一种亚式风格可重置执行价格期权设计

2014-10-27 10:06陈鹏李笋��
经济数学 2014年3期

陈鹏++李笋��

摘 要 本文设计了一种亚式风格的可重置执行价格期权;严格证明了可重置执行边界的存在性,以及连续区域与重置区域的单连通性;利用HartmanWatson分布,写出了可重置期权的定价公式,并利用此公式给出了可重置执行边界的一种新的数值算法.

关键词 市场流动性;亚式可重置期权;重置执行边界;重置执行红利;新型递归积分法

中图分类号 F224.7 文献标识码 A

One Resettable Striking Price Options

Design of Asian Style

CHEN Peng ,LI Sun

(College of Mathematic and Econometrics Hunan University , Changsha, Hunan 410082,China)

Abstract This paper designed one kind of resettable strike price options with Asian style, and proved strictly the existence of resetting boundary and the simple connectedness of continuation region and resetting region. Making use of HartmanWatson distribution, the pricing formula of resettable strike price options was written out, and a new numerical algorithm for resetting boundary utilizing this formula was given.

Key words market liquidity; Asian resettable options; resetting boundary; resetting premium; new recursive integral method

1 引 言

当今世界,金融衍生产品主要以美式产品为主,因为它们比欧式品有更大的交易灵活性,受到越来越多投资者青睐.美式产品很丰富,除了传统的普通美式看涨、看跌期权,人们创造了各种奇异性的美式期权.比如,在金融期权领域有:美式亚式期权[1]、俄罗斯期权[2]、美式巴黎期权[3]、以色列期权[4]、不列颠期权[5]、各种抵押贷款等[6];在实物期权领域有各种早期执行机会[7]、变更条约条款[8]等.尽管美式品日益成为主流,但部分投资者,仍然会选择欧式品,比如大宗原料、能源进出口条约,因为这里头很大部分购买者是风险对冲者,他们不关心价格的波动,只要能对冲掉风险就好;而另一部分人是纯正的期权投资者,甘愿暴露在价格波动的风险下,但又承担不了美式期权昂贵的价格.

以普通欧式看涨为例,若在接近到期日前资产价格S远低于执行价格K,则欧式期权价值几乎为零,因为市场翻转的机会不大.纯正的看涨权购买者陷入流动性风险,因为想卖掉期权也很难.为增加市场流动性,金融工程师们设计了诸如shout floor[9],reset strike put(call)[10],multiple reset rights[11],geometric average trigger reset options[12]、the British put option等等具有内生可抗流动性风险条款的新期权.这些期权中大部分本质上来说是另外一种美式期权,只不过它赌的不完全是资产在未来某一个时刻价格,还有随机化的参数.这样的期权具有更大的奇异性,需要更多的定价技巧.

本文设计的新期权属于可变更合约条款类期权,这一类产品设计思想是通过改变原始合约条款中的某些参数值,赋予投资者更多的选择权利.在香港市场上常见的产品有shout floor、reset strike put(call),其中,reset strike put 就是在普通看跌期权基础上,让期权购买者在合约期限内有限次改变交割价格的一种新期权,它能让已经进入“死态”的期权“复活”,所以比普通的看跌权更昂贵.重置条款既可以是手动的,也可以是自动的[8,12],后者本质上还是欧式权,而前者却是美式权.重置条款也可以选择其他参数,比如延长交易时间,这在实物期权领域很常见;利率相关产品也可以考虑更改借贷款利率.[9-11]考虑了将交割价格置换为当前价格的设计,本文设计的新期权在文献[10]基础上扩展,将交割价格置换为过去一段时间的平均值,这样可以减少将来后悔的可能,这正是亚式风格期权设计的思想.新产品能继承文献[10]中产品关于增强市场流动性的功能,同时,因为是亚式设计,故比reset strike call更便宜[3].这就是本文选题的出发点.本文采用手动停止设计,本质是美式期权.

2 模型假设

假设市场上存在两种可交易资产,风险资产和无风险资产.无风险资产Bt一般假定就是货币市场账户,它的动力学方程为:

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[22]熊炳忠,马柏林.基于贝叶斯MCMC算法的美式期权定价[J].经济数学,2013,30(2):55-62.

[23]邢迎春.CARA效用函数下美式期权的定价[J].经济数学,2011,28(1):18-20.

[24]梅树立.求解非线性BlackScholes模型的自适应小波精细积分法[J].经济数学,2012,29(4):8-14.

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[3] 郭宇权.金融衍生产品数学模型[M].第2版.北京:世界图书出版公司北京公司,2010:243.

[4] Y KIFER. Game options [J]. Finance and Stochastics, 2000, 4(4):443-463.

[5] G PESKIR, F SAMEE. The british put option [J]. Appl Math. Finance, 2011, 18(6): 537-563.

[6] J XIA, X Y ZHOU. Stock loans [J]. Mathematical Finance, 2007, 17(2):307-317.

[7] Chi Man LEUNG, Yue Kuen KWOK. PatentInvestment Games under Asymmetric Information [J]. European Journal of Operational Research, 2012, 223(2):441-451.

[8] Chi Man LEUNG, Yue Kuen KWOK. Employee stock option valuation with repricing features[J].Quantitative Finance, 2008, 8(6):561-569.

[9] T H F CHEUK, T C F VORST. Shout floors[J]. Financial engineering review, 2003,1(2):15-35.

[10]Min DAI, Yue Kuen KWOK, Lixin WU. Optimal shouting policies of options with strike reset right[J]. Mathematical Finance, 2004, 14(3): 383-401.

