Optimal proportional reinsurance and investment with transaction costs, I: Maximizing the terminal wealth

We consider a problem of optimal reinsurance and investment with multiple risky assets for an insurance company whose surplus is governed by a linear diffusion. The insurance company’s risk can be reduced through reinsurance, while in addition the company invests its surplus in a financial market with one risk-free asset and n risky assets. In this paper, we consider the transaction costs when investing in the risky assets. Also, we use Conditional Value-at-Risk (CVaR) to control the whole risk. We consider the optimization problem of maximizing the expected exponential utility of terminal wealth and solve it by using the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Explicit expression for the optimal value function and the corresponding optimal strategies are obtained.     

Optimal investment with transaction costs based on exponential utility function: a parabolic double obstacle problem

This paper concerns optimal investment problem with proportional transaction costs and finite time horizon based on exponential utility function. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. Numerical examples are obtained by the binomial method.     

Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C2,1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized.     

Optimal trading strategy for European options with transaction costs

The European option with transaction costs is studied. The cost of making a transaction is taken to be proportional by a factor λ to the value (in dollars) of stock traded. When there are no transaction costs (i.e. when λ=0) the well-known Black-Scholes strategy tells how to hedge the option. Since no non-trivial perfect hedging strategy exists when λ>0 (see (Ann. Appl. Probab. 5(2) (1995) 327)), we instead try to maximize the expected utility attainable. We seek to understand the effect transaction costs have on the maximum attainable expected utility over all strategies, when λ is small but non-zero. It turns out that transaction costs diminish the expected utility by an amount which has the order of magnitude λ^2/3. We will compute that correction explicitly modulo an error which is small compared to λ^2/3. We will exhibit an explicit strategy whose expected utility differs from the maximum attainable expected utility by an error small in comparison to λ^2/3.     

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].     

Optimal proportional reinsurance and dividend payments with transaction costs and internal competition

We study the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside company and transaction costs when dividends occur. The management of the company controls the reinsurance rate, the timing and the amount of dividends paid out to maximize the expected total dividends paid out to the shareholders until ruin time. By solving the corresponding quasi-variational inequality, we obtain the optimal return function and the optimal strategy.     

跳扩散模型中具有成比例交易费的最优投资消费模型

研究在跳扩散模型中一类最优投资消费问题．假定市场由无风险债券和一种风险股票构成且具有成比例的交易费,在限制卖空股票和借款的条件下,证明了该问题的值函数为相应HJB方程惟一的带状态空间约束的粘性解．     

Optimal dividend-equity issuance strategy in a dual model with fixed and proportional transaction costs

In this paper, we consider the problem of optimal dividend payout and equity issuance for a company whose liquid asset is modeled by the dual of classical risk model with diffusