Lévy过程在金融保险中的应用
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摘要
相对于Black-Scholes模型而言,由带跳的Levy过程驱动的信用风险模型更符合市场上的金融数据经验验证.带跳的Levy过程具有非对称的尖峰厚尾性质和不连续性,克服了正态分布的对称性,而且可以很好地描述突发事件带来的影响.正因为如此,带跳的Levy过程在金融保险中受到越来越广泛的应用.本文主要利用Levy过程及相关理论研究以下两方面问题.
一是研究信用衍生品公平定价问题.在信用衍生品市场中,信用衍生品是一种有效转移,分散以及对冲风险的重要工具.本文在Cox型约化信用风险模型框架下,利用Levy过程刻画违约强度的跳跃结构,假设违约强度过程分别为从属过程和由从属过程驱动的Levy型Vasicek模型,且违约时间是条件独立的,提出新的组合信用风险模型.在此模型下,给出了多个相关风险资产的联合生存概率,并给出一些信用衍生品担保费的公平定价.二是研究最优再保险问题.许多学者在经典风险模型下研究单一险种的最优再保险问题.由于保险公司的风险业务经营规模不断扩大,保险公司经营的险种多元化.本文引入稀疏相关风险模型,利用Levy过程及其相关理论研究了稀疏相关风险模型的最优再保险问题,给出了最优再保险策略以及相关最佳自留索赔.
本文的主要内容安排如下：
第一部分主要研究信用衍生品公平定价问题.首先在Cox型约化信用风险模型框架下,通过假设强度过程分别为从属过程和由从属过程驱动的Levy型Vasicek模型,且违约时间之间条件独立,建立新的组合信用风险模型；接着,在此模型下,给出了多个相关公司的联合生存概率的闭型表达式和第n次违约概率的闭型表达式.然后,推导了带对手风险的信用违约互换,一篮子信用违约互换(Basket Credit Default Swaps,简称,一篮子CDS)和抵押贷款信用违约互换(Loan Credit Default Swaps,简称,Loan CDS)的公平担保费的中性定价公式,
第二部分主要利用Lévy过程及其相关理论研究稀疏相关风险模型的最优再保险问题.对于最优比例再保险问题,分别在调节系数最大化和均值方差原理两种优化准则下,给出了最优比例再保险策略及相应的最佳自留索赔比例.类似地,对于最优超额损失再保险问题,在期望指数效用最大化和调节系数最大化两种优化准则下,得到了最优超额损失再保险策略及相应的最佳自留索赔额.
Compared with the Black-Scholes model, the risk model driven by Lévy processeswith jumps seems to provide a better ft to the market data of fnance and insurance.In the risk theory, Lévy processes with jumps can capture two features. One featureis leptokurtosis and unsymmetrical, and the other is discontinuities in the trajectory.Thus, Lévy processes can not only overcome the default of continuities of normaldistribution, but also describe the efect of the jumps well. Therefore, Lévy processeswith jumps has become more and more important in modeling fnance and insurance.In this dissertation, we mainly study the following two problems by Lévy processestheory.
One is to study the problem of the neutral pricing in the credit derivatives. Creditderivatives are increasingly important in fnancial markets. They provide methodsto hedge credit risks that arise during everyday trading, as well as more chances topromote investment return. In this paper we propose a new model with dependentrisks in the Cox framework, while the jumps of the default intensity are describedby Lévy processes. Specially, we consider that the default intensity processes aresubordinator processes and certain Vasicek processes driven by subordinator processes,respectively. Correspondingly, we give the closed forms of joint survival probability andthe fair spread of some credit derivatives. The other is to study the optimal reinsuranceproblem under the risk model with the thinning dependence structure. Many authorshave studied the optimal reinsurance problem under the classic risk model. As thescale of the insurance expands, the business varieties of insurance companies diversify.We introduce the risk model with the thinning dependence structure and study theoptimal proportional reinsurance strategy and the excess of loss reinsurance strategyby Lévy processes theory.
The dissertation is organized as follows:
In the frst part, we mainly study the problem of the neutral pricing in the creditderivatives. We consider a correlated reduced form credit risk model. While the de-fault intensity processes are assumed to be a subordinator processes and certain Vasicek processes which is the solution of some stochastic equation driven by subordinator pro-cesses, respectively. We obtain the explicit expression for the joint Laplace transformof the default intensity processes and the cumulative default intensity processes, andthen we get the closed form for the joint distribution of default times. We also give theclose form of the nth default probability for a portfolio, and get the explicit expressionfor the fair spread of credit default swap (CDS) with the counterparty risk under theproposed credit risk model. We also present the numerical solutions for the fair spread.
