单点多层重设看涨期权的鞅定价
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摘要
期权作为二十世纪金融行业创新的一个代表产物,在1973年的美国芝加哥期权交易所第一次开始交易后,对于整个金融衍生品市场及金融数学理论所带来的冲击是当时无人所能预想到的。现今,期权交易市场已经成为整个金融市场里不可或缺的一个重要组成部分。基于以上原因,对于期权定价理论与实证的研究推动了金融创新的新高潮。
本文主要介绍一种新型的重设期权,并且结合概率论中的鞅理论、随机微分方程、测度论等数学工具,对于该期权在风险中性概率测度下的理论公允价格给出了多重积分形式的表达式。
对于这一新型期权,即多点重设期权,是在前人对于标准重设期权讨论的基础上,进行了创新,在原有的重设方案的基础上,多加了一层重设。这样的创新主要是受到了可转换债券的启发：由于可转债一般会有这样的规定,在一段时间内股价不超出某一限制则允许进行转换。本文将这一规定进行了一定修正后加入重设规则的构建中,使得重设条件与股价在一段时间内的最大值有关,如果股价在这一段持续的时间内超过一定水平则将取消重设资格。与原先只以某一时刻的价格水平来决定是否进行重设的方案相比,新型期权的设计大大降低了投机者利用原来单一的重设结构来操作股价,乘机套利的机会。本文的研究目标也对于这样的期权采用风险中性条件下,等价鞅测度变化的方法给出其理论公允值。这种新型期权突破原来重设期权的固有方式,在求解上也要借助一些复杂的随机微分方程理论,利用原生资产首达时间的概率密度函数,反复利用联合概率分布和条件概率的乘法公式进行计算,做了具有一定原创性的工作。
"Option" is considered to be a successful example of creative practice world finance market of the 20th Century. Although options have been firstly traded in American Chicago Board Option Exchange since 1973, almost nobody could, at that time, foresee that they would bring about huge influence on practice and financial and banking theory in the following several decades. Now option market has become an important part of international finance market. Under such background, research and application of option pricing theory, currently, set off a new climax of innovation in the field of financial theory.
A new kind of reset option will be introduced in this paper, and we are going to give out the expression in multi-integration form with the method of martingale theory and stochastic analysis
This new kind of option, single point multiple level reset option, based on the research of the theory before, I constitute a brand new kind of reset option. Inspired by the convertible bonds'exchange regulation, a new stage of the reset is added to the option besides of the basic principle, which can be more helpful to prevent the arbitrage of the speculator controlling the asset price by means of great amount contract than the original reset option. The aim of our research is actually to show the fair price of the option in the risk neutral probability measure with Girsanov theorem. The new option just break the ordinary thought of the reset option, and much knowledge of stochastic differential equation and the Multiplication Formula in probability space will be necessary for solving the pricing problem, setting out some creative work.
本文主要介绍一种新型的重设期权,并且结合概率论中的鞅理论、随机微分方程、测度论等数学工具,对于该期权在风险中性概率测度下的理论公允价格给出了多重积分形式的表达式。
对于这一新型期权,即多点重设期权,是在前人对于标准重设期权讨论的基础上,进行了创新,在原有的重设方案的基础上,多加了一层重设。这样的创新主要是受到了可转换债券的启发：由于可转债一般会有这样的规定,在一段时间内股价不超出某一限制则允许进行转换。本文将这一规定进行了一定修正后加入重设规则的构建中,使得重设条件与股价在一段时间内的最大值有关,如果股价在这一段持续的时间内超过一定水平则将取消重设资格。与原先只以某一时刻的价格水平来决定是否进行重设的方案相比,新型期权的设计大大降低了投机者利用原来单一的重设结构来操作股价,乘机套利的机会。本文的研究目标也对于这样的期权采用风险中性条件下,等价鞅测度变化的方法给出其理论公允值。这种新型期权突破原来重设期权的固有方式,在求解上也要借助一些复杂的随机微分方程理论,利用原生资产首达时间的概率密度函数,反复利用联合概率分布和条件概率的乘法公式进行计算,做了具有一定原创性的工作。
"Option" is considered to be a successful example of creative practice world finance market of the 20th Century. Although options have been firstly traded in American Chicago Board Option Exchange since 1973, almost nobody could, at that time, foresee that they would bring about huge influence on practice and financial and banking theory in the following several decades. Now option market has become an important part of international finance market. Under such background, research and application of option pricing theory, currently, set off a new climax of innovation in the field of financial theory.
A new kind of reset option will be introduced in this paper, and we are going to give out the expression in multi-integration form with the method of martingale theory and stochastic analysis
This new kind of option, single point multiple level reset option, based on the research of the theory before, I constitute a brand new kind of reset option. Inspired by the convertible bonds'exchange regulation, a new stage of the reset is added to the option besides of the basic principle, which can be more helpful to prevent the arbitrage of the speculator controlling the asset price by means of great amount contract than the original reset option. The aim of our research is actually to show the fair price of the option in the risk neutral probability measure with Girsanov theorem. The new option just break the ordinary thought of the reset option, and much knowledge of stochastic differential equation and the Multiplication Formula in probability space will be necessary for solving the pricing problem, setting out some creative work.