[11]Min DAI, Yue Kuen KWOK, Lixin WU. Options with multiple reset rights[J]. International Journal of Theoretical and Applied Finance, 2003, 6(6): 637-653.

[12]T S DAI, Y Y FANG, Y D LYUU. Analytics for geometric average trigger reset options[J]. Applied Economics Letters, 2005, 12(13): 835-840.

[13]H JONSSON, A G KUKUSH, D S SILVESTROV. Threshold structure of optimal stopping strategies for american type option(II)[J].Theory of Probability and Mathematical Statistics,2006,(72):47-58.

[14]S D JACKA. Optimal stopping and the American put[J]. Math. Finance,1991, 1(2) :1-14.

[15]R GESKE. The valuation of compound options[J]. J. Financial Econom, 1979,7(1): 63-81.

[16]S D HODGES, M J P SELBY. On the evaluation of compound options[J]. Management Science, 1987,33(3):347-355.

[17]S GERHLD. The hartmanwatson distribution revisited: asymptotics for pricing asian options[J]. Journal of Applied Probability, 2011, 48(3):597-899.

[18]G PESKIR. From stochastic calculus to mathematical finance[M].Berlin: Springer Berlin Heidelberg, 2006:535-546.

[19]S P ZHU. A new analyticalapproximation formula for the optimal exercise boundary of american put options [J]. International Journal of Theoretical and Applied Finance, 2006,9(7):1141-1177.

[20]J E ZHANG , T C LI. Pricing and hedging american options analytically: A Perturbation Method[J]. Mathematical Finance, 2010, 20(1): 59-87.

[21]S P ZHU. An exact and explicit solution for the valuation of american put options[J]. Quant. Finan., 2006,6(3): 229-242.

[22]熊炳忠,马柏林.基于贝叶斯MCMC算法的美式期权定价[J].经济数学,2013,30(2):55-62.

[23]邢迎春.CARA效用函数下美式期权的定价[J].经济数学,2011,28(1):18-20.

[24]梅树立.求解非线性BlackScholes模型的自适应小波精细积分法[J].经济数学,2012,29(4):8-14.

[25]科森多尔.随机微分方程[M].第6版.北京:世界图书出版公司北京公司,2006:139-140.endprint

[3] 郭宇权.金融衍生产品数学模型[M].第2版.北京:世界图书出版公司北京公司,2010:243.

[4] Y KIFER. Game options [J]. Finance and Stochastics, 2000, 4(4):443-463.

[5] G PESKIR, F SAMEE. The british put option [J]. Appl Math. Finance, 2011, 18(6): 537-563.

[6] J XIA, X Y ZHOU. Stock loans [J]. Mathematical Finance, 2007, 17(2):307-317.

[7] Chi Man LEUNG, Yue Kuen KWOK. PatentInvestment Games under Asymmetric Information [J]. European Journal of Operational Research, 2012, 223(2):441-451.

[8] Chi Man LEUNG, Yue Kuen KWOK. Employee stock option valuation with repricing features[J].Quantitative Finance, 2008, 8(6):561-569.

[9] T H F CHEUK, T C F VORST. Shout floors[J]. Financial engineering review, 2003,1(2):15-35.

[10]Min DAI, Yue Kuen KWOK, Lixin WU. Optimal shouting policies of options with strike reset right[J]. Mathematical Finance, 2004, 14(3): 383-401.

[11]Min DAI, Yue Kuen KWOK, Lixin WU. Options with multiple reset rights[J]. International Journal of Theoretical and Applied Finance, 2003, 6(6): 637-653.

[12]T S DAI, Y Y FANG, Y D LYUU. Analytics for geometric average trigger reset options[J]. Applied Economics Letters, 2005, 12(13): 835-840.

[13]H JONSSON, A G KUKUSH, D S SILVESTROV. Threshold structure of optimal stopping strategies for american type option(II)[J].Theory of Probability and Mathematical Statistics,2006,(72):47-58.

[14]S D JACKA. Optimal stopping and the American put[J]. Math. Finance,1991, 1(2) :1-14.

[15]R GESKE. The valuation of compound options[J]. J. Financial Econom, 1979,7(1): 63-81.

[16]S D HODGES, M J P SELBY. On the evaluation of compound options[J]. Management Science, 1987,33(3):347-355.

[17]S GERHLD. The hartmanwatson distribution revisited: asymptotics for pricing asian options[J]. Journal of Applied Probability, 2011, 48(3):597-899.

[18]G PESKIR. From stochastic calculus to mathematical finance[M].Berlin: Springer Berlin Heidelberg, 2006:535-546.

[19]S P ZHU. A new analyticalapproximation formula for the optimal exercise boundary of american put options [J]. International Journal of Theoretical and Applied Finance, 2006,9(7):1141-1177.

[20]J E ZHANG , T C LI. Pricing and hedging american options analytically: A Perturbation Method[J]. Mathematical Finance, 2010, 20(1): 59-87.

[21]S P ZHU. An exact and explicit solution for the valuation of american put options[J]. Quant. Finan., 2006,6(3): 229-242.

[22]熊炳忠,马柏林.基于贝叶斯MCMC算法的美式期权定价[J].经济数学,2013,30(2):55-62.

[23]邢迎春.CARA效用函数下美式期权的定价[J].经济数学,2011,28(1):18-20.

[24]梅树立.求解非线性BlackScholes模型的自适应小波精细积分法[J].经济数学,2012,29(4):8-14.

[25]科森多尔.随机微分方程[M].第6版.北京:世界图书出版公司北京公司,2006:139-140.endprint