保险与金融中CEV模型的最优化问题

发布时间：2018-04-27 16:10

本文选题：CEV模型 + 最优化策略　；参考：《河北师范大学》2014年博士论文

【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.
[Abstract]:The constant difference elasticity coefficient (CEV) model is a generalization of the geometric Brown motion (GBM) model. It is first used to calculate the pricing, sensitivity analysis and implicit volatility of options and other assets. In recent years, the CEV model has been used to optimize the investment problem. This paper mainly discusses the CEV model under the finite time level in the insurance and finance. The two major applications are the optimal consumption and investment of the general investors. The first is the insurer's optimal reinsurance and investment.
In investment behavior, financial advisers usually recommend that young investors should invest more assets in risky assets than older investors because young investors have longer time to invest. This indicates that investors are faced with different remaining time and their investment strategies are not appropriate. In the same way, in order to be practical, we consider a finite time level. At a limited time, the remaining time is different from different moments, so the strategy is no longer independent of time as it is at an infinite time level.
Investment and consumption are an important part of the total social demand. The relationship between investment and consumption is one of the most important and most complex relationships in economic theory and empirical research. For investors, the purpose of investing the assets is to enhance their consumption ability. In the third chapter, the problem of optimal consumption and investment and investment are studied in this paper. A person investing part of a total asset into a riskless asset (deposited in a bank or buying a bond), and the remaining part is invested in a risky asset (stock or fund), and the investor is also consumed and maximized the expected index consumption utility until the fixed terminal moment (the index utility function is the only one that meets the "zero benefit" principle. In this chapter, the HJB equation of this problem is established by dynamic programming, and then the explicit expression of the optimal consumption and investment strategy is obtained by means of power transformation and variable substitution.
The problem of the best reinsurance and investment of the insurer is a good combination of insurance and financial activities. It has strong practicability and research value. Reinsurance is a powerful means for insurance companies to avoid risk. Investment is an important guarantee for the insurance company to improve its income. The fourth chapter of this article has considered the excess loss of insurance companies. In order to avoid the risk of large amount of claims, the insurance company will reinsurance, and in order to ensure the value and value of the capital, it is necessary to allocate the funds in the hands of a riskless asset and a kind of venture capital. The goal of exponential utility is to obtain the corresponding HJB equation by using the method of dynamic programming, and then the optimal excess loss reinsurance and investment strategy are obtained by solving the HJB equation.
The fifth chapter and the sixth chapter deals with the combination of proportional reinsurance and investment behavior. The fifth chapter is concerned with the mean variance problem, which is to minimize the risk while achieving the expected benefit. This is an optimization problem with expected returns and risks (mean and variance). The route adopted in this chapter is the first. First, the two targets are fused together by the introduction of the Lagrange multiplier and transformed into a single objective problem, and its HJB equation is established. Then the HJB equation is solved, and the optimal strategy with the Lagrange multiplier is obtained. Finally, the effective boundary and effective strategy of the original problem are obtained by the Lagrange duality theorem.
In the study of the previous chapters, in order to highlight the focus of the problem, we assume that investors only invest in a riskless asset and a risk asset. The sixth chapter, in order to be closer to the reality, not only consider a single risk asset, but also invest in a variety of risky assets, and combine the proportional reinsurance, the goal is to maximize the fixed terminal. In this chapter, based on the establishment of the corresponding HJB equation, this chapter obtains the explicit expression of the optimal proportional reinsurance and investment strategy by using the technique of power transformation and variable substitution.

【学位授予单位】：河北师范大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F830;F840;O211.6;F224

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