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Optimal reinsurance-investment problem with default risk for an insurer under the constant elasticity of variance model

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IMA JOURNAL OF MANAGEMENT MATHEMATICS 2024年第1期36卷 135-160页

作者： Yan, Yiqi Rong, Ximin Zhao, Hui Tianjin Univ Technol & Educ Sch Sci Tianjin 300222 Peoples R China Tianjin Univ Sch Math Tianjin 300072 Peoples R China Tianjin Univ Ctr Appl Math Tianjin 300072 Peoples R China

Accepted by: Giorgio ConsigliIn this paper we study the optimal reinsurance-investment problem for an insurer with constant absolute risk aversion (CARA) utility. Because the insurer needs to avoid large claims and make profits in the actual operation process, he/she is allowed to buy proportional reinsurance and invest in a risk-free bond, a stock and a default bond. Moreover, the price process of the stock follows the constant elasticity of variance (CEV) model and the correlation between the risk process and the stock's price is considered. For the objective of expected utility maximization, we establish the Hamilton-Jacobi-Bellman equation and derive the optimal reinsurance-investment strategy explicitly for the pre-default case and the post-default case via the dynamic programming. Finally, numerical examples and sensitivity analyses are provided to illustrate the effects of model parameters on the optimal strategy. We find that (i) the default risk has no impact on the optimal strategy of the stock and the parameters of the stock's price have no impact on the optimal money amount invested in the defaultable bond. (ii) The optimal reinsurance strategy depends on the volatility of the stock's price under the CEV model. (iii) The correlation between risk model and the stock's price does have effects on the insurer's optimal reinsurance-investment strategy. These findings may provide guidance to the management of insurers in practice.

关键词： investment-reinsurance constant elasticity of variance model default risk correlation risk dynamic programming

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On the constant elasticity of variance model for the utility maximization problem with multiple risky assets

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IMA JOURNAL OF MANAGEMENT MATHEMATICS 2017年第2期28卷 299-320页

作者： Zhao, Hui Rong, Ximin Tianjin Univ Sch Sci Tianjin 300072 Peoples R China

In this paper, we study the portfolio selection problem with a risk-free asset and multiple risky assets under the constant elasticity of variance (CEV) model. The aim is to maximize the different utilities of an investor's terminal wealth. The Hamilton-Jacobi-Bellman equation associated with the optimization problem is established via stochastic control theory and we obtain the explicit solutions for the exponential and power utility functions, respectively. We find that for a portfolio selection problem concerning risky assets with the CEV price processes, the n-dimensional case is quite different from the one-dimensional case. Furthermore, the properties of optimal strategies are analysed and a numerical analysis is presented to illustrate our results.

关键词： constant elasticity of variance model Hamilton-Jacobi-Bellman equation portfolio selection power and exponential utilities maximization Legendre transform portfolio selection

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Mean-variance portfolio selection under a constant elasticity of variance model

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OPERATIONS RESEARCH LETTERS 2014年第5期42卷 337-342页

作者： Shen, Yang Zhang, Xin Siu, Tak Kuen Univ New S Wales Australian Sch Business Sch Risk & Actuarial Studies Sydney NSW 2052 Australia Univ New S Wales Australian Sch Business CEPAR Sydney NSW 2052 Australia Nankai Univ Sch Math Sci Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China Southeast Univ Dept Math Nanjing 210096 Jiangsu Peoples R China City Univ London Cass Business Sch London EC1Y 8TZ England Macquarie Univ Fac Business & Econ Dept Appl Finance & Actuarial Studies Sydney NSW 2109 Australia

This paper discusses a mean-variance portfolio selection problem under a constant elasticity of variance model. A backward stochastic Riccati equation is first considered. Then we relate the solution of the associated stochastic control problem to that of the backward stochastic Riccati equation. Finally, explicit expressions of the optimal portfolio strategy, the value function and the efficient frontier of the mean variance problem are expressed in terms of the solution of the backward stochastic Riccati equation. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Mean-variance portfolio selection constant elasticity of variance model Backward stochastic Riccati equation Efficient frontier

