期权定价二叉树算法收敛阶研究
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摘要
期权是最重要的金融衍生品之一,自从期权交易产生以来,学者一直致力于如何正确确定期权的价格。期权定价是期权交易的核心内容,具有重要的理论价值和实际应用价值。期权定价在金融产品创新、套期保值、风险管理等领域扮演至关重要的角色。上世纪70年代,布莱克和斯科尔斯(Black and Scholes,1973)以及莫顿(Merton,1973)在期权定价领域取得重大突破,他们的理论被称为布莱克-斯科尔斯模型或布莱克-斯科尔斯-莫顿模型。在此之后,分析金融学进入了一个高速发展时期,一系列期权定价理论相继问世。
期权定价模型主要包括两大类：连续时间模型和离散时间模型。在期权定价理论基础之上,本文运用随机分析、组合数学等工具证明二叉树算法计算期权的收敛阶。本文对二叉树算法及收敛阶理论做比较充分的综述,针对典型算法进行拓展研究并证明收敛阶。本文另外一个较大贡献就是证明了一些奇异期权二叉树算法收敛阶(幂期权、缺口期权等)。最后,本文研究了幂期权希腊字母二叉树算法收敛阶。
本文从算法的角度研究连续时间期权定价模型收敛阶的数值解,本文的研究意义在于证明二叉树算法收敛阶,而收敛阶可以精确刻画算法的收敛速度。在理论上对算法的可靠性和计算效率提供依据,同时对算法的改进提供一个依据。
Option is one of the most important financial derivatives, option pricing plays a key role in option trading. Theory of option pricing can do it back to Black and Scholes (1973) and Morton (1973), since then, there have been emerging extensive studies in option pricing theory. Binomial approach is one of the popular methods for pricing option. Recently, there are increasingly interests in proving the convergent rates of the binomial methods.
In this thesis, we utilize stochastic analysis, combinatorial mathematics and other mathematical tools to prove the convergence rates of binomial tree algorithms for pricing European-style options and exotic options.
This thesis provides comprehensive literature review for the theory of convergence rates for binomial tree methods, and proves the convergence rates of many variants binomial tree methods which are lack in the existent literature. Another contribution of this thesis is to prove the convergence rate of some exotic options (power options, gap options, etc.). Finally, the convergence rates for computing the Greeks of power options are proved.
期权定价模型主要包括两大类：连续时间模型和离散时间模型。在期权定价理论基础之上,本文运用随机分析、组合数学等工具证明二叉树算法计算期权的收敛阶。本文对二叉树算法及收敛阶理论做比较充分的综述,针对典型算法进行拓展研究并证明收敛阶。本文另外一个较大贡献就是证明了一些奇异期权二叉树算法收敛阶(幂期权、缺口期权等)。最后,本文研究了幂期权希腊字母二叉树算法收敛阶。
本文从算法的角度研究连续时间期权定价模型收敛阶的数值解,本文的研究意义在于证明二叉树算法收敛阶,而收敛阶可以精确刻画算法的收敛速度。在理论上对算法的可靠性和计算效率提供依据,同时对算法的改进提供一个依据。
Option is one of the most important financial derivatives, option pricing plays a key role in option trading. Theory of option pricing can do it back to Black and Scholes (1973) and Morton (1973), since then, there have been emerging extensive studies in option pricing theory. Binomial approach is one of the popular methods for pricing option. Recently, there are increasingly interests in proving the convergent rates of the binomial methods.
In this thesis, we utilize stochastic analysis, combinatorial mathematics and other mathematical tools to prove the convergence rates of binomial tree algorithms for pricing European-style options and exotic options.
This thesis provides comprehensive literature review for the theory of convergence rates for binomial tree methods, and proves the convergence rates of many variants binomial tree methods which are lack in the existent literature. Another contribution of this thesis is to prove the convergence rate of some exotic options (power options, gap options, etc.). Finally, the convergence rates for computing the Greeks of power options are proved.
引文
① 参见巴曙松(2011)。
② 参见Nawalkha S. K. and Chambers D. R. (1995).
① Jabbour G. E., Kramin M. V. and Young S. T. (2001)。
① 参见Bachelier L. (1900).
① 参见Brennan, M. J. and Schwartz, E. S. (1978)=
② 参见Tavella, D. and Randall, C. (2000).
