保险与金融中CEV模型的最优化问题
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摘要
常方差弹性系数(CEV)模型是几何布朗运动(GBM)模型的一个推广.它最早常用于计算期权等资产的定价,敏感性分析和隐含波动率等问题,近年来, CEV模型开始用于最优化投资问题.本文主要讨论了有限时间水平下CEV模型在保险和金融中两大方面的应用,一是一般投资人的最优消费和投资问题;一是保险人的最优再保险和投资问题.
在投资行为中,财务顾问通常推荐年轻的投资人相对于年老的投资人来说应该投入较多的资产到风险资产中,这是因为年轻的投资人拥有更长的时间来进行投资行为.这说明投资者所面临不的剩余时间不同,他们的投资策略就不应相同.为了切合实际,我们考虑的是有限时间水平.在有限时间水平下,从不同的时刻出发,剩余时间也不同,这样所得到的策略不再像无限时间水平下那样与时间无关.
投资与消费是社会总需求的重要组成部分,投资与消费关系又是经济理论和实证研究中最重要、最复杂的关系之一.对投资人来说,他把资产进行投资的目的就是增强自己的消费能力.本文在第三章研究了最优化消费和投资问题,投资人把总资产的一部分投资到无风险资产(存入银行或购买债券),剩下的部分投入到风险资产(股票或基金),同时投资者还进行消费,且以最大化直到固定终端时刻的期望指数消费效用(指数效用函数是唯一一个满足“零效益”原则的效用函数)为目的,寻求最优化消费和投资策略.本章首先利用动态规划的方法建立了本问题的HJB方程,然后通过幂变换和变量替换等手段,求得了最优消费和投资策略的明确表达式.
保险人的最优再保险和投资问题是保险问题和金融活动的一个很好的结合,具有很强的实用性和研究价值.再保险是保险公司规避风险的一个有力手段,投资则是保险公司提高收益的重要保障.本文的第四章考虑了保险公司的超额损失再保险和投资策略.保险公司收取保费,就要承担索赔.为了避免大额索赔出现的危险,于是进行再保险;同时为了确保资金的保值与增值,需要把手中的资金进行一种无风险资产和一种风险资产的投资分配.针对最大化固定终端时刻指数效用的目标,利用动态规划的方法求出了相应的HJB方程,然后通过求解HJB方程,得到了最优的超额损失再保险和投资策略.
第五章和第六章研究的是比例再保险和投资行为的结合.其中第五章关心的是均值-方差问题,也就是在达到预期效益的同时要确保风险最小化.这是一个以期望收益和风险（即均值和方差）为双目标的最优化问题.本章采用的路线是首先通过拉格朗日乘数的引入把两个目标融合在一起,化为一个单目标问题,并建立其HJB方程,然后求解HJB方程,得到了带有拉格朗日乘数的最优策略,最后借由Lagrange对偶定理,求出了原问题的有效边界和有效策略.
在前面几章的研究中,为了突出问题的重点,都假设了投资者只投资到一种无风险资产和一种风险资产.第六章为了更加贴近实际,不再只考虑单一风险资产,而是投资于多种风险资产,同时结合比例再保险,目标依然是最大化固定终端时刻的期望指数效用.本章在建立了相应的HJB方程的基础上,借由幂变换和变量替换的技巧,得到了其最优比例再保险和投资策略的明确表达式.
The CEV model is an extension model of geometric Brownian motion (GBM). It was usu-ally applied to calculating the theoretical price, sensitivities and implied volatility of options. Inrecent years, the CEV model began to be applied in optimal investment research. In this paper,we discuss two aspects application of CEV model with finite horizon. One is the investor’s opti-mal consumption and investment problem, the other is the insurer’s reinsurance and investmentproblem.
In the action of investment, the financial advisers always recommend that younger in-vestors allocate a greater share of wealth to risky asset than older investors. The cause is thatthe younger investors have longer period to hold on to the investment. This indicates that thedifferent remaining horizon that the investors faced with the investors’ strategies will different.Therefore, we consider with the case that the horizon is finite. And in the finite horizon case,unlike the infinite case, starting from each initial time the remaining horizon is different andthus the strategies may change depending on how close it is to the terminal time.
