Abstract: Recently, GRASSELLI (2017) proposed a new SV model, called the 4/2 (that is, 1/2+3/2) SV model, which assumes that the instaneous variance is a linear combination of the 1/2 and the 3/2 terms. So, it combines the properties of both Heston’s SV and 3/2 models. In addition, the 4/2 model has some new features that are not contemplated in Heston’s SV model and 3/2 model. In view of the advantages of the 4/2 SV model, in this paper we try to consider the optimal investment and risk control strategies for two competing insurers under relative performance criteria. Specially, we set up a combined financial and insurance market consisting of one risk-free asset, one risky asset, and two dependent risk process representing the liabilities per unit (or per policy), for k=1,2. We apply the standard Cramér-Lundberg diffusion approximation model for the risk process R_k, and assume that each insurer can directly control her liability exposure, or alternatively, each insurer can decide the total amount of liabilities measured by units (or the number of policies) L_k times R_k. One can easily see that such an assumption is equivalent to allowing the insurer to purchase proportional reinsurance to manage her risk exposure from underwriting. Under the framework of Nash equilibrium theory, the relative performance concerns of insurers is used to describe their game behaviors,a non-zero-sum game model is constructed which maximizes the expected exponential utility of his terminal surplus relative to that of his competitor. Applying the techniques of stochastic dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equations for both insurers are obtained, and the Nash equilibrium strategies of both insurers are established by solving HJB equations. Moreover, two special cases are discussed. Finally, some numerical examples are conducted to illustrate the effects of several model parameters on the Nash equilibrium strategies and draw some economic interpretations from these results.
Numerical examples demonstrate that the relative performance concerns of the insurer increase the retained insurance policies and the amount invested in the risky asset, which implies that the competition would lead the insurers to be much more risk-seeking.
Several possible extensions of our work deserve further investigation. The first extension is to apply other utility functions in establishing objective functions in the game framework. Under such a formulation, explicit expressions for Nash equilibrium strategies might be difficult to derive. However, we could apply suitable numerical approximation methods when solving the system of HJB equations. Furthermore, this paper only considers games with perfect revelation, or perfect observation. It is well known that such assumption is too stringent, and the partial observation assumption is more realistic. In such a case, it would be interesting to investigate the game with asymmetric information. We leave these suggestions for future research.

Key words: investment and risk control, non-zero-sum game, Nash equilibrium, Hamilton-Jacobi-Bellman equation

摘要： 近年,GRASSELLI(2017)提出的4/2随机波动率模型构建了一种新型波动动态框架,其扩散项系Heston模型与3/2模型扩散项的线性组合。该混合结构不仅具有Heston模型和3/2模型的基本特征,还有一些它们所不具备的新特性,因此能够更好地描述金融市场中风险资产价格的动态变化。本文基于4/2随机波动率模型的优势,研究了两个处于竞争关系的保险公司之间的最优投资和风险控制问题。具体来说,在保险风险建模方面,采用扩散近似风险模型刻画保单赔付动态过程。金融市场环境设定为混合波动率框架,包含无风险资产与符合4/2随机波动特征的风险资产。保险公司通过双重策略实现风险管理:一方面动态调整承保规模控制保险风险暴露,另一方面优化金融资产配置结构,最终达成公司价值稳健增长的战略目标。同时考虑到市场竞争,基于相对财富视角刻画保险公司间竞争行为,构建双主体非零和投资—风险控制动态博弈模型,以实现终端时刻相对财富期望效用最大化。运用动态规划方法推导得到 Hamilton-Jacobi-Bellman(HJB)方程,并通过求解获取了博弈均衡策略,进一步讨论了本文模型的两种特殊情形。最后,通过数值算例给出了参数的敏感性分析,并进行了经济意义解释。

关键词: 投资与风险控制, 非零和博弈, 纳什均衡, Hamilton-Jacobi-Bellman方程