Vasicek利率模型下带有零息票债券的投资-消费模型

应用随机最优控制理论研究Vasicek利率模型下的投资-消费问题,其中假设无风险利率是服从Vasicek利率模型的随机过程,且与股票价格过程存在一般相关性.假设金融市场由一种无风险资产、一种风险资产和一种零息票债券所构成,投资者的目标是最大化中期消费与终端财富的期望贴现效用.应用变量替换方法得到了幂效用下最优投资-消费策略的显示表达式,并分析了最优投资-消费策略对市场参数的灵敏度.     

Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

In this paper, based on equilibrium control law proposed by Björk and Murgoci (2010), we study an optimal investment and reinsurance problem under partial information for insurer with mean–variance utility, where insurer’s risk aversion varies over time. Instead of treating this time-inconsistent problem as pre-committed, we aim to find time-consistent equilibrium strategy within a game theoretic framework. In particular, proportional reinsurance, acquiring new business, investing in financial market are available in the market. The surplus process of insurer is depicted by classical Lundberg model, and the financial market consists of one risk free asset and one risky asset with unobservable Markov-modulated regime switching drift process. By using reduction technique and solving a generalized extended HJB equation, we derive closed-form time-consistent investment–reinsurance strategy and corresponding value function. Moreover, we compare results under partial information with optimal investment–reinsurance strategy when Markov chain is observable. Finally, some numerical illustrations and sensitivity analysis are provided.     

Robust optimal reinsurance–investment strategy with price jumps and correlated claims

This paper considers the robust optimal reinsurance–investment strategy selection problem with price jumps and correlated claims for an ambiguity-averse insurer (AAI). The correlated claims mean that future claims are correlated with historical claims, which is measured by an extrapolative bias. In our model, the AAI transfers part of the risk due to insurance claims via reinsurance and invests the surplus in a financial market consisting of a risk-free asset and a risky asset whose price is described by a jump–diffusion model. Under the criterion of maximizing the expected utility of terminal wealth, we obtain closed-form solutions for the robust optimal reinsurance–investment strategy and the corresponding value function by using the stochastic dynamic programming approach. In order to examine the influence of investment risk on the insurer’s investment behavior, we further study the time-consistent reinsurance–investment strategy under the mean–variance framework and also obtain the explicit solution. Furthermore, we examine the relationship among the optimal reinsurance–investment strategies of the AAI under three typical cases. A series of numerical experiments are carried out to illustrate how the robust optimal reinsurance–investment strategy varies with model parameters, and result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.     

Optimal Basket Liquidation for CARA Investors is Deterministic

Abstract

We consider the problem faced by an investor who must liquidate a given basket of assets over a finite time horizon. The investor's goal is to maximize the expected utility of the sales revenues over a class of adaptive strategies. We assume that the investor's utility has constant absolute risk aversion (CARA) and that the asset prices are given by a very general continuous-time, multiasset price impact model. Our main result is that (perhaps surprisingly) the investor does no worse if he narrows his search to deterministic strategies. In the case where the asset prices are given by an extension of the nonlinear price impact model of Almgren [(2003) Applied Mathematical Finance, 10, pp. 1–18], we characterize the unique optimal strategy via the solution of a Hamilton equation and the value function via a nonlinear partial differential equation with singular initial condition.     

基于动态VaR约束与随机波动率模型的最优投资策略

研究Stein-Stein随机波动率模型下带动态VaR约束的最优投资组合选择问题. 假设投资者的目标是最大化终端财富的期望幂效用,可投资于无风险资产和一种风险资产, 风险资产的价格过程由Stein-Stein随机波动率模型刻画. 同时, 投资者期望能在投资过程中利用动态VaR约束控制所面对的风险.运用Bellman动态规划方法和Lagrange乘子法, 得到了该约束问题最优策略的解析式及特殊情形下最优值函数的解析式; 并通过理论分析和数值算例, 阐述了动态VaR约束与随机波动率对最优投资策略的影响.     