In second part, we investigate the reinsurance strategy in the risk model with thin-ning dependence structure. For the optimization proportional reinsurance problem, weinvestigate the optimal reinsurance strategy that maximizes the adjustment coefcientand minimizes the variance of the surplus under the given expected proft, respectively.We derive the optimal solutions and the numerical illustrations to show the impact ofthe dependence among the classes of business on the optimal reinsurance arrangements.Similarly, for the optimization excess of loss reinsurance problem, we investigate theoptimal reinsurance strategy that maximizes the expected exponential utility and theadjustment coefcient, respectively. Correspondingly, we derive the optimal solutionsand give a numerical example to analyze the impacts of the thinning factor on eachstrategy.
一是研究信用衍生品公平定价问题.在信用衍生品市场中,信用衍生品是一种有效转移,分散以及对冲风险的重要工具.本文在Cox型约化信用风险模型框架下,利用Levy过程刻画违约强度的跳跃结构,假设违约强度过程分别为从属过程和由从属过程驱动的Levy型Vasicek模型,且违约时间是条件独立的,提出新的组合信用风险模型.在此模型下,给出了多个相关风险资产的联合生存概率,并给出一些信用衍生品担保费的公平定价.二是研究最优再保险问题.许多学者在经典风险模型下研究单一险种的最优再保险问题.由于保险公司的风险业务经营规模不断扩大,保险公司经营的险种多元化.本文引入稀疏相关风险模型,利用Levy过程及其相关理论研究了稀疏相关风险模型的最优再保险问题,给出了最优再保险策略以及相关最佳自留索赔.
本文的主要内容安排如下：
第一部分主要研究信用衍生品公平定价问题.首先在Cox型约化信用风险模型框架下,通过假设强度过程分别为从属过程和由从属过程驱动的Levy型Vasicek模型,且违约时间之间条件独立,建立新的组合信用风险模型；接着,在此模型下,给出了多个相关公司的联合生存概率的闭型表达式和第n次违约概率的闭型表达式.然后,推导了带对手风险的信用违约互换,一篮子信用违约互换(Basket Credit Default Swaps,简称,一篮子CDS)和抵押贷款信用违约互换(Loan Credit Default Swaps,简称,Loan CDS)的公平担保费的中性定价公式,
第二部分主要利用Lévy过程及其相关理论研究稀疏相关风险模型的最优再保险问题.对于最优比例再保险问题,分别在调节系数最大化和均值方差原理两种优化准则下,给出了最优比例再保险策略及相应的最佳自留索赔比例.类似地,对于最优超额损失再保险问题,在期望指数效用最大化和调节系数最大化两种优化准则下,得到了最优超额损失再保险策略及相应的最佳自留索赔额.
Compared with the Black-Scholes model, the risk model driven by Lévy processeswith jumps seems to provide a better ft to the market data of fnance and insurance.In the risk theory, Lévy processes with jumps can capture two features. One featureis leptokurtosis and unsymmetrical, and the other is discontinuities in the trajectory.Thus, Lévy processes can not only overcome the default of continuities of normaldistribution, but also describe the efect of the jumps well. Therefore, Lévy processeswith jumps has become more and more important in modeling fnance and insurance.In this dissertation, we mainly study the following two problems by Lévy processestheory.
One is to study the problem of the neutral pricing in the credit derivatives. Creditderivatives are increasingly important in fnancial markets. They provide methodsto hedge credit risks that arise during everyday trading, as well as more chances topromote investment return. In this paper we propose a new model with dependentrisks in the Cox framework, while the jumps of the default intensity are describedby Lévy processes. Specially, we consider that the default intensity processes aresubordinator processes and certain Vasicek processes driven by subordinator processes,respectively. Correspondingly, we give the closed forms of joint survival probability andthe fair spread of some credit derivatives. The other is to study the optimal reinsuranceproblem under the risk model with the thinning dependence structure. Many authorshave studied the optimal reinsurance problem under the classic risk model. As thescale of the insurance expands, the business varieties of insurance companies diversify.We introduce the risk model with the thinning dependence structure and study theoptimal proportional reinsurance strategy and the excess of loss reinsurance strategyby Lévy processes theory.