① 关于有限差分方法计算期权价格时其稳定性和收敛性的详细讨论参见姜礼尚(2008)第94-102页。
① 具体内容可参见Kwok Y.K. (2008)第332-352页,邵宇(2008)第678-688页。
② 具体内容可参见Boyle, P. P. (1977)第323-338页。
① 最近蒙特卡罗模拟方法经过改良后,也被应用于计算美式期权,具体参见Boyle,P.P.,Broadie,M.and Glasserman,P(1997),Broadie,M.,Glasseman,P.and Jain,G.(1997)等.
① 关于蒙特卡罗模拟方法计算多维资产期权详细资料参见Hull, J. C. (2012)第20.6节。
② 参见宋逢明(2006)第114页。
① 这个条件不仅具有数学上的必要性,同时具有金融学意义。这意味着,标的资产(股票)的回报不能绝对占优无风险债券的收益,同样,无风险债券的收益也不能绝对占优于股票的回报。简单证明如下：假设该条件不成立,例如,当u≤R时,投资者在初始时刻具有零财富,可以在股票市场上通过卖空股票、在无风险债券市场上买入无风险债券,从而获得无风险套利；同理,当d≥R时,投资者可以在无风险债券市场上卖空无风险债券、在股票市场上买入股票来获得无风险套利。事实上,这与无风险套利原理相违背的,所以有该条件成立。该条件的详细讨论可以参考TomasBjork(2009),或者Steven E. Shreve (2004, Ⅰ).
① 参见Kwok Y. K. (2008)。
① 参见Brennan, M. J. and Schwartz, E. S. (1978).
② 参见Tavella, D. and Randall, C. (2000)。
① 其实这就是Rendleman and Bartter (1979)算法模型,具体参见下一章。
② 具体参数p,u,d如何求解可以参考下一节或Cox J. C., Ross S., and Rubinstein M. (1979).
① 详细论证见下一节或Chung S. L. and Shih P. T. (2007)
① 参见Cox J. C.,Ross S. A. and Rubinstein M. (1979)。
② 具体参见Cox J. C.,Ross S. A. and Rubinstein M. (1979)第248-249页。
① 参见Rendleman R. J. and Batter B. J. (1979).
② 参见Rendleman R. J. and Batter B. J. (1979)第1098页第(10)、(11)式。
③ 参见Rendleman R. J. and Batter B. J. (1979)第1101页。
④ 关于二叉树算法中风险中性概率的讨论参见Lyuu Y.D.(2004)第95页,或McDonald R.L. (2013)第326-330页。
① 参见.Jarrow R. A. and Rudd A. (1983).
② 参见Jarrow R. A. and Turnbull S. (2000).
③ 参见Trigeorgsi L. (1991).
① 参见Tian Y S. (1993).
② BIN模型即Tian模型,参见Tian Y. S. (1993)第566页。
③ 参见Tian Y. S. (1993)第564页公式(1)。
① 参见Chriss N. (1996)
② 参见Chriss N. (1996)第233-238页。
① 参见Leisen D. and Reimer M. (1996)。
② 参见Leisen D. and Reimer M. (1996)第325页。实际上,Leisen and Reimer第一次解释了估计结果不精确的来源,在其随后构造的模型中,使得敲定价格正好处于二叉树的中间位置,从而使得敲定价格依赖于二叉树,提高收敛的光滑性。Diener and Diener (2004)对于Leisen and Reimer的论断给予了严格的数学证明,我们将在下一章中详细讨论。
③ 限于篇幅在此不再详细讨论,详情请参见Leisen D. and Reimer M. (1996)第331-332页。另外Huang E. G.(2006)第七章第1节有关于Leisen and Reimer算法的介绍和程序代码。
④ 参见Wilmott P.(1998).
① 参见Tian Y. S. (1999).
② 关于收敛阶和收敛光滑性的定义可以参考下一章或Leisen D. and Reimer M. (1996), 或 Chang L. B. and Palmer K. (2007)第94页。由于Tian(1999)算法具备良好的光滑性,因而可以通过外推法(extrapolation technique)提高收敛阶。参见原文第828-832页。
① 参见Avellaneda M.and Laurence P.(1999).
