基于分数—跳过程的巨灾债券定价
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摘要
最近几十年来,每年几乎都有巨灾的发生,并且发生的频率以及造成的损害程度都有大幅度的上升。面对损失程度和频率的大幅递增,保险公司的承保能力和政府财政救济能力的限制,使得我们不得不分散转移来自保险市场上的风险,所以人们把目光转移到了资本市场,进行保险证券化研究。
对于我国来说,自然灾害种类繁多并且发生频率较大、损失惨重:76唐山地震、98长江洪水、08湖南郴州冰雪灾害、08四川汶川地震、09-10西南灾害、10青海玉树地震等等。我国保险体系并不发达,大部分救灾都依赖于政府财政救济,这不利于财政的平稳,所以进行巨灾风险证券化研究就迫在眉睫。
本文的主要研究内容是对巨灾债券进行设计和定价。论文结构为:简要的了解全球以及我国的巨灾风险情况,提出巨灾证券化的必要性,以及最近这些年国内外学者对巨灾证券定价的研究情况介绍;给出在本文当中用到的数学随机过程方面的知识以及金融基础知识,比如常用的利率模型;简单介绍巨灾债券的运作机制,让读者有一个大致的了解,并且着重介绍比较经典的巨灾证券衍生品的定价模型;最后一部分是本文重点,本部分在前人的基础上进行创新:假设巨灾风险损失指数的价值过程满足一个分数布朗运动加泊松跳的过程,并且引入随机利率模型,对巨灾债券进行设计和定价,其中债券的设计分为两种形式。
The latest decades have witnessed catastrophes in almost every year, and the huge increasing frequency and intensity. In face of the increasing loss, due to the limitation of insurance companies and governmental relief, we have to figure out other methods to transfer catastrophe risks, therefore, capital market has attracted world’s attention. Insurance securitization becomes a hot issue.
In China, natural disasters are various and frequent with huge loss. From the worst flooding along Yangzi River in 1998 in a century, to the snow disaster in the south of China in the beginning of 2008, to 5.12 Wenchuan earthquake in the same year, to the recent months’especially big drought in the south west of China and to the 4.14 Yushu earthquake days ago, these raged natural disasters have brought huge economic loss as well as mental injury to Chinese people. However, because of weak risk management system, people are always ready to receive governmental aid or charity donation for free without purchasing insurance against natural disasters after disasters.
In this paper, we mainly discuss the designing and pricing of catastrophe bond. The article is organized as follows. The section of“Introduction”focuses on literature review. We briefly introduce the status quo of some other countries’catastrophe risks as well as China, and the research situation about the pricing of catastrophe bond at home and abroad. The section“Fundamental Knowledge”introduces some basic knowledge about stochastic process and financial engineering such as frequently-used interest rate model. The section“The operating mechanism and classical pricing theory of Catastrophe Bond”simply introduces the operating mechanism and some classical pricing models of Catastrophe derivatives. The last section”the designing and pricing of Catastrophe Bond”is the core and innovation. Based on previous research, we suppose that the value of the loss index on catastrophe risk is driven by a fractional Brownian motion with Poisson jumps, and we introduce the stochastic rate model. We design two kinds of catastrophe bonds and price these bonds based on our assumptions.
对于我国来说,自然灾害种类繁多并且发生频率较大、损失惨重:76唐山地震、98长江洪水、08湖南郴州冰雪灾害、08四川汶川地震、09-10西南灾害、10青海玉树地震等等。我国保险体系并不发达,大部分救灾都依赖于政府财政救济,这不利于财政的平稳,所以进行巨灾风险证券化研究就迫在眉睫。
本文的主要研究内容是对巨灾债券进行设计和定价。论文结构为:简要的了解全球以及我国的巨灾风险情况,提出巨灾证券化的必要性,以及最近这些年国内外学者对巨灾证券定价的研究情况介绍;给出在本文当中用到的数学随机过程方面的知识以及金融基础知识,比如常用的利率模型;简单介绍巨灾债券的运作机制,让读者有一个大致的了解,并且着重介绍比较经典的巨灾证券衍生品的定价模型;最后一部分是本文重点,本部分在前人的基础上进行创新:假设巨灾风险损失指数的价值过程满足一个分数布朗运动加泊松跳的过程,并且引入随机利率模型,对巨灾债券进行设计和定价,其中债券的设计分为两种形式。
The latest decades have witnessed catastrophes in almost every year, and the huge increasing frequency and intensity. In face of the increasing loss, due to the limitation of insurance companies and governmental relief, we have to figure out other methods to transfer catastrophe risks, therefore, capital market has attracted world’s attention. Insurance securitization becomes a hot issue.
