Finance(5),December: 1141--1155.
[15] MecBeth ,J. and Merville, L. 1980.Tests of the Black--Scholes and Cox Call option valuation models,Journal of Finance,(2),May:285--300
[16] Smith, c.w, Jr. 1976.Option Pricing: a review. Journal of Financial Economics 3(1-2),Jcinury-March:3-51
[17] Hull J. Option, Futures and other Derivative Securities [M], Second Edition.PrenticeHall, 1993
[18] Abrahamazks. Arnuities under Random rate of interest[J] Insurancel :Mathematics & Economics,2001,1:1-11.
[19] Westcott. DA. Moment of compound interest function under fluctuating interest rates [J]. Scandinavian Actuarial Journal, 1981,4:237-249
[20] Boylepp.Rates of return as of risk and insurance, 1976,4 693-713
[21] 张铁．一个新型的期权定价二差数参数模型[J]，系统工程理论与实践，2000．
[22] 关莉，李耀堂．修正的Black-scholes期权定价模型[J]．云南大学学报，2001，23(2)：84-86
[23] (美)Sheldon Natenbeg．option volatilitg & pricing．中译书《期权价格波动率与定价理论》．刘鸿儒主编，经济科学出版社．2000．
[24] 姜礼尚．期权定价的数学模型和方法．高等教育出版社．2003．
[25] 林元烈．应用随机过程．清华大学出版社．2002．
[26] 孔爱国．现代投资学．上海人民出版社．2003．
[27] 范龙振，胡畏．金融工程学．上海人民出版社．2003．
[28] 雍炯敏，刘道百．数学金融学．上海人民出版社．2003．
[29] 严士健，王隽骧，刘秀芳．概率论基础．科学出版社．1982．
[30] 黄志远．随机分析学基础(第二版)．科学出版社．2001．
[31] 姜礼尚，陈亚浙．数学物理方程讲义．高等教育出版社．1986．