欧式巨灾任选股票期权定价及其对冲
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摘要
巨灾风险是指突发性、不可预料、无法回避而且危害性极大的自然灾难性事件所产生的风险.近几十年来,世界范围内自然灾害(地震、海啸、飓风、水旱灾、泥石流等)频繁发生,给人类带来极大的生命和财产损失,损失爆发的频率和严重程度以惊人的速度上升.由此引发的巨额保险赔付让传统的保险经营方式承受着极大的挑战,制约了保险业的发展.巨灾期权是一种保险风险转移的金融工具,他利用风险证券化将保险市场与证券市场有效结合,用资本市场的实力来分散(再)保险公司的巨灾风险,使金融、保险实现一体化.
尽管巨灾期权或巨灾股票期权深受保险业和投资者的喜爱,但是他们并没有在市场上很好的发展起来,究其原因主要是难以准确定价期权合约,以及供投资者选择的证券化产品数量不足.基于此,本文通过引入一类新型的巨灾任选股票期权,并研究其定价与风险管理问题,主要内容有：
第二章在常数利率条件下假设巨灾损失总额满足复合Poisson过程,以及市场股票价格满足几何布朗运动的情形下,研究巨灾简单任选期权和复杂任选期权的定价,给出了他们的定价公式和对冲避险策略.在假定巨灾损失额服从正态分布的条件下进一步给出巨灾任选期权的评价,以及数值计算了风险发生频率入,巨灾损失参数θ,δ对期权合约的影响.
第三章在随机利率条件下研究巨灾简单任选期权和复杂任选期权的定价问题,应用远期测度变换方法给出了巨灾任选股票期权的显示解,并在利率满足Vasicek模型下分析了利率模型中各参数值变动对期权价格的影响.
第四章总结了本文的主要工作及有待进一步研究的问题.
The catastrophe risks are generated by these events of natural catastrophe which are of sudden, nopredicting, noavoiding and have great harmfulness. In recent decades,the worldwide natural dis-asters (like earthquake, tsunami, hurricane, flood drought and debris flows etc.)occur frequently, and cause huge loss of lives and property. The occurred frequency of loss and severity of the catas-trophe are increasing wonderfully. Caused by the huge insurance compensation let the traditional insurance management way bear the great challenge, and restrict the development of the insurance industry. Catastrophe option is a financial instrument for shifting the insurance risks, combines the insurance market with stock market effectively by securitizing the catastrophe risks,and spreads the catastrophe risks of insurance or reinsurance industry by means of the capital market. These result in integration between finance and insurance.
Although the catastrophe options or the catastrophe stock options are favourited by the in-surance industry and many investors, they are not very well developed in the market. The main reasons include the difficult pricing the option accurately and the short of amount of securitised products for investor choosing. Based on these mentioned above,an new product of catastrophe option, called by catastrophe chooser option Written on stock whose pricing and hedging are con-sidered in this thesis. Main contributions are as follows:
In chapter2, the pricings of both European catastrophe simple chooser option and complex chooser option are considered under the stock's price following the diffusion model and constant interest rate framework with the catastrophe losses generated by a compound Poisson process. The closed-form solutions of the price and hedging strategy are obtained. Furthermore, the pricing closed-form solution of the European chooser options are also gained and some numerical exam-ples such as the affect of the parameters λ, θ,δ to the option are provided under the size of the losses satisfying the normal distribution.
In chapter3,we derived the analytic price formulas for the European catastrophe chooser op-tions within the stochastic interest rate framework by using forward martingale method, and the explicit solutions of the chooser options are obtained. Some behaviours of parameters in the Va-sicek model are analyzed with numerical examples.
Main conclusions and the further research works are summarized in chapter4.
尽管巨灾期权或巨灾股票期权深受保险业和投资者的喜爱,但是他们并没有在市场上很好的发展起来,究其原因主要是难以准确定价期权合约,以及供投资者选择的证券化产品数量不足.基于此,本文通过引入一类新型的巨灾任选股票期权,并研究其定价与风险管理问题,主要内容有：
第二章在常数利率条件下假设巨灾损失总额满足复合Poisson过程,以及市场股票价格满足几何布朗运动的情形下,研究巨灾简单任选期权和复杂任选期权的定价,给出了他们的定价公式和对冲避险策略.在假定巨灾损失额服从正态分布的条件下进一步给出巨灾任选期权的评价,以及数值计算了风险发生频率入,巨灾损失参数θ,δ对期权合约的影响.
第三章在随机利率条件下研究巨灾简单任选期权和复杂任选期权的定价问题,应用远期测度变换方法给出了巨灾任选股票期权的显示解,并在利率满足Vasicek模型下分析了利率模型中各参数值变动对期权价格的影响.