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A Mean-variance Problem in the constant elasticity of variance(CEV) model

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Communications in Mathematical Research 2015年第3期31卷 242-252页

作者： Hou Ying-li Liu Guo-xin Jiang Chun-lan College of Mathematics and Information Science Hebei Normal University School of Science Hebei University of Technology Department of Mathematics and Physics Shijiazhuang Tiedao University

In this paper, we focus on a constant elasticity of variance （CEV） modeland want to find its optimal strategies for a mean-variance problem under two constrainedcontrols： reinsurance/new business and investment （no-shorting）. First, aLagrange multiplier is introduced to simplify the mean-variance problem and thecorresponding Hamilton-Jacobi-Bellman （HJB） equation is established. Via a powertransformation technique and variable change method, the optimal strategies withthe Lagrange multiplier are obtained. Final, based on the Lagrange duality theorem,the optimal strategies and optimal value for the original problem （i.e., the efficientstrategies and efficient frontier） are derived explicitly.

关键词： constant elasticity of variance model mean-variance optimal strategy

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Optimal portfolio strategy of wealth process: a Lévy process model-based method

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2024年第6期55卷 1089-1103页

作者： Yi, Haoran Shan, Yuanchuang Shu, Huisheng Zhang, Xuekang Donghua Univ Coll Sci Shanghai Peoples R China Anhui Polytech Univ Sch Math Phys & Finance Wuhu Peoples R China

This paper is concerned with the optimal portfolio problem for a company that can invest in two risky assets, where a novel Levy-process-driven model is constructed to describe the dynamics of the wealth process by using a constant elasticity of variance model and a jump-diffusion process. A delicately designed value function is proposed under the mean-variance criterion to reflect the optimal portfolio for the stochastic volatility model. By using the verification theorem, the desired optimal portfolio strategy is proposed by the solution to certain Hamilton-Jacobi-Bellman equations. Furthermore, the corresponding expressions are achieved by using the stochastic analysis theory. Finally, a numerical simulation example is provided to verify the effectiveness of the proposed optimal portfolio strategy.

关键词： Optimal portfolio mean-variance criterion constant elasticity of variance model jump-diffusion process

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Optimal investment strategies for the HARA utility under the constant elasticity of variance model

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INSURANCE MATHEMATICS & ECONOMICS 2012年第3期51卷 667-673页

作者： Jung, Eun Ju Kim, Jai Heui Pusan Natl Univ Dept Math Pusan 609735 South Korea

We give an explicit expression for the optimal investment strategy, under the constant elasticity of variance (CEV) model, which maximizes the expected HARA utility of the final value of the surplus at the maturity time. To do this, the corresponding HJB equation will be transformed into a linear partial differential equation by applying a Legendre transform. And we prove that the optimal investment strategy corresponding to the HARA utility function converges a.s. to the one corresponding to the exponential utility function. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Stochastic optimal control constant elasticity of variance model HARA utility function HJB equation Legendre transform

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Optimal consumption and portfolios with the hyperbolic absolute risk aversion preference under the CEV model

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2022年第24期51卷 8799-8821页

作者： Chang, Hao Li, Xueyan Chen, Xingjiang Tiangong Univ Sch Math Sci Tianjin 300387 Peoples R China Auckland Univ Technol Sch Engn Comp & Math Sci Auckland New Zealand

This paper studies the optimal investment-consumption decision under the constant elasticity of variance (CEV) model for an individual seeking to maximize the expected utility from cumulative consumption plus the expected utility from terminal wealth. Due to the fact that different individuals may have different risk preferences, we assume that the risk preference of an individual satisfies a hyperbolic absolute risk aversion (HARA) utility function. Generally speaking, power utility function, logarithmic utility function and exponential utility function widely used in investment theory are usually special cases of HARA utility function. By using the principle of dynamic programming and Legendre transform-dual technique, we obtain the explicit expression of the optimal investment-consumption decision. In addition, we derive the results under other utility functions as well and analyze some characteristics of the optimal portfolios and the optimal consumption decisions. A numerical simulation is presented to illustrate our results. Research results suggest that the optimal investment decisions between with consumption behavior and without it have considerable differences.