Investment and consumption are important component parts of the total demand of wholesociety. The relationship between investment and consumption is very important and complex.For an investor, the aim of investment is to improve his ability of consumption. In section3,we study the optimal consumption and investment policy of a constant absolute risk aversion(CARA) investor. The investor’s interesting is to maximizing the fix terminal expectation ofexponential utility function. The investor can trade in one risk-free asset and a risky asset.First, the Hamilton-Jacobi-Bellman (HJB) equation for the value function of the optimizationproblem is established by the dynamic programming approach. The optimal consumption andinvestment policy is derived via power transformation technique and variable change method.
Insurer’s optimal reinsurance and investment problem is a strongly practical and theoreti-cal combination of insurance and finance. Reinsurance is an important technique for a insurerto refrain from his risk. And investment is the powerful guarantee for insurer to improve hisbenefit. In section4, we discuss the the optimal excess-of-loss reinsurance and investment strat-egy.The insurance company charges premium as while as undertakes claim. In order to avoidthe huge amount of claim account, the insurance company pays reinsurance premium to an an-other insurance company. At the same time, the insurer invest his wealth in one risk-free assetand a risky asset to assure his benefit. For the purpose of maximizing the fix terminal expec-tation of exponential utility function, the corresponding HJB equation is obtained by dynamic programming principle. Via solving the HJB equation, the explicit expression of the optimalexcess-of-loss reinsurance and investment strategy is formulated.
In the rest of this paper, we pay attention to proportional reinsurance and investment prob-lem. In section5, we focus on a mean-variance problem. This is a problem with two objectswhich is to achieve expect profit as while as minimizing the risk. First, a Lagrange multiplieris introduced to simplify the mean-variance problem and the corresponding Hamilton-Jacobi-Bellman (HJB) equation is established. Via power transformation technique and variable changemethod, the optimal strategies with the Lagrange multiplier are obtained. Final, based on theLagrange duality theorem, the optimal strategies and optimal value for the original problem (i.e.the efficient strategies and efficient frontier) are derived explicitly.
In the above sections,we suppose that the investor only trade in a risk-free asset and onerisky asset. In section6, the investor can invest his wealth in a risk-free asset and multiple riskyassets. Meanwhile, the proportional reinsurance is considered. In order to maximizing the fixterminal expectation of exponential utility function, the corresponding HJB equation is formu-lated by dynamic programming principle.By the means of power transformation technique andvariable change method, the explicit expression of the optimal proportional reinsurance and in-vestment strategy is obtained.
在投资行为中,财务顾问通常推荐年轻的投资人相对于年老的投资人来说应该投入较多的资产到风险资产中,这是因为年轻的投资人拥有更长的时间来进行投资行为.这说明投资者所面临不的剩余时间不同,他们的投资策略就不应相同.为了切合实际,我们考虑的是有限时间水平.在有限时间水平下,从不同的时刻出发,剩余时间也不同,这样所得到的策略不再像无限时间水平下那样与时间无关.
投资与消费是社会总需求的重要组成部分,投资与消费关系又是经济理论和实证研究中最重要、最复杂的关系之一.对投资人来说,他把资产进行投资的目的就是增强自己的消费能力.本文在第三章研究了最优化消费和投资问题,投资人把总资产的一部分投资到无风险资产(存入银行或购买债券),剩下的部分投入到风险资产(股票或基金),同时投资者还进行消费,且以最大化直到固定终端时刻的期望指数消费效用(指数效用函数是唯一一个满足“零效益”原则的效用函数)为目的,寻求最优化消费和投资策略.本章首先利用动态规划的方法建立了本问题的HJB方程,然后通过幂变换和变量替换等手段,求得了最优消费和投资策略的明确表达式.
保险人的最优再保险和投资问题是保险问题和金融活动的一个很好的结合,具有很强的实用性和研究价值.再保险是保险公司规避风险的一个有力手段,投资则是保险公司提高收益的重要保障.本文的第四章考虑了保险公司的超额损失再保险和投资策略.保险公司收取保费,就要承担索赔.为了避免大额索赔出现的危险,于是进行再保险;同时为了确保资金的保值与增值,需要把手中的资金进行一种无风险资产和一种风险资产的投资分配.针对最大化固定终端时刻指数效用的目标,利用动态规划的方法求出了相应的HJB方程,然后通过求解HJB方程,得到了最优的超额损失再保险和投资策略.