第四章总结了本文的主要工作及有待进一步研究的问题.
The catastrophe risks are generated by these events of natural catastrophe which are of sudden, nopredicting, noavoiding and have great harmfulness. In recent decades,the worldwide natural dis-asters (like earthquake, tsunami, hurricane, flood drought and debris flows etc.)occur frequently, and cause huge loss of lives and property. The occurred frequency of loss and severity of the catas-trophe are increasing wonderfully. Caused by the huge insurance compensation let the traditional insurance management way bear the great challenge, and restrict the development of the insurance industry. Catastrophe option is a financial instrument for shifting the insurance risks, combines the insurance market with stock market effectively by securitizing the catastrophe risks,and spreads the catastrophe risks of insurance or reinsurance industry by means of the capital market. These result in integration between finance and insurance.
Although the catastrophe options or the catastrophe stock options are favourited by the in-surance industry and many investors, they are not very well developed in the market. The main reasons include the difficult pricing the option accurately and the short of amount of securitised products for investor choosing. Based on these mentioned above,an new product of catastrophe option, called by catastrophe chooser option Written on stock whose pricing and hedging are con-sidered in this thesis. Main contributions are as follows:
In chapter2, the pricings of both European catastrophe simple chooser option and complex chooser option are considered under the stock's price following the diffusion model and constant interest rate framework with the catastrophe losses generated by a compound Poisson process. The closed-form solutions of the price and hedging strategy are obtained. Furthermore, the pricing closed-form solution of the European chooser options are also gained and some numerical exam-ples such as the affect of the parameters λ, θ,δ to the option are provided under the size of the losses satisfying the normal distribution.
In chapter3,we derived the analytic price formulas for the European catastrophe chooser op-tions within the stochastic interest rate framework by using forward martingale method, and the explicit solutions of the chooser options are obtained. Some behaviours of parameters in the Va-sicek model are analyzed with numerical examples.
Main conclusions and the further research works are summarized in chapter4.
引文
[1]Sigma. Natural Catastrophes and man-made disasters in 2007:high loss in European[R], Sigma Publication No.1, Swiss Re, Zurich,2008.
[2]Black F., Scholes M. The pricing of options and corporate liabilities[J]. Journal of Political Economy,1973(81):637-654.
[3]Merton R. C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Economics,1976(3):125-144.
[4]Froot K. A. The Market for Catastrophe Risk:A Clinical Examination[J]. Journal of Financial Economics,2001(3):529-571.
[5]Punter A. Securitisation of insurance risk[J]. In 2001 Hazard Conference, Aon Limited,2001.
[6]Cox S. H, Schwebach R. G. Insurance futures and Hedging Insurance price risk[J]. Journal of Risk and Insurance,1992(59):628-644.
[7]Cummins J. D., Geman H. Pricing Catastrophe futures and call spreads:An arbitrage ap-proach[J]. Journal of Fixed Income,1995(4):46-57.
[8]Chang C., Chang J., Yu M. T. Pricing catastrophe insurance futures call spreads:a randomized operational time approach[J]. Journal of Risk and Insurance.1996(63):599-617.
[9]Louberge H., Kellezi E., Gilli M. Using catastrophe-lined securities to diversify insurance risk:a financial analysis of CAT bonds[J], Journal of Risk and Insurance,1999(22):125-146.
[10]Lee J. P., Yu M. T. Pricing default-risky CAT bonds with moral Hazard and basis risk[J]. Journal of Risk and Insurance,2002(69):25-44.
[11]Vaugirard V. Pricing catastrophe bonds by arbitrage[J]. The Quarterly Review of Economics and Finance Approach,2003(43):119-132.
[12]Dassios A., Jiang J. W. Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity[J]. Financial Stochastics,2003(7):73-95.
[13]Cox H, Fairchild J, Pedersen H. Valuation of structured risk management products[J]. Insur-ance:Mathematics and Economics,2004(34):259-272.
[14]Jaimungal S., Wang T. Catastrophe options with stochastic interest rates and compound pois-son losses[J]. Insurance:Mathematics and Economics,2006(38):469-483.
[15]Fujita T., Ishimura N., Tanaka D. An arbitrage approach to the pricing of catastrophe options involving the Cox process[J]. Hitotsubashi Journal of Economics,2008(49):67-74.
[16]Lin S. K., Chang C. C. Catastrophe Equity put in Markov Jump-diffusion model[J],2007 ASTIN Colloquium Paper Presentations, Orlando, FL, USA(50th),2007, June:19-22.
(17] Bakshi G., Madan D. Average rate claims with emphasis on catastrophe loss options[J]. Jour-nal of Financial and Quantitative Analysis,2002(37):93-115.