In China, natural disasters are various and frequent with huge loss. From the worst flooding along Yangzi River in 1998 in a century, to the snow disaster in the south of China in the beginning of 2008, to 5.12 Wenchuan earthquake in the same year, to the recent months’especially big drought in the south west of China and to the 4.14 Yushu earthquake days ago, these raged natural disasters have brought huge economic loss as well as mental injury to Chinese people. However, because of weak risk management system, people are always ready to receive governmental aid or charity donation for free without purchasing insurance against natural disasters after disasters.
In this paper, we mainly discuss the designing and pricing of catastrophe bond. The article is organized as follows. The section of“Introduction”focuses on literature review. We briefly introduce the status quo of some other countries’catastrophe risks as well as China, and the research situation about the pricing of catastrophe bond at home and abroad. The section“Fundamental Knowledge”introduces some basic knowledge about stochastic process and financial engineering such as frequently-used interest rate model. The section“The operating mechanism and classical pricing theory of Catastrophe Bond”simply introduces the operating mechanism and some classical pricing models of Catastrophe derivatives. The last section”the designing and pricing of Catastrophe Bond”is the core and innovation. Based on previous research, we suppose that the value of the loss index on catastrophe risk is driven by a fractional Brownian motion with Poisson jumps, and we introduce the stochastic rate model. We design two kinds of catastrophe bonds and price these bonds based on our assumptions.
引文
[1]赵正堂.保险风险证券化研究[M].厦门:厦门大学出版社,2008.
[2]田玲.巨灾风险债券运作模式和定价机理研究[M].武汉:武汉大学出版社,2009.
[3] Cox,Samuel H.and Robert G.Schwebach.Insurance Futures and Hedging Insurance Price Risk[J].Journal of Risk and Insurance.1992,59.
[4] Yong Wang,Palljer H.Axiomatic Characterization of Insurance Prices[J].Insurance [J]:Mathematics and Economics.1997,21.
[5] Cummins,J.David,Helyette Geman.An Asian Option Approach ti the Valuation of Insurance Futures[J].Review of Futures Markets.1994,13.
[6] Cummins,J.David,Helyette Geman.Pricing Catastrophe Insurance Futures and Call Spreads[J].Journal of Fixed Income.1995,4.
[7] Litzenberger,R.H.,D.R. Beaglehole,C.E.Reynold,Assessing Catastrophe-Reinsu -rance-Linked Securities as a New Asset Class[J].Journal of Portfolio Management.1996,23.
[8] Helyette Geman,M.Yor.Stochastic Time Change in Catastrophe Option Pricing[J].Insurance:Mathematics and Economics.1997,21.
[9] Zajdenweber,D.The Valuation of Catastrophe-Reinsurance-Linked Securities Paper Presented at American Risk and Insurance Association Meeting.1998.
[10] Henri Loubergé,Evis Kellezi and Manfred Gill.Using Catastrophe-Linked Secu- rities to Diversify Insurance Risk: A Financial Analysis of Cat Bonds[J].Journal of Insurance Issues,1999,22(2).
[11] Knut K.Aase.An equilibrium model of catastrophe insurance futures and spread[J].Unpublished working paper,Bergen,Norway:The Norwegian School of Economics and Business Administration,1995.
[12] Christensen,C.V..A new Model for pricing catastrophe insurance derivatives [J]. Gentre for Analytical Finance,Working Paper Series.1999,28.
[13] Christensen,C.V..The PCS Option an improvement of the CAT-future[J]. working papers,University of Aarhus,2000.
[14] Sam Cox,Hal Pedersen.Catastrophe Risk Bonds[J].North American Actuarial Journal.2000,4.
[15] Knut K.Aase.A Markov Model for The Pricing of Catastrophe Insurance Futures and Spreads[J].Journal of Risk and Insurance.2001,68.
[16] Grundl H,Schmeiser H.Pricing Double-Trigger Reinsurance Contracts:Financial versus Actuarial Approach [J].Journal of Risk and Insurance.2002,69.
[17] Jinping Lee and Min-The Yu.Pricing Default-Risky Cat bonds with Moral Hazard and Basis Risk[J] .Journal of Risk and Insurance.2002,69(1).
[18]李冰清,田存志.保险产品的定价:CAPM的应用[J].当代财经.2001,11.
[19]韩天雄,陈建华.巨灾风险证券化产品的定价问题[J].保险研究专题.2003,12.
[20]田玲,向飞.基于风险定价框架的巨灾债券定价模型比较研究[J].武汉大学学报.2006,59(2).
[21]陶正如.基于工程地震风险评估的巨灾债券定价模型[J].中国地震局工程力学研究所博士学位论文.2007.
[22] S.M.劳斯.何声武,谢盛荣,程依明译.随机过程[M].北京:中国统计出版社,1997.