第五章和第六章研究的是比例再保险和投资行为的结合.其中第五章关心的是均值-方差问题,也就是在达到预期效益的同时要确保风险最小化.这是一个以期望收益和风险（即均值和方差）为双目标的最优化问题.本章采用的路线是首先通过拉格朗日乘数的引入把两个目标融合在一起,化为一个单目标问题,并建立其HJB方程,然后求解HJB方程,得到了带有拉格朗日乘数的最优策略,最后借由Lagrange对偶定理,求出了原问题的有效边界和有效策略.
在前面几章的研究中,为了突出问题的重点,都假设了投资者只投资到一种无风险资产和一种风险资产.第六章为了更加贴近实际,不再只考虑单一风险资产,而是投资于多种风险资产,同时结合比例再保险,目标依然是最大化固定终端时刻的期望指数效用.本章在建立了相应的HJB方程的基础上,借由幂变换和变量替换的技巧,得到了其最优比例再保险和投资策略的明确表达式.
The CEV model is an extension model of geometric Brownian motion (GBM). It was usu-ally applied to calculating the theoretical price, sensitivities and implied volatility of options. Inrecent years, the CEV model began to be applied in optimal investment research. In this paper,we discuss two aspects application of CEV model with finite horizon. One is the investor’s opti-mal consumption and investment problem, the other is the insurer’s reinsurance and investmentproblem.
In the action of investment, the financial advisers always recommend that younger in-vestors allocate a greater share of wealth to risky asset than older investors. The cause is thatthe younger investors have longer period to hold on to the investment. This indicates that thedifferent remaining horizon that the investors faced with the investors’ strategies will different.Therefore, we consider with the case that the horizon is finite. And in the finite horizon case,unlike the infinite case, starting from each initial time the remaining horizon is different andthus the strategies may change depending on how close it is to the terminal time.
Investment and consumption are important component parts of the total demand of wholesociety. The relationship between investment and consumption is very important and complex.For an investor, the aim of investment is to improve his ability of consumption. In section3,we study the optimal consumption and investment policy of a constant absolute risk aversion(CARA) investor. The investor’s interesting is to maximizing the fix terminal expectation ofexponential utility function. The investor can trade in one risk-free asset and a risky asset.First, the Hamilton-Jacobi-Bellman (HJB) equation for the value function of the optimizationproblem is established by the dynamic programming approach. The optimal consumption andinvestment policy is derived via power transformation technique and variable change method.
Insurer’s optimal reinsurance and investment problem is a strongly practical and theoreti-cal combination of insurance and finance. Reinsurance is an important technique for a insurerto refrain from his risk. And investment is the powerful guarantee for insurer to improve hisbenefit. In section4, we discuss the the optimal excess-of-loss reinsurance and investment strat-egy.The insurance company charges premium as while as undertakes claim. In order to avoidthe huge amount of claim account, the insurance company pays reinsurance premium to an an-other insurance company. At the same time, the insurer invest his wealth in one risk-free assetand a risky asset to assure his benefit. For the purpose of maximizing the fix terminal expec-tation of exponential utility function, the corresponding HJB equation is obtained by dynamic programming principle. Via solving the HJB equation, the explicit expression of the optimalexcess-of-loss reinsurance and investment strategy is formulated.
In the rest of this paper, we pay attention to proportional reinsurance and investment prob-lem. In section5, we focus on a mean-variance problem. This is a problem with two objectswhich is to achieve expect profit as while as minimizing the risk. First, a Lagrange multiplieris introduced to simplify the mean-variance problem and the corresponding Hamilton-Jacobi-Bellman (HJB) equation is established. Via power transformation technique and variable changemethod, the optimal strategies with the Lagrange multiplier are obtained. Final, based on theLagrange duality theorem, the optimal strategies and optimal value for the original problem (i.e.the efficient strategies and efficient frontier) are derived explicitly.
In the above sections,we suppose that the investor only trade in a risk-free asset and onerisky asset. In section6, the investor can invest his wealth in a risk-free asset and multiple riskyassets. Meanwhile, the proportional reinsurance is considered. In order to maximizing the fixterminal expectation of exponential utility function, the corresponding HJB equation is formu-lated by dynamic programming principle.By the means of power transformation technique andvariable change method, the explicit expression of the optimal proportional reinsurance and in-vestment strategy is obtained.
引文
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