This paper is devoted to derive the time-consistent reinsurance–investment strategy for an insurer and a reinsurer under mean–variance criterion. We aim to maximize the weighted sum of the insurer’s ...and the reinsurer’s objectives with different risk averse coefficients. The claim process is assumed to follow a Brownian motion with drift and the insurer can purchase proportional reinsurance from the reinsurer. Moreover, both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. In order to reduce risks, the insurer and the reinsurer can invest in different risky assets whose price processes are described by the constant elasticity of variance (CEV) models. Consideration of the CEV model and the profits of both the insurer and the reinsurer, the proof of the verification theorem becomes difficult and the solution becomes complex. We first formulate a general problem and prove the verification theorem. By solving an extended Hamilton–Jacobi–Bellman (HJB) equation, we obtain the time-consistent reinsurance–investment strategy and the corresponding value function explicitly. Finally, sensitivity analysis and numerical simulation are presented to show the effects of parameters on the time-consistent strategy and illustrate the economic meaning.
We consider mean-reverting CIR/CEV processes with delay and jumps used as models on the financial markets. These processes are solutions of stochastic differential equations with jumps, which have no ...explicit solutions. We prove the non-negativity property of the solution of the above models and propose an explicit positivity preserving numerical scheme, using the semi-discrete method, that converges in the strong sense to the exact solution. We also make some minimal numerical experiments to illustrate the proposed method.
In this paper, we consider an optimal proportional reinsurance and investment problem for an insurer whose objective is to maximise the expected exponential utility of terminal wealth. Suppose that ...the insurer's surplus process follows a Brownian motion with drift. The insurer is allowed to purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. The correlation between risk model and the risky asset's price is considered. By applying dynamic programming approach, we derive the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The asymptotic expansions of the solution to the partial differential equation (PDE) derived from the HJB equation are presented as the parameter appearing in the exponent of the diffusion coefficient tends to 0. We use perturbation theory for partial differential equations (PDEs) to obtain the asymptotic solutions of the optimal reinsurance and investment strategies. Finally, we provide numerical examples and sensitivity analyses to illustrate the effects of model parameters on the optimal reinsurance and investment strategies.
This article considers the optimal reinsurance-portfolio problem that the insurer invests in two related risky assets described by different types: constant elasticity of variance model and ...jump-diffusion process model, besides a risk-free asset. There is a correlation between the diffusion processes of the two models. Meanwhile, the company purchases proportional reinsurance. Specially, assume the claim process follows a Lévy process and the reinsurance's premium principle has not to be certain. Then based on stochastic control theory, a novel form of the optimal value function for solving the Hamilton-Jacobi-Bellman equation is constructed. Finally, the expressions of the optimal results are obtained under maximizing the expected exponential utility of terminal wealth. In addition, we listed several examples of the common premium principles. Numerical simulations are supplied for sensitivity analysis of parameters.
We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices ...and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.
We survey the financial markets whose risks are caused by uncertain volatilities. The financial markets focus on the assets which are effectively allocated in one risk-free asset and one risky asset, ...whose price process is governed by the constant elasticity of variance (CEV for short) model which contains the G-Brownian motion rather than the classical Brownian motion. Such the CEV model which includes the G-Brownian motion utilized to financial markets is the extension of the classical CEV model. Applying the concept of arbitrage and the properties of G-expectation, we consider stock price dynamics which exclude arbitrage opportunities. Moreover, the interval of no-arbitrage price for the general European contingent claims is found in the Markovian case.
On the premise of considering the interests of insurance companies and reinsurance companies at the same time, this paper studies the investment and reinsurance game between them. Suppose that the ...compensation process faced by an insurance company is described by Brownian motion with drift. Insurance companies can purchase proportional reinsurance from reinsurance companies, and both companies can invest in a risk-free asset, a risky asset whose price process follows the constant elasticity of variance (CEV) model, and a defaultable bond. With the goal of maximizing the expected utility of weighted terminal wealth, the corresponding Hamilton–Jacobi–Bellman (HJB) equations are established and solved by using the principle of dynamic programming, and the analytical expressions of the equilibrium investment-reinsurance strategies of insurers and reinsurers are derived respectively. Finally, the influence of each model parameter on the equilibrium strategy is analyzed by numerical examples.
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