An Optimal Investment Strategy for Insurers in Incomplete Markets

Abstract

In this paper we consider the problem of an insurance company where the wealth of the insurer is described by a Cramér-Lundberg process. The insurer is allowed to invest in a risky asset with stochastic volatility subject to the influence of an economic factor and the remaining surplus in a bank account. The price of the risky asset and the economic factor are modeled by a system of correlated stochastic differential equations. In a finite horizon framework and assuming that the market is incomplete, we study the problem of maximizing the expected utility of terminal wealth. When the insurer's preferences are exponential, an existence and uniqueness theorem is proven for the non-linear Hamilton-Jacobi-Bellman equation (HJB). The optimal strategy and the value function have been produced in closed form. In addition and in order to show the connection between the insurer's decision and the correlation coefficient we present two numerical approaches: A Monte-Carlo method based on the stochastic representation of the solution of the insurer problem via Feynman-Kac's formula, and a mixed Finite Difference Monte-Carlo one. Finally the results are presented in the case of Scott model.

Expected exponential utility maximization of insurers with a Linear Gaussian stochastic factor model

“…[18,61,65] employed Black-Scholes models. And, [7,8,23] adopted stochastic factor models that can be expected to compensate for the disadvantages of Black-Scholes model.…”

“…In particular, we state Badaoui-Fernández [7], Badaoui et al [8], Fernández et al [18], Hata-Yasuda [23], Wang [61], Yang-Zhang [65], Zhou [70]. These treated the problems of optimal investment by insurance companies when the utility function is of exponential type.…”

Optimal investment and reinsurance of insurers with lognormal stochastic factor model

“…in order to forecast the return of risky assets, we realize that stochastic factor models seem to be more realistic than Black-Scholes models. Hence, stochastic factor models are also adopted in [3,4,12].…”

“…In particular, we state Badaoui-Fernández [3], Badaoui et al [4], Fernández et al [7], Hata and Yasuda [12], Liang et al [18] and Xu et al [25]. Fernández et al [7] studied maximization of the expected exponential utility with Black-Scholes model.…”

“…Fernández et al [7] studied maximization of the expected exponential utility with Black-Scholes model. Indeed, Badaoui et al [3,4] considered a stochastic volatility model as counterparts of [7] where Badaoui and Fernández [3] assumed that the risky assets and factor processes are not correlated and Badaoui et al [4] allowed that the risky assets and factor processes are correlated. Hata et al [12] discussed a linear Gaussian stochastic factor model, and constructed the explicit optimal strategy.…”