② 符号cosh(·)表示双曲余弦函数,即cosh(x)=ex+e-x/2
③ 参见Jabbour G. E.,Kramin M. V.and Young S.T.(2001).
① 参见Leisen D. P. J.and Reimer M. (1996)第326-327页。
① 参见Chang L. B. and Palmer K. (2007)第94页。收敛的光滑性是一个非常重要的问题,这是因为,如果收敛是光滑的,则可以通过外推法(extrapolation technique)提高收敛阶。关于外推在二叉树算法中的应用参见Broadie M. and J. B. Detemple (1996), Tian Y, S. (1999), Heston S. and Zhou G. (2000)。本章关于收敛阶的证明均采用Leisen and Reimer (1996),第5、6章的证明采用(:hang and Palmer (2007)的定义。
② 参见Leisen D. P. J. and M. Reimer (1996)第328-330页.
① 作者证明了三个模型中mn2=mn3=Δn=O(Δt2),参见Leisen D.P. J.and M.Reimer(1996)附录B第344-346页。
② 参见Leisen D.P.J.and Reimer M.(1996)第333-335页。
③ 参见Walsh J.B.(2002,2003).
④ 参见Diener F. and Diener M.(2004).
⑤ 参见Diener F. and Diener M.(2004)第282-288页。
① 参见Chang L. B. and Palmer K. (2007)第95页推论1。
② 参见Chang L. B. and Palmer K. (2007)第96页推论2。
③ 参见Chung S. L. and Shih P. T. (2007)。
① 参见Chung S.L.and Shih P.T(2007)第509页定理1,该定理仅仅是收敛性的证明,并未涉及收敛阶,本文第三章第二节也有所提及。作者在其后文中者对收敛阶进行了详细的论证。
② 参见Joshi M.S.(2009).
③ 在该算法中,作者令μ=logK-logS0/T这恰好使得当上跳次数j≥n+1时,S0ujdn-j>K更加详细的讨论参见原文第173-174页。原文只对互补二叉树分布函数展开,由期权定价二叉树计算公式(3.1.4)可以看出,二者收敛阶、收敛光滑性相同。
④ 参见Joshi M.S.(2010)。
① 参见Diener F. and Diener M. (2004)第277页定理2.1。
① Boyle P. P. and Lau S. H. (1994)。
② 参见Boyle P. P. and Lau S. H. (1994)第8页。其后,Lin and Palmer证明该算法计算障碍期权时的收敛阶为1,参见Lin J. R. and Palmer K. (2013)第323页。
① 具体算法参见Tian Y. S. (1999)第832-837页,后文也对离散障碍期权如何计算进行了讨论。
② 参见Ritchken P. (1995).
③ 参见Lyuu Y. D.(1998)第77页。
④ 参见Chung S. L. and Shih P. T. (2007)电子附录(e-companion)第2-4页。
⑤ 参见Lin and Palmer (2013)。
① 参见Heynen R. C. and Kat H. M. (1996)、Esser A. (2003)或Tompkins R. G. (2000).
① 可参见Uspensky J.V. (1937)第119-129页。
① 参见Chung S. L. and Shih P. T. (2007)第518页附录A.1。
① 参见P.G.Zhang(1998)第214页公式(10.26 a)。
② 参见Ibrahim S. N. I., O'Hara J. Q and Constantinou N. (2013)第597页公式(2.11)、(2.12)、(2.13)、(2.14)。
① 此引理引自Lin and Palmer (2013),参见其文献第324页引理4.1,证明见其文献附录C。
① 标准下降敲入看涨障碍期权公式证明参见Reimer and Sandmann (1995)第2-7页,或Lin and Palmer(2013)附录B。显然,根据我们的定义,相应的幂期权价格计算公式不同之处仅在于收益函数部分。
① 参见Reiner E. and Rubinstein M.(1991)或Haug E.G. (2006)
① 由Black-Scholes微分方程(参见第二章定理2.6式(2.17))可以直接推出：
① 参见Chung and Shackleton (2002)第147-148页。
① 详细证明参见Chung, et al (2010)第586页附录。
① 参见Chung et al (2010)附录第587-590页。
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② 参见Nawalkha S. K. and Chambers D. R. (1995).
① Jabbour G. E., Kramin M. V. and Young S. T. (2001)。
① 参见Bachelier L. (1900).