关键词： Investment and consumption constant elasticity of variance model hyperbolic absolute risk aversion utility Legendre transform-dual theory stochastic optimal control theory

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A Sixth-Order CEV Option Valuation Algorithm on Non-uniform Spatial Grids 22nd

A Sixth-Order CEV Option Valuation Algorithm on Non-uniform ...

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22nd International Conference on Computational Science and its Applications (ICCSA)

作者： Thakoor, Nawdha Univ Mauritius Dept Math Moka Mauritius

ISBN: (纸本)9783031105364;9783031105357

Due to its ability to fit skew implied volatility profiles of market option prices, the constant elasticity of variance (CEV) model overcomes a short-coming of the constant volatility assumption used by the Black-Scholes log-normal model. Although the CEV model has an analytical formula for European options, the presence of the non-central Chi-square distribution in the formula brings instability in the computation of European option prices for certain parameter ranges. For efficient valuation of the option prices for all realistic option parameters, a sixth-order finite difference method is proposed in this work. The computational procedure employs a seven-point discretisation of the space derivatives on non-uniform grids together with a mesh-refinement technique around the option's strike price. This new technique is very fast and is shown to achieve sixth-order numerical convergence ratios. The stability of the finite difference method is demonstrated through its capability to generate accurate prices for strongly negative elasticity factors. The numerical method is thus a superior technique to existing methods for numerical computations of CEV option prices.

关键词： Option valuation constant elasticity of variance model Finite difference algorithm Non-uniform grids

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Time-consistent strategies between two competitive DC pension plans with the return of premiums clauses and salary risk

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2024年第22期53卷 7807-7828页

作者： Nie, Gaoqin Chen, Xingjiang Chang, Hao Capital Univ Econ & Business Sch Stat Beijing Peoples R China Auckland Univ Technol Sch Engn Comp & Math Sci Auckland New Zealand Tiangong Univ Sch Math Sci Tianjin Peoples R China Capital Univ Econ & Business Sch Stat Beijing 100070 Peoples R China

In practice, there is always competition among different pension managers. The competitive pension managers are concerned about their relative performance to make better decisions. In addition, to protect the rights of plan members who die before retirement, most of the defined contribution (DC) pension plans contain return of premiums clauses. In this article, we introduce the return of premiums clauses into two competitive DC pension plans with salary risk. Each pension manager cares about not only his own terminal wealth but also his relative wealth versus his competitor. Both the pension managers are allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process follows the constant elasticity of variance (CEV) model. We investigate the time-consistent investment strategies of the two pension managers under the mean-variance (MV) criterion. The aim of each pension manager is to maximize the mean and minimize the variance of his relative terminal wealth. The explicit expressions of the time-consistent strategies and the efficient frontiers are obtained via applying the extended stochastic optimal control theory. Some special cases are also discussed in detail. Finally, a numerical simulation demonstrates the results obtained.

关键词： Time-consistent strategy relative performance return of premiums clauses mean-variance criterion constant elasticity of variance model

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An efficient numerical method for pricing American put options under the CEV model

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 389卷 113311-113311页

作者： Lee, Jung-Kyung Anyang Univ Coll Liberal Arts 22 Samdeok Ro 37Beon Gil Anyang Si 14028 Gyeonggi Do South Korea

The constant elasticity of variance (CEV) model is a practical approach to option pricing by fitting to the implied volatility smile. However, pricing American options is computationally intensive because no analytical formulas are available. In this paper, we present numerical methods to find the optimal exercise boundary with respect to an American put option under the CEV model. This problem corresponds to the free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. After determining the optimal exercise boundary, we calculate American put option values under the CEV model using the finite-difference method. Finally, we use the proposed numerical method to obtain several results that are then compared with the results of other methods. (C) 2020 Elsevier B.V. All rights reserved.

关键词： American option constant elasticity of variance model Optimal exercise boundary Transformed function Cubic spline interpolation

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