[23] Avinash K.Dixit,Robert.Pindyck.朱勇等译.不确定条件下的投资[M].北京:中国人民大学出版社,2002.
[24] Claus Vorm Christensen,Hanspeter Schmidli.Pricing catastrophe insurance products based on actually reported claims[J].Insurance:Mathematics and Economics 2000(27).
[25]王安兴.利率模型[M].上海:上海财经大学出版社,2007.
[26]李勇权.巨灾保险风险证券化研究[M].北京:中国财经经济出版社,2005.
[27] Joseph Stampfli,Victor Goodman.蔡明超译.金融数学-模型与套期保值[M].北京:机械工业出版社,2004.
[28] Wang,S.S.Cat Bond Pricing Using Probability Transforms.Geneva Association Insurance and the State of the Art in Cat Bond Pricing[J].Working Paper Series,No278,2004.
[29]杨晔.巨灾债券的定价模型比较研究[J].统计与决策.2008(6).
[30]周俊.巨灾指数模型及其衍生品套利定价方法[J].吉首大学学报.2007,28[J](5).
[31] Angelos Dassios,Ji-Wook Jang.Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity[J].Finance and Stochastics.2003,7(1).
[32] Sebastian Jaimungal,Tao Wang.Catastrophe options with Stochastic Interest Rates and Compound Possion Losses[J] . Insurance : Mathematics and Economics.2006,38(3).
[33] Yijia Lin , Samuel H . Cox . Securitization of catastrophe mortality risks[J].Insurance:Mathematics and Economics.2008(42).
[34]李永,刘娟.中国保险风险证券化研究[M].上海:上海财经大学出版社,2008.
[35] Bladt M,Rydberg T H.An actuarial to option pricing under the physical measure and without market assumption[J].Insurance:Mathematics and Economics.1998,22(1).
[36] Sylvie Bouriaux,Richard MacMinn.Securitization of Catastrophe Risk:New Developments in Insurance-Linked Securities and Derivatives[J].Journal of Insurance Issues.2009,32(1).
[37] Cox,H.,J.Fairchild,H.Pedersen.Valuation of Structured Risk Management Products[J].Insurance:Mathematics and Economics.2004,34.
[38]魏敏杰,田玲.巨灾风险证券化及在中国保险市场的应用[J].科技进步与对策.2003,10.
[39]李湛湘.关于我国实行巨灾保险证券化的思考[J].保险职业学院学报.2009,23(1).
[40]雪飞胜,陈超.股票价格服从跳-扩散过程的两值期权定价模型[J].数学理论与应用.2000,20(3).
[41]隋梅真,张元庆.分数布朗运动和泊松过程共同驱动下的欧式期权定价[J].山东建筑大学学报.2008,23(1).
[42]吴云,何建敏.两值期权的定价模型及其求解研究[J].管理工程学报.2002,16(4).
[43] X. Mao,Stochastic differential equations and their applies..Chichster:Horwood Publ,1997.
[44] Francesca Biagini et al.Stochastic Calculus for Fractional Brownian Motion and Applications [M].Published in association with the Applied Probability Trust,2008.
[2]田玲.巨灾风险债券运作模式和定价机理研究[M].武汉:武汉大学出版社,2009.
[3] Cox,Samuel H.and Robert G.Schwebach.Insurance Futures and Hedging Insurance Price Risk[J].Journal of Risk and Insurance.1992,59.
[4] Yong Wang,Palljer H.Axiomatic Characterization of Insurance Prices[J].Insurance [J]:Mathematics and Economics.1997,21.
[5] Cummins,J.David,Helyette Geman.An Asian Option Approach ti the Valuation of Insurance Futures[J].Review of Futures Markets.1994,13.
[6] Cummins,J.David,Helyette Geman.Pricing Catastrophe Insurance Futures and Call Spreads[J].Journal of Fixed Income.1995,4.
[7] Litzenberger,R.H.,D.R. Beaglehole,C.E.Reynold,Assessing Catastrophe-Reinsu -rance-Linked Securities as a New Asset Class[J].Journal of Portfolio Management.1996,23.
[8] Helyette Geman,M.Yor.Stochastic Time Change in Catastrophe Option Pricing[J].Insurance:Mathematics and Economics.1997,21.
[9] Zajdenweber,D.The Valuation of Catastrophe-Reinsurance-Linked Securities Paper Presented at American Risk and Insurance Association Meeting.1998.
[10] Henri Loubergé,Evis Kellezi and Manfred Gill.Using Catastrophe-Linked Secu- rities to Diversify Insurance Risk: A Financial Analysis of Cat Bonds[J].Journal of Insurance Issues,1999,22(2).
[11] Knut K.Aase.An equilibrium model of catastrophe insurance futures and spread[J].Unpublished working paper,Bergen,Norway:The Norwegian School of Economics and Business Administration,1995.