The dissertation is organized as follows:
In the frst part, we mainly study the problem of the neutral pricing in the creditderivatives. We consider a correlated reduced form credit risk model. While the de-fault intensity processes are assumed to be a subordinator processes and certain Vasicek processes which is the solution of some stochastic equation driven by subordinator pro-cesses, respectively. We obtain the explicit expression for the joint Laplace transformof the default intensity processes and the cumulative default intensity processes, andthen we get the closed form for the joint distribution of default times. We also give theclose form of the nth default probability for a portfolio, and get the explicit expressionfor the fair spread of credit default swap (CDS) with the counterparty risk under theproposed credit risk model. We also present the numerical solutions for the fair spread.
In second part, we investigate the reinsurance strategy in the risk model with thin-ning dependence structure. For the optimization proportional reinsurance problem, weinvestigate the optimal reinsurance strategy that maximizes the adjustment coefcientand minimizes the variance of the surplus under the given expected proft, respectively.We derive the optimal solutions and the numerical illustrations to show the impact ofthe dependence among the classes of business on the optimal reinsurance arrangements.Similarly, for the optimization excess of loss reinsurance problem, we investigate theoptimal reinsurance strategy that maximizes the expected exponential utility and theadjustment coefcient, respectively. Correspondingly, we derive the optimal solutionsand give a numerical example to analyze the impacts of the thinning factor on eachstrategy.
引文
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[2] Giesecke, K.. A simple exponential model for dependent defaults. Journal of FixedIncome,2003,13,74-83.
[3] Jarrow, Robert A., Yu, F.. Counterparty risk and the pricing of defaultable secu-rities. Journal of Finance,2001,56(5),555-576.
[4] Frey, R., Backhaus, J.. Portfolio credit risk models with interacting default inten-sities: a Markovian approach. Working paper,2004.
[5] Herbertsson, A., Rootzen, H.. Pricing kth-to-default swaps under default conta-gion: the matrix-analytic approach. Working paper,2006.
[6] Leung, S. Y., Kwok, Y. K.. Credit default swap valuation with counterparty risk.The Kyoto Economic Review,2005,74,25-45.
[7] Yu, F.. Correlated defaults in intensity-based models. Mathematical Finance,2007,17,155-173.
[8] Zheng H., Jiang L.. Basket CDS pricing with interacting intensities. Working paperof Imperial College,2008.
[9] Li, D.. On default correlation: a copula function approach. Journal of Fixed In-come,2000,9,4354.
[10] Hull, J., White, A. Valuation of a CDO and an nth to Default CDS WithoutMonte Carlo Simulation. Journal of Derivatives,2004,12,2,8-23.
[11] Sch¨onbucher, P. J., Schubert, D.. Copula-dependent Default Risk in IntensityModels. Working paper, Department of Statistics, Bonn Unversity,2001.
[12] Luciano, E., Schoutens, W.. A multivariate jump-driven fnancial asset model.Quantitative Finance,2006,6(5),385-402.
[13] Jang, J.. Copula Dependent Collateral Default Intensity And Its Application toCDS rate. Working Paper,2010.
[14] Nelsen R.B.. An introduction to copulas. Lectures Notes in Statistics, New York:Springer-Verlag,1999.
[15] Cont, R., Tankov, P.. Financial Modelling with Jump Processes. Financial Math-ematics Series. Chapman&Hall/CRC Press, London,2004.
[16] Gregory, J.. Credit Derivatives: The Defnitive Guide,2003.
[17] Glasserman, P.. Tail approximations for portfolio credit risk. J. Derivatives,2004,24-42.
[18] Zheng, H.. Efcient hybrid methods for portfolio credit derivatives. QuantitativeFinance,2006,6,349-357.
[19] Shek, H., Uematsu, S., Wei, Z. Valuation of Loan CDS and CDX. Working Paper,2007. Available at SSRN: http://ssrn.com/abstract=1008201.
[20] Liang, J., Ma, J. M., Wang, T., et al. Valuation of Portfolio Credit Derivativeswith Default Intensities Using the Vasicek Model. Asia-Pacifc Financial Markets,2011,18,33-54.
[21] Jarrow, R. A., Stuart M. Turnbull. Pricing derivatives on fnancial securities sub-ject to credit risk. Journal of Finance,1995,50(1),53-86.
[22] Hull, J., White, A.. Valuing Credit Default Swaps II: Modeling Default Correla-tion, The Journal of derivatives,2001,8(3),1222.
[23] Bielecki, T. R., Rutkowski, M,. Credit Risk: Modeling, Valuation and Hedging.Springer-Verlag, Berlin,2002.
[24] O’Kane, D., Turnbull,S.. Valuation of Credit Default Swaps. Quantitative CreditResearch Quarterly, Lehman Brothers,2003.
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