引文
1. Paul Wilmott. Derivatives the theory and practical of financial engineering.
2. Applebaum, D.,2004. Levy Processes and Stochastic Calculus. Cambridge University Press.
3. DecreusefondL. UstunelA S. Stochastic analysis of fractional Brownian motion[J]. Potential Analysis,1999,10:177-214.
3. Black F, Scholes M J. The pricing of options and corporate liabilities. Journal of political Economics,1973,81:637-659
4. Susanne Kruse, Ulrich Nogel. On the pricing of forward start options in Heston's model on stochastic volatility [J]. Finance Stochast,2005,9233-250
5. Hu, Y. and B. Oksendal, Fractional white noise calculus and application to finance[J]. Pure Mathematics,1999,10-99.
6. Sottinen T, Valkeila E. Fractional Brownian motion as a model in finance[J]. Dept of Math University of OSLO, Preprint,2007,25.
7. Merton M C. Option Pricing When Underlying Stock Returns Are Disconti-nuous, Journal of Financial Economics,1976,3:125-144.
8. Carr, P., Wu L.2004. Time Changed Levy Processes and Option Pricing. Journal of Financial Economics 71,113-141.
9. Mao-wei hung, Chang. Analytical Valuation of Catastrophe Equity Options With Negative Exponential Jumps, Insurance:Mathematics and Economics 44 (2009)59-69
10. Clark P. K. A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices. Econometrica,1973,41(1):135-155.
11. Ait-sahalia Y Z. Nonparametric pricing of interest rate derivative securities. Journal of Econometric,1996,64:23-47.
12. Wiggin, J. B., Option values under stochastic volatility:Theory and empirical estimates, J. Financial Economics,19(1987),351-372.
13. Chen, C.&Zou, J. Z., Risk management for the financial derivative security, Advances in Investment Decision Analysis,2001.
14. Davis, M. H. A. Panas, V. P.& Zariphopoulou, T. European option pricing with transaction costs. SIAM J. Control Optim.31(1993),470-493.
15. Karatzas I. Martingale And Duality Methods For Utility Maximization In An Incomplete Market. Control and Optimization 1991,29(3):702-730.
16. Benth F. E, Karlsen K. H. A PDE Representation of the Density of the Minimal Entropy Martingale Measure in Stochastic Volatility Markets. Stochastics:An International Journal of Probability and Stochastic Processes,2005,77(2): 109-137.
17. Jouini E. Market imperfection, Equilibrium and Arbitrage, Financial Mathematics. Spring-Verlag, Berlin1997
18. Rubinstin, M., Implied binomial trees, Journal of Finance,69(1994),771-818.
19. Carr P, Wu L. Time — Changed Levy Processes and Option Pricing. Journal of Financial Economics,2004,71:113-141
20. Barrieu P, Bellamy N. Optimal Hitting Time and Real Options in a Single Jump Diffusion Model. Applied Probability Trust,2005,3:1-12
21. Cox J C, Ross S A. Rubinstein, M. Option Pricing:a Simplified Approach. Journal of Financial Economics,1979,3:145-166.
22. Duffle D. An Extension Of The Black-Scholes Model Of Security Valuation. Journal Of Economics Theory,1988,46:194-204.
23.金治明,数学金融学基础.
24.肖艳秋,邹捷中.分数布朗运动环境下的期权定价与测度变换[J].数学的实践与认识,2008,38(20)：58-62.
25.何韵妍,杨立洪,何春雄.多点重设型看跌期权的鞅定价[J].科学技术与工程,2008,8(14)：3872-3878.
26.叶中行、林建忠,数理金融,科学出版社,1998.
27.刘韶跃,杨向群.分数布朗运动环境中欧式未定权益的定价[J].应用概率统计,2004,20(4)：429-434.
28.叶中行,林建中,数理金融-资产定价与金融决策理论[M],北京：科学出版社,2005,131-170.
29.雍炯敏,数学金融学中的若干问题,数学的实践与认识,2(1999),97-108.
30.欧辉,杨向群.重设型牛市认购权证的鞍定价.湖南师范大学自然科学学报,2004,12,27(4)：3-6.
31.姜礼尚,期权定价的数学模型和方法,高等教育出版社,2004
32.许永庆,李时银.带有信用风险的重设型卖权的定计公式,数学研究,2005,38(1)：103-111.
33.达雷尔·达菲著,潘存武译.动态资产定价理论(第三版),上海财经大学出版社,2003.
34.邓国和,杨向群.随机波动率与双指数跳扩散组合模型的美式期权定价,应用数学学报,2009-02-006.
35.门东平,王军.含跳跃股价波动过程的期权定价,科学技术与工程,2008,15-4272-03.
36.唐晓.带有汇率的期权,[硕士学位论文],大连,大连理工大学2005.