[18]Chang C. W, Chang J-S. K., Lu W. L. Pricing Catastrophe options with stochastic claim arrival intensity in claim time[J], Journal of Blanking and Finance,2010(34):24-32.
[19]牛津津.巨灾期权定价模型研究[D],哈尔滨工程大学硕士学位论文,2008.
[20]王博.巨灾期权定价方法的探讨[J],统计与决策,2009(9)：33-35.
[21]丁波,巴曙松.中国地震巨灾期权定价机制的实证研究[J],农村金融研究,2010(6)：29-35.
[22]谢世清.谈巨灾期权及其演变[J],经济理论与经济管理,2010(6)：36-42.
[23]周洪海.基于跳扩散过程的重置巨灾看跌期权定价[D],新疆大学硕士学位论文,2008.
124] Rubinstein M. Options for the undecided in from Black-Scholes to Blcak-Holes:New fron-tiers in options[J]. Risk Magazine,1992,187-189.
[25]Detemple J., Emmerling T. American Chooser Options[J]. Journal of Economic Dynamics Control,2009(33):128-153.
[26]黄国安,邓国和,霍海峰.跳扩散过程下欧式任选期权定价[J],山西大学学报(自然科学版),2008.31(3)：438-442.
[27]黄国安.随机利率环境下欧式任选期权定价[J],广西师范大学硕士学位论文,2011.
[28]Wu D. S. Catastrophe Bond and Risk Modeling:A Review and Calibration using Chinese Earthquake Loss Data[J]. Human and Ecological Risk Assessment,2010(16):512-523.
[29]Genz A. Numerical computation of multivariation normal probabilities[J]. Journal of Com-putational Graph and Statics,1992(1):141-183.
[30]Merton R. C. Theory of rational options pricing[J]. Bell Journal of Economics,1973(4):141-183.
[31]Amin K. I., Jarrow R. Pricing options on risky asset in a stochastic interest rate economy[J]. Mathematical Finance,2003(13):445-466.
[32]Bakshi G., Gao C., Chen Z. W. Pricing and hedging long-term options[J]. Journal of Econo-metrics,2009(94):277-318.
[33]Scott L. O. Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates:application of Fourier inversion methods[J]. Mathematical Finance.1997(7): 413-424.
[34]Nicolato E. Options pricing stochastic volatility models of Ornstein-Uhlenbeck type[J]. Mathematical Finance,2003(13):445-466.
[35]Deng G H. Pricing European option in a double exponential jump-diffusion model with two market structure risks and its comparisons[J]. Applies Mathematics-A Journal of Chinese University,2007(22):127-137.
[36]Deng G H., Huang L H. A poisson-Gaussian model to price European options on the ex-tremum of several risky assets within the HJK framework[J]. Journal Systems Sciences and Complex,2010(23):769-783.
[37]Deng G H. Pricing reset options with multiple reset features under stochastic interest rates and jump-diffusion model[M]. Proceedings of the 29th CCC,2010, July 29-31, Beijing,5559-5562.
[38]王莉君,张曙光Vasucek随机利率下的重置期权定价[J],高等应用数学学报A辑,2002,17(4)：471-478.
[39[Gerber H., Karoui N., Pocher J. Change of numeraire, change of probability measure,and option pricing[J]. Journal of Applied Probability,1995(32):443-459.
[40]Vasicek O. An equilibrium characterisation of the term structure[J]. Journal of Financial Eco-nomics,1997(5):177-188.
[2]Black F., Scholes M. The pricing of options and corporate liabilities[J]. Journal of Political Economy,1973(81):637-654.
[3]Merton R. C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Economics,1976(3):125-144.
[4]Froot K. A. The Market for Catastrophe Risk:A Clinical Examination[J]. Journal of Financial Economics,2001(3):529-571.
[5]Punter A. Securitisation of insurance risk[J]. In 2001 Hazard Conference, Aon Limited,2001.
[6]Cox S. H, Schwebach R. G. Insurance futures and Hedging Insurance price risk[J]. Journal of Risk and Insurance,1992(59):628-644.
[7]Cummins J. D., Geman H. Pricing Catastrophe futures and call spreads:An arbitrage ap-proach[J]. Journal of Fixed Income,1995(4):46-57.
[8]Chang C., Chang J., Yu M. T. Pricing catastrophe insurance futures call spreads:a randomized operational time approach[J]. Journal of Risk and Insurance.1996(63):599-617.
[9]Louberge H., Kellezi E., Gilli M. Using catastrophe-lined securities to diversify insurance risk:a financial analysis of CAT bonds[J], Journal of Risk and Insurance,1999(22):125-146.
[10]Lee J. P., Yu M. T. Pricing default-risky CAT bonds with moral Hazard and basis risk[J]. Journal of Risk and Insurance,2002(69):25-44.
