Optimal investment and risk control for an insurer with stochastic factor

doi:10.1016/j.orl.2017.04.002

Operations Research Letters

Volume 45, Issue 3, May 2017, Pages 259-265

https://doi.org/10.1016/j.orl.2017.04.002 Get rights and content

Abstract

We study an optimal investment and risk control problem for an insurer under stochastic factor. The insurer allocates his wealth across a riskless bond and a risky asset whose drift and volatility depend on a factor process. The risk process is modeled by a jump–diffusion with state-dependent jump measure. By maximizing the expected power utility of the terminal wealth, we characterize the optimal strategy of investment and risk control, analyze classical solutions of HJB PDE and prove the verification theorem.

Introduction

Since the seminal work of Merton [11], portfolio optimization problems have been the subject of considerable investigation. The recent development focuses on the optimal investment problem with stochastic volatility, see e.g. Fouque, et al. [6]. The stochastic volatility model directly relaxes the log-normal assumptions on the price process dynamics which is able to capture empirically observed features of price processes and has been successfully used in several contexts, including stochastic interest rates, see e.g. Brennan and Xia [4], and stochastic volatility, see e.g. Zariphopoulou [20] for a related survey.

The paper considers an optimal investment and risk control problem for an insurer under stochastic factor. The stochastic factor models the evolution of macroeconomic variables such as interest rates, broad share price indices or measures of economic activity or growth. For the case without stochastic factor, Zou and Cadenillas [22] study an optimal investment and risk control for an insurer by selecting the insurance policies. This work is also related to the optimal reinsurance which is raised by the case where the insurer wants to control the reinsurance payout. It has been extensively studied by [9], [18] in the (jump) diffusion market. Zhuo, et al. [21] take the regime-switching risk into the optimal reinsurance. Peng and Wang [15] consider the optimal strategy of investment and risk control for an insurer who has some inside information on the insurance business. Further, for the general Lévy market model without risk control, Nutz [13] studies power utility maximization for exponential Lévy models with portfolio constraints. For the random utility, Nutz [12] studies utility maximization for power utility random fields with and without intermediate consumption in a general semimartingale model with closed portfolio constraints. For the random coefficient driven optimal investment without risk control, Benth, et al. [2] and Delong and Klüppelberg [5] deal with the Merton’s case in a Black–Scholes market where the volatility is described as a pure-jump OU process. In our model, an insurer allocates his wealth across a riskless bond and a risky asset where drift and volatility of its price dynamics depend on a diffusion factor. The risk process is described as a general jump–diffusion with state-dependent jump measure. We also allow the correlation among the risky asset price, risk control process and stochastic factor. Differently from the works reviewed above, the appearance of stochastic factor in the model leads that HJB PDE is a fully nonlinear PDE. We then analyze classical solutions to this equation via a power transformation and then the original HJB PDE can be transformed to a linear one. Since the coefficients of our equation on unbounded domain only satisfy local conditions, they are unbounded, have unbounded derivatives and do not satisfy linear growth constraint. Hence standard existence and uniqueness results (see e.g. Chapter 6 of Friedman [7] and Section 2.9 of Krylov [10]) do not apply here. Becherer and Schweizer [1, see Proposition 2.3] and Health and Schweizer [8, see Theorem 1 and Lemma 2] provide new sufficient conditions for guaranteeing the existence and uniqueness of global classical solutions of the PDE under some type of local conditions. We then apply the above technique to analyze the global classical solution of the transformed equation.

The rest of the paper is organized as follows. Section 2 formulates the model. Section 3 derives the HJB PDE. Section 4 characterizes the optimal strategies for investment and risk control. Section 5 analyzes the classical solution of HJB PDE and proves the verification theorem. Section 6 presents a numerical analysis.

Section snippets

The model

We fix $T > 0$ to be the finite target horizon and consider a complete filtered probability space $(Ω, G, G, P)$ . This space also supports a 3-dimensional Brownian motion $(W_{t}, {\hat{W}}_{t}, {\bar{W}}_{t})$ for $t \in [0, T]$ and an independent Poisson random measure $N (d u, d t)$ on $U \times [0, T]$ . Here $U$ is a topological space and the reference filtration $G = {(G_{t})}_{t \in [0, T]}$ is given by the augmented natural filtration generated jointly by Brownian motion and Poisson random measure. We next describe the market model considered in this paper which

Dynamic optimization for an insurer

This section formulates the optimal portfolio problem of the insurer with risk control and derives the HJB PDE. Recall that the average premium per liability for the insurer is $p (Y_{t})$ , and $η_{t}$ the $G$ -adapted total outstanding number of policies (liabilities) at time $t$ introduced in the above section. Then the revenue from selling insurance policies over the time period of $(t, t + d t)$ is given by $p (Y_{t}) η_{t} d t$ . Denote by $ϕ_{t}$ the time- $t$ amount of the money invested in the risky asset. Then the surplus