基于Heston随机波动率模型和风险偏好视角的资产负债管理

本文研究基于Heston随机波动率模型的资产负债管理问题。假设金融市场由一个无风险资产和一个风险资产构成,投资者的目标是最大化其终端财富的期望效用。应用随机控制方法,得到了该问题最优资产配置策略的解析表达式和相应值函数的解析解,通过数值算例分析了Heston模型主要参数以及债务对最优资产配置策略的影响。结果表明:配置到风险资产的比例对Heston模型中的参数非常敏感;为了对冲债务风险,负债的引入使得配置到风险资产的比例比无负债情形下的高;在风险厌恶系数变大时,无论投资者是否有负债,其投资到风险资产的比例则越来越低。     

Outperformance and Tracking: Dynamic Asset Allocation for Active and Passive Portfolio Management

ABSTRACT

Portfolio management problems are often divided into two types: active and passive, where the objective is to outperform and track a preselected benchmark, respectively. Here, we formulate and solve a dynamic asset allocation problem that combines these two objectives in a unified framework. We look to maximize the expected growth rate differential between the wealth of the investor’s portfolio and that of a performance benchmark while penalizing risk-weighted deviations from a given tracking portfolio. Using stochastic control techniques, we provide explicit closed-form expressions for the optimal allocation and we show how the optimal strategy can be related to the growth optimal portfolio. The admissible benchmarks encompass the class of functionally generated portfolios (FGPs), which include the market portfolio, as the only requirement is that they depend only on the prevailing asset values. Finally, some numerical experiments are presented to illustrate the risk–reward profile of the optimal allocation.     

Optimal mean–variance asset-liability management with stochastic interest rates and inflation risks

This paper considers an optimal asset-liability management problem with stochastic interest rates and inflation risks under the mean–variance framework. It is assumed that there are \(n+1\) assets available in the financial market, including a risk-free asset, a default-free zero-coupon bond, an inflation-indexed bond and \(n-2\) risky assets (stocks). Moreover, the liability of the investor is assumed to follow a geometric Brownian motion process. By using the stochastic dynamic programming principle and Hamilton–Jacobi–Bellman equation approach, we derive the efficient investment strategy and efficient frontier explicitly. Finally, we provide numerical examples to illustrate the effects of model parameters on the efficient investment strategy and efficient frontier.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

Optimal consumption and investment problem with random horizon in a BMAP model

In this paper, we consider the consumption and investment problem with random horizon in a Batch Markov Arrival Process (BMAP) model. The investor invests her wealth in a financial market consisting of a risk-free asset and a risky asset. The price processes of the riskless asset and the risky asset are modulated by a continuous-time Markov chain, which is the phase process of a BMAP. The possible consumption or investment are restricted to a sequence of random discrete time points which are determined by the same BMAP. The investor has only consumption opportunities at some of these random time points, has both consumption and investment opportunities at some other random time points, and can do nothing at the remaining random time points. The object of the investor is to select the consumption–investment strategy that maximizes the expected total discounted utility. The purpose of this paper is to analyze the impact of the consumption–investment opportunity and the economic state on the value functions and consumption–investment strategies. The general solution and the exact solution under the assumption that the consumption and the terminal wealth are evaluated by the power utility are obtained. Finally, a numerical example is presented.     

Optimal investment for the defined-contribution pension with stochastic salary under a CEV model

In this paper, we study the optimal investment strategy of defined-contribution pension with the stochastic salary. The investor is allowed to invest in a risk-free asset and a risky asset whose price process follows a constant elasticity of variance model. The stochastic salary follows a stochastic differential equation, whose instantaneous volatility changes with the risky asset price all the time. The HJB equation associated with the optimal investment problem is established, and the explicit solution of the corresponding optimization problem for the CARA utility function is obtained by applying power transform and variable change technique. Finally, we present a numerical analysis.     

Time-consistent investment-proportional reinsurance strategy with random coefficients for mean–variance insurers

In this study, we consider an insurer who manages her underlying risk by purchasing proportional reinsurance and investing in a financial market consisting of a risk-free bond and a risky asset. The objective of the insurer is to identify an investment–reinsurance strategy that minimizes the mean–variance cost function. We obtain a time-consistent open-loop equilibrium strategy and the corresponding efficient frontier in explicit form using two systems of backward stochastic differential equations. Furthermore, we apply our results to Vasiček’s stochastic interest rate model and Heston’s stochastic volatility model. In both cases, we obtain a closed-form solution.     

Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer

In this paper, we study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance. The insurer’s risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process. In addition, the insurer can purchase new business (such as reinsurance). The controls (investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset. We use the stochastic linear-quadratic (LQ) control theory to derive the optimal value and the optimal strategy. The corresponding Hamilton–Jacobi–Bellman (HJB) equation no longer has a classical solution. With the framework of viscosity solution, we give a new verification theorem, and then the efficient strategy (optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.     

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset–liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear–quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.     

随机利率条件下最优投资消费与寿险购买策略

随着我国利率市场化的深入发展, 利率的随机波动对投资者的最优投资消费策略将产生重要影响. 与此同时, 随着我国寿险市场的渐趋完善, 寿险购买也越来越受到投资者的重视, 投资者的最优策略也将发生改变. 现研究由 Vasicek 模型来刻画的随机利率条件下最优投资消费与寿险购买策略. 投资者的目标在于选择最优投资消费与寿险购买策略使期望效用最大化. 通过运用 Legendre 转换方法求出最优投资消费与寿险购买的显性解. 通过数值分析的方法, 实证分析相关变量的变化对投资者最优投资与寿险购买策略的影响.     

Robust optimal investment and reinsurance problem for a general insurance company under Heston model

In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.     

Optimal consumption–investment strategy under the Vasicek model: HARA utility and Legendre transform

This paper studies the optimal consumption–investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption–investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton–Jacobi–Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption–investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results.     

Robust investment–reinsurance optimization with multiscale stochastic volatility

This paper investigates the investment and reinsurance problem in the presence of stochastic volatility for an ambiguity-averse insurer (AAI) with a general concave utility function. The AAI concerns about model uncertainty and seeks for an optimal robust decision. We consider a Brownian motion with drift for the surplus of the AAI who invests in a risky asset following a multiscale stochastic volatility (SV) model. We formulate the robust optimal investment and reinsurance problem for a general class of utility functions under a general SV model. Applying perturbation techniques to the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation associated with our problem, we derive an investment–reinsurance strategy that well approximates the optimal strategy of the robust optimization problem under a multiscale SV model. We also provide a practical strategy that requires no tracking of volatility factors. Numerical study is conducted to demonstrate the practical use of theoretical results and to draw economic interpretations from the robust decision rules.     

贝叶斯方法在养老基金资产负债管理中的应用

把一个静态资产负债管理模型———均值方差模型应用到定额给付养老金计划的资产负债管理中,在允许无风险借贷的条件下研究养老金在无风险资产和风险资产间的分配问题,用定量分析的方法求出了最优投资组合的一般形式;又针对投资收益率特征参数未知的情况,提出了矩估计和贝叶斯估计两种方法求解最优资本配置比例,将两种方法的结果与一般形式对比,分析了影响最优投资组合的因素,得知养老基金在风险资产中的投资比例与基金经理对风险的厌恶程度、风险资产的风险益酬、风险资产收益率的波动性成负相关关系;并且随决策者掌握的历史信息增加,在风险资产上的投资比例也随之增加,投资行为逐渐趋于理性化;对上述结果进行仿真,验证了结论的有效性。     

Robust optimal investment–reinsurance strategies for an insurer with multiple dependent risks

This paper considers a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who is uncertain about the model parameters. We assume that the surplus of the insurance company can be allocated to the financial market consisting of one risk-free asset and one risky asset whose price process satisfies square root factor process. Under the objective of maximizing the expected utility of the terminal surplus, by adopting the technique of stochastic control, closed-form expressions of the robust optimal strategy and the corresponding value function are derived. The verification theorem is also provided. Finally, by presenting some numerical examples, the impact of some parameters on the optimal strategy is illustrated and some economic explanations are also given. We find that the robust optimal reinsurance strategies under the generalized mean–variance premium are very different from that under the variance premium principle. In addition, ignoring model uncertainty risk will lead to significant utility loss for the AAI.