[11]Vaugirard V. Pricing catastrophe bonds by arbitrage[J]. The Quarterly Review of Economics and Finance Approach,2003(43):119-132.
[12]Dassios A., Jiang J. W. Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity[J]. Financial Stochastics,2003(7):73-95.
[13]Cox H, Fairchild J, Pedersen H. Valuation of structured risk management products[J]. Insur-ance:Mathematics and Economics,2004(34):259-272.
[14]Jaimungal S., Wang T. Catastrophe options with stochastic interest rates and compound pois-son losses[J]. Insurance:Mathematics and Economics,2006(38):469-483.
[15]Fujita T., Ishimura N., Tanaka D. An arbitrage approach to the pricing of catastrophe options involving the Cox process[J]. Hitotsubashi Journal of Economics,2008(49):67-74.
[16]Lin S. K., Chang C. C. Catastrophe Equity put in Markov Jump-diffusion model[J],2007 ASTIN Colloquium Paper Presentations, Orlando, FL, USA(50th),2007, June:19-22.
(17] Bakshi G., Madan D. Average rate claims with emphasis on catastrophe loss options[J]. Jour-nal of Financial and Quantitative Analysis,2002(37):93-115.
[18]Chang C. W, Chang J-S. K., Lu W. L. Pricing Catastrophe options with stochastic claim arrival intensity in claim time[J], Journal of Blanking and Finance,2010(34):24-32.
[19]牛津津.巨灾期权定价模型研究[D],哈尔滨工程大学硕士学位论文,2008.
[20]王博.巨灾期权定价方法的探讨[J],统计与决策,2009(9)：33-35.
[21]丁波,巴曙松.中国地震巨灾期权定价机制的实证研究[J],农村金融研究,2010(6)：29-35.
[22]谢世清.谈巨灾期权及其演变[J],经济理论与经济管理,2010(6)：36-42.
[23]周洪海.基于跳扩散过程的重置巨灾看跌期权定价[D],新疆大学硕士学位论文,2008.
124] Rubinstein M. Options for the undecided in from Black-Scholes to Blcak-Holes:New fron-tiers in options[J]. Risk Magazine,1992,187-189.
[25]Detemple J., Emmerling T. American Chooser Options[J]. Journal of Economic Dynamics Control,2009(33):128-153.
[26]黄国安,邓国和,霍海峰.跳扩散过程下欧式任选期权定价[J],山西大学学报(自然科学版),2008.31(3)：438-442.
[27]黄国安.随机利率环境下欧式任选期权定价[J],广西师范大学硕士学位论文,2011.
[28]Wu D. S. Catastrophe Bond and Risk Modeling:A Review and Calibration using Chinese Earthquake Loss Data[J]. Human and Ecological Risk Assessment,2010(16):512-523.
[29]Genz A. Numerical computation of multivariation normal probabilities[J]. Journal of Com-putational Graph and Statics,1992(1):141-183.
[30]Merton R. C. Theory of rational options pricing[J]. Bell Journal of Economics,1973(4):141-183.
[31]Amin K. I., Jarrow R. Pricing options on risky asset in a stochastic interest rate economy[J]. Mathematical Finance,2003(13):445-466.
[32]Bakshi G., Gao C., Chen Z. W. Pricing and hedging long-term options[J]. Journal of Econo-metrics,2009(94):277-318.
[33]Scott L. O. Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates:application of Fourier inversion methods[J]. Mathematical Finance.1997(7): 413-424.
[34]Nicolato E. Options pricing stochastic volatility models of Ornstein-Uhlenbeck type[J]. Mathematical Finance,2003(13):445-466.
[35]Deng G H. Pricing European option in a double exponential jump-diffusion model with two market structure risks and its comparisons[J]. Applies Mathematics-A Journal of Chinese University,2007(22):127-137.
[36]Deng G H., Huang L H. A poisson-Gaussian model to price European options on the ex-tremum of several risky assets within the HJK framework[J]. Journal Systems Sciences and Complex,2010(23):769-783.
[37]Deng G H. Pricing reset options with multiple reset features under stochastic interest rates and jump-diffusion model[M]. Proceedings of the 29th CCC,2010, July 29-31, Beijing,5559-5562.
[38]王莉君,张曙光Vasucek随机利率下的重置期权定价[J],高等应用数学学报A辑,2002,17(4)：471-478.
[39[Gerber H., Karoui N., Pocher J. Change of numeraire, change of probability measure,and option pricing[J]. Journal of Applied Probability,1995(32):443-459.
[40]Vasicek O. An equilibrium characterisation of the term structure[J]. Journal of Financial Eco-nomics,1997(5):177-188.