① 参见Brennan, M. J. and Schwartz, E. S. (1978)=
② 参见Tavella, D. and Randall, C. (2000).
① 关于有限差分方法计算期权价格时其稳定性和收敛性的详细讨论参见姜礼尚(2008)第94-102页。
① 具体内容可参见Kwok Y.K. (2008)第332-352页,邵宇(2008)第678-688页。
② 具体内容可参见Boyle, P. P. (1977)第323-338页。
① 最近蒙特卡罗模拟方法经过改良后,也被应用于计算美式期权,具体参见Boyle,P.P.,Broadie,M.and Glasserman,P(1997),Broadie,M.,Glasseman,P.and Jain,G.(1997)等.
① 关于蒙特卡罗模拟方法计算多维资产期权详细资料参见Hull, J. C. (2012)第20.6节。
② 参见宋逢明(2006)第114页。
① 这个条件不仅具有数学上的必要性,同时具有金融学意义。这意味着,标的资产(股票)的回报不能绝对占优无风险债券的收益,同样,无风险债券的收益也不能绝对占优于股票的回报。简单证明如下：假设该条件不成立,例如,当u≤R时,投资者在初始时刻具有零财富,可以在股票市场上通过卖空股票、在无风险债券市场上买入无风险债券,从而获得无风险套利；同理,当d≥R时,投资者可以在无风险债券市场上卖空无风险债券、在股票市场上买入股票来获得无风险套利。事实上,这与无风险套利原理相违背的,所以有该条件成立。该条件的详细讨论可以参考TomasBjork(2009),或者Steven E. Shreve (2004, Ⅰ).
① 参见Kwok Y. K. (2008)。
① 参见Brennan, M. J. and Schwartz, E. S. (1978).
② 参见Tavella, D. and Randall, C. (2000)。
① 其实这就是Rendleman and Bartter (1979)算法模型,具体参见下一章。
② 具体参数p,u,d如何求解可以参考下一节或Cox J. C., Ross S., and Rubinstein M. (1979).
① 详细论证见下一节或Chung S. L. and Shih P. T. (2007)
① 参见Cox J. C.,Ross S. A. and Rubinstein M. (1979)。
② 具体参见Cox J. C.,Ross S. A. and Rubinstein M. (1979)第248-249页。
① 参见Rendleman R. J. and Batter B. J. (1979).
② 参见Rendleman R. J. and Batter B. J. (1979)第1098页第(10)、(11)式。
③ 参见Rendleman R. J. and Batter B. J. (1979)第1101页。
④ 关于二叉树算法中风险中性概率的讨论参见Lyuu Y.D.(2004)第95页,或McDonald R.L. (2013)第326-330页。
① 参见.Jarrow R. A. and Rudd A. (1983).
② 参见Jarrow R. A. and Turnbull S. (2000).
③ 参见Trigeorgsi L. (1991).
① 参见Tian Y S. (1993).
② BIN模型即Tian模型,参见Tian Y. S. (1993)第566页。
③ 参见Tian Y. S. (1993)第564页公式(1)。
① 参见Chriss N. (1996)
② 参见Chriss N. (1996)第233-238页。
① 参见Leisen D. and Reimer M. (1996)。
② 参见Leisen D. and Reimer M. (1996)第325页。实际上,Leisen and Reimer第一次解释了估计结果不精确的来源,在其随后构造的模型中,使得敲定价格正好处于二叉树的中间位置,从而使得敲定价格依赖于二叉树,提高收敛的光滑性。Diener and Diener (2004)对于Leisen and Reimer的论断给予了严格的数学证明,我们将在下一章中详细讨论。
③ 限于篇幅在此不再详细讨论,详情请参见Leisen D. and Reimer M. (1996)第331-332页。另外Huang E. G.(2006)第七章第1节有关于Leisen and Reimer算法的介绍和程序代码。
④ 参见Wilmott P.(1998).
① 参见Tian Y. S. (1999).
② 关于收敛阶和收敛光滑性的定义可以参考下一章或Leisen D. and Reimer M. (1996), 或 Chang L. B. and Palmer K. (2007)第94页。由于Tian(1999)算法具备良好的光滑性,因而可以通过外推法(extrapolation technique)提高收敛阶。参见原文第828-832页。
① 参见Avellaneda M.and Laurence P.(1999).