[12] Christensen,C.V..A new Model for pricing catastrophe insurance derivatives [J]. Gentre for Analytical Finance,Working Paper Series.1999,28.
[13] Christensen,C.V..The PCS Option an improvement of the CAT-future[J]. working papers,University of Aarhus,2000.
[14] Sam Cox,Hal Pedersen.Catastrophe Risk Bonds[J].North American Actuarial Journal.2000,4.
[15] Knut K.Aase.A Markov Model for The Pricing of Catastrophe Insurance Futures and Spreads[J].Journal of Risk and Insurance.2001,68.
[16] Grundl H,Schmeiser H.Pricing Double-Trigger Reinsurance Contracts:Financial versus Actuarial Approach [J].Journal of Risk and Insurance.2002,69.
[17] Jinping Lee and Min-The Yu.Pricing Default-Risky Cat bonds with Moral Hazard and Basis Risk[J] .Journal of Risk and Insurance.2002,69(1).
[18]李冰清,田存志.保险产品的定价:CAPM的应用[J].当代财经.2001,11.
[19]韩天雄,陈建华.巨灾风险证券化产品的定价问题[J].保险研究专题.2003,12.
[20]田玲,向飞.基于风险定价框架的巨灾债券定价模型比较研究[J].武汉大学学报.2006,59(2).
[21]陶正如.基于工程地震风险评估的巨灾债券定价模型[J].中国地震局工程力学研究所博士学位论文.2007.
[22] S.M.劳斯.何声武,谢盛荣,程依明译.随机过程[M].北京:中国统计出版社,1997.
[23] Avinash K.Dixit,Robert.Pindyck.朱勇等译.不确定条件下的投资[M].北京:中国人民大学出版社,2002.
[24] Claus Vorm Christensen,Hanspeter Schmidli.Pricing catastrophe insurance products based on actually reported claims[J].Insurance:Mathematics and Economics 2000(27).
[25]王安兴.利率模型[M].上海:上海财经大学出版社,2007.
[26]李勇权.巨灾保险风险证券化研究[M].北京:中国财经经济出版社,2005.
[27] Joseph Stampfli,Victor Goodman.蔡明超译.金融数学-模型与套期保值[M].北京:机械工业出版社,2004.
[28] Wang,S.S.Cat Bond Pricing Using Probability Transforms.Geneva Association Insurance and the State of the Art in Cat Bond Pricing[J].Working Paper Series,No278,2004.
[29]杨晔.巨灾债券的定价模型比较研究[J].统计与决策.2008(6).
[30]周俊.巨灾指数模型及其衍生品套利定价方法[J].吉首大学学报.2007,28[J](5).
[31] Angelos Dassios,Ji-Wook Jang.Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity[J].Finance and Stochastics.2003,7(1).
[32] Sebastian Jaimungal,Tao Wang.Catastrophe options with Stochastic Interest Rates and Compound Possion Losses[J] . Insurance : Mathematics and Economics.2006,38(3).
[33] Yijia Lin , Samuel H . Cox . Securitization of catastrophe mortality risks[J].Insurance:Mathematics and Economics.2008(42).
[34]李永,刘娟.中国保险风险证券化研究[M].上海:上海财经大学出版社,2008.
[35] Bladt M,Rydberg T H.An actuarial to option pricing under the physical measure and without market assumption[J].Insurance:Mathematics and Economics.1998,22(1).
[36] Sylvie Bouriaux,Richard MacMinn.Securitization of Catastrophe Risk:New Developments in Insurance-Linked Securities and Derivatives[J].Journal of Insurance Issues.2009,32(1).
[37] Cox,H.,J.Fairchild,H.Pedersen.Valuation of Structured Risk Management Products[J].Insurance:Mathematics and Economics.2004,34.
[38]魏敏杰,田玲.巨灾风险证券化及在中国保险市场的应用[J].科技进步与对策.2003,10.
[39]李湛湘.关于我国实行巨灾保险证券化的思考[J].保险职业学院学报.2009,23(1).
[40]雪飞胜,陈超.股票价格服从跳-扩散过程的两值期权定价模型[J].数学理论与应用.2000,20(3).
[41]隋梅真,张元庆.分数布朗运动和泊松过程共同驱动下的欧式期权定价[J].山东建筑大学学报.2008,23(1).
[42]吴云,何建敏.两值期权的定价模型及其求解研究[J].管理工程学报.2002,16(4).
[43] X. Mao,Stochastic differential equations and their applies..Chichster:Horwood Publ,1997.
[44] Francesca Biagini et al.Stochastic Calculus for Fractional Brownian Motion and Applications [M].Published in association with the Applied Probability Trust,2008.