Optimal strategies

This section focuses on the characterization of the optimal strategies of investment and risk control of the insurer. By Theorem 11.2.3., pag. 232 in Oksendal [14], it suffices to consider the Markov control in our case. Recall the Hamiltonian given by (10). Then the first-order condition of the Hamiltonian w.r.t. $π$ gives that, for $(π, ℓ) \in R^{2}$ and $(y, φ) \in R \times R$ , $\frac{\partial H (π, ℓ; y, φ)}{\partial π} = γ θ (y) + γ (γ - 1) (σ^{2} (y) π - ρ_{2} ℓ σ (y) ϕ (y)) + ρ_{1} γ σ (y) a (y) φ = 0 .$ The solution of the above first-order condition equation admits, for $(y, φ) \in R \times R$

HJB PDE and verification theorem

In this section, we analyze existence and uniqueness of the global classical solution of HJB PDE (9) and then we will prove the corresponding verification theorem.

Recall the optimal strategy $(π^{*}, ℓ^{*})$ given by (18). Plugging it into (9) and we have the following updated HJB PDE given by, on $(t, y) \in [0, T) \times R$ , $0 = \frac{\partial B (t, y)}{\partial t} + \frac{1}{2} a^{2} (y) \frac{\partial^{2} B (t, y)}{\partial y^{2}} + (b (y) - \frac{ρ_{1} γ}{γ - 1} \frac{a (y) θ (y)}{σ (y)}) \frac{\partial B (t, y)}{\partial y} - \frac{ρ_{1}^{2} γ}{2 (γ - 1)} a^{2} (y) \frac{{(\frac{\partial B (t, y)}{\partial y})}^{2}}{B (t, y)} + γ B (t, y) (r (y) + p_{c} (y) ℓ^{*} (y) + \frac{ρ_{2} ϕ (y) θ (y)}{σ (y)} ℓ^{*} (y) - \frac{θ^{2} (y)}{2 (γ - 1) σ^{2} (y)} + \frac{γ - 1}{2} (1 - ρ_{2}^{2}) ϕ^{2} (y) {| ℓ^{*} (y) |}^{2}) + B (t, y) \int_{U}$

Numerical analysis

Recall Example 2.1 and choose uniformly elliptic Scott volatility here, i.e. $σ (y) = \sqrt{ϑ_{1} (y)} = \sqrt{ε_{1} + e^{γ_{1} y}}$ and $ϕ (y) = \sqrt{ϑ_{2} (y)} = \sqrt{ε_{2} + e^{γ_{2} y}}$ for $y \in R$ . The condition (17) implies that for $y \in R, j (y) g (y, 1) \leq p_{c} + ρ_{2} θ \sqrt{\frac{ε_{2} + e^{γ_{2} y}}{ε_{1} + e^{γ_{1} y}}} ≕ κ (y)$ . Consider the risk aversion parameter $γ = 0.5$ . This follows from Eq. (18) that the optimal strategy for the risky asset is given by, for $(y, φ) \in R \times R$ , $π^{*} (y, φ) = ρ_{2} \sqrt{\frac{ε_{2} + e^{γ_{2} y}}{ε_{1} + e^{γ_{1} y}}} ℓ^{*} (y) + \frac{2 ρ_{1} a}{\sqrt{ε_{1} + e^{γ_{1} y}}} φ + \frac{2 θ}{(ε_{1} + e^{γ_{1} y})},$ where $ℓ^{*} (y)$ is given by (31) in the supplementary material at //staff.ustc.edu.cn/%7Elijunbo/research/ORL-complete-version.pdf

Acknowledgments

This research was partially supported by NSF of China (No. 11471254, 11426236), The Key Research Program of Frontier Sciences, CAS (No. QYZDB-SSW-SYS009) and Fundamental Research Funds for the Central Universities (No. WK3470000008). The authors also gratefully acknowledge the constructive and insightful comments provided by the anonymous reviewer, Associate Editor and Area Editor which contributed to improve the quality of the manuscript greatly.

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View full text

Optimal investment and risk control for an insurer with stochastic factor

Abstract

Introduction

Section snippets

The model

Dynamic optimization for an insurer

Optimal strategies

HJB PDE and verification theorem

Numerical analysis

Acknowledgments

J. Econom. Theory

Insurance Math. Econom.

Insurance Math. Econom.

Insurance Math. Econom.

Classical solutions to reaction–diffusion systems for hedging problems with interacting Itô and point processes

Ann. Appl. Probab.

Mertons portfolio optimization problem in Black–Scholes market with non-Gaussian stochastic volatility of Ornstein–Uhlenbeck type

Math. Finance

Kernel correlated Lévy field driven forward rate and application to derivative pricing

Appl. Math. Optim.

Stochastic interest rates and the bond-stock mix

Eur. Finance Rev.

Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients

Ann. Appl. Probab.

Portfolio optimization and stochastic volatility asymptotics

Math. Finance

Stochastic Differential Equations and Applications