37.薛红,彭玉成.鞅在未定权益定价中的应用,工程数学学报,2000,8,17(3)：35-138.
2. Applebaum, D.,2004. Levy Processes and Stochastic Calculus. Cambridge University Press.
3. DecreusefondL. UstunelA S. Stochastic analysis of fractional Brownian motion[J]. Potential Analysis,1999,10:177-214.
3. Black F, Scholes M J. The pricing of options and corporate liabilities. Journal of political Economics,1973,81:637-659
4. Susanne Kruse, Ulrich Nogel. On the pricing of forward start options in Heston's model on stochastic volatility [J]. Finance Stochast,2005,9233-250
5. Hu, Y. and B. Oksendal, Fractional white noise calculus and application to finance[J]. Pure Mathematics,1999,10-99.
6. Sottinen T, Valkeila E. Fractional Brownian motion as a model in finance[J]. Dept of Math University of OSLO, Preprint,2007,25.
7. Merton M C. Option Pricing When Underlying Stock Returns Are Disconti-nuous, Journal of Financial Economics,1976,3:125-144.
8. Carr, P., Wu L.2004. Time Changed Levy Processes and Option Pricing. Journal of Financial Economics 71,113-141.
9. Mao-wei hung, Chang. Analytical Valuation of Catastrophe Equity Options With Negative Exponential Jumps, Insurance:Mathematics and Economics 44 (2009)59-69
10. Clark P. K. A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices. Econometrica,1973,41(1):135-155.
11. Ait-sahalia Y Z. Nonparametric pricing of interest rate derivative securities. Journal of Econometric,1996,64:23-47.
12. Wiggin, J. B., Option values under stochastic volatility:Theory and empirical estimates, J. Financial Economics,19(1987),351-372.
13. Chen, C.&Zou, J. Z., Risk management for the financial derivative security, Advances in Investment Decision Analysis,2001.
14. Davis, M. H. A. Panas, V. P.& Zariphopoulou, T. European option pricing with transaction costs. SIAM J. Control Optim.31(1993),470-493.
15. Karatzas I. Martingale And Duality Methods For Utility Maximization In An Incomplete Market. Control and Optimization 1991,29(3):702-730.
16. Benth F. E, Karlsen K. H. A PDE Representation of the Density of the Minimal Entropy Martingale Measure in Stochastic Volatility Markets. Stochastics:An International Journal of Probability and Stochastic Processes,2005,77(2): 109-137.
17. Jouini E. Market imperfection, Equilibrium and Arbitrage, Financial Mathematics. Spring-Verlag, Berlin1997
18. Rubinstin, M., Implied binomial trees, Journal of Finance,69(1994),771-818.
19. Carr P, Wu L. Time — Changed Levy Processes and Option Pricing. Journal of Financial Economics,2004,71:113-141
20. Barrieu P, Bellamy N. Optimal Hitting Time and Real Options in a Single Jump Diffusion Model. Applied Probability Trust,2005,3:1-12
21. Cox J C, Ross S A. Rubinstein, M. Option Pricing:a Simplified Approach. Journal of Financial Economics,1979,3:145-166.
22. Duffle D. An Extension Of The Black-Scholes Model Of Security Valuation. Journal Of Economics Theory,1988,46:194-204.
23.金治明,数学金融学基础.
24.肖艳秋,邹捷中.分数布朗运动环境下的期权定价与测度变换[J].数学的实践与认识,2008,38(20)：58-62.
25.何韵妍,杨立洪,何春雄.多点重设型看跌期权的鞅定价[J].科学技术与工程,2008,8(14)：3872-3878.
26.叶中行、林建忠,数理金融,科学出版社,1998.
27.刘韶跃,杨向群.分数布朗运动环境中欧式未定权益的定价[J].应用概率统计,2004,20(4)：429-434.
28.叶中行,林建中,数理金融-资产定价与金融决策理论[M],北京：科学出版社,2005,131-170.
29.雍炯敏,数学金融学中的若干问题,数学的实践与认识,2(1999),97-108.
30.欧辉,杨向群.重设型牛市认购权证的鞍定价.湖南师范大学自然科学学报,2004,12,27(4)：3-6.
31.姜礼尚,期权定价的数学模型和方法,高等教育出版社,2004
32.许永庆,李时银.带有信用风险的重设型卖权的定计公式,数学研究,2005,38(1)：103-111.
33.达雷尔·达菲著,潘存武译.动态资产定价理论(第三版),上海财经大学出版社,2003.
34.邓国和,杨向群.随机波动率与双指数跳扩散组合模型的美式期权定价,应用数学学报,2009-02-006.
35.门东平,王军.含跳跃股价波动过程的期权定价,科学技术与工程,2008,15-4272-03.
36.唐晓.带有汇率的期权,[硕士学位论文],大连,大连理工大学2005.
37.薛红,彭玉成.鞅在未定权益定价中的应用,工程数学学报,2000,8,17(3)：35-138.