The Optimal Problem Of CEV Model In Insurance And Finance

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.