② 符号cosh(·)表示双曲余弦函数,即cosh(x)=ex+e-x/2
③ 参见Jabbour G. E.,Kramin M. V.and Young S.T.(2001).
① 参见Leisen D. P. J.and Reimer M. (1996)第326-327页。
① 参见Chang L. B. and Palmer K. (2007)第94页。收敛的光滑性是一个非常重要的问题,这是因为,如果收敛是光滑的,则可以通过外推法(extrapolation technique)提高收敛阶。关于外推在二叉树算法中的应用参见Broadie M. and J. B. Detemple (1996), Tian Y, S. (1999), Heston S. and Zhou G. (2000)。本章关于收敛阶的证明均采用Leisen and Reimer (1996),第5、6章的证明采用(:hang and Palmer (2007)的定义。
② 参见Leisen D. P. J. and M. Reimer (1996)第328-330页.
① 作者证明了三个模型中mn2=mn3=Δn=O(Δt2),参见Leisen D.P. J.and M.Reimer(1996)附录B第344-346页。
② 参见Leisen D.P.J.and Reimer M.(1996)第333-335页。
③ 参见Walsh J.B.(2002,2003).
④ 参见Diener F. and Diener M.(2004).
⑤ 参见Diener F. and Diener M.(2004)第282-288页。
① 参见Chang L. B. and Palmer K. (2007)第95页推论1。
② 参见Chang L. B. and Palmer K. (2007)第96页推论2。
③ 参见Chung S. L. and Shih P. T. (2007)。
① 参见Chung S.L.and Shih P.T(2007)第509页定理1,该定理仅仅是收敛性的证明,并未涉及收敛阶,本文第三章第二节也有所提及。作者在其后文中者对收敛阶进行了详细的论证。
② 参见Joshi M.S.(2009).
③ 在该算法中,作者令μ=logK-logS0/T这恰好使得当上跳次数j≥n+1时,S0ujdn-j>K更加详细的讨论参见原文第173-174页。原文只对互补二叉树分布函数展开,由期权定价二叉树计算公式(3.1.4)可以看出,二者收敛阶、收敛光滑性相同。
④ 参见Joshi M.S.(2010)。
① 参见Diener F. and Diener M. (2004)第277页定理2.1。
① Boyle P. P. and Lau S. H. (1994)。
② 参见Boyle P. P. and Lau S. H. (1994)第8页。其后,Lin and Palmer证明该算法计算障碍期权时的收敛阶为1,参见Lin J. R. and Palmer K. (2013)第323页。
① 具体算法参见Tian Y. S. (1999)第832-837页,后文也对离散障碍期权如何计算进行了讨论。
② 参见Ritchken P. (1995).
③ 参见Lyuu Y. D.(1998)第77页。
④ 参见Chung S. L. and Shih P. T. (2007)电子附录(e-companion)第2-4页。
⑤ 参见Lin and Palmer (2013)。
① 参见Heynen R. C. and Kat H. M. (1996)、Esser A. (2003)或Tompkins R. G. (2000).
① 可参见Uspensky J.V. (1937)第119-129页。
① 参见Chung S. L. and Shih P. T. (2007)第518页附录A.1。
① 参见P.G.Zhang(1998)第214页公式(10.26 a)。
② 参见Ibrahim S. N. I., O'Hara J. Q and Constantinou N. (2013)第597页公式(2.11)、(2.12)、(2.13)、(2.14)。
① 此引理引自Lin and Palmer (2013),参见其文献第324页引理4.1,证明见其文献附录C。
① 标准下降敲入看涨障碍期权公式证明参见Reimer and Sandmann (1995)第2-7页,或Lin and Palmer(2013)附录B。显然,根据我们的定义,相应的幂期权价格计算公式不同之处仅在于收益函数部分。
① 参见Reiner E. and Rubinstein M.(1991)或Haug E.G. (2006)
① 由Black-Scholes微分方程(参见第二章定理2.6式(2.17))可以直接推出：
① 参见Chung and Shackleton (2002)第147-148页。
① 详细证明参见Chung, et al (2010)第586页附录。
① 参见Chung et al (2010)附录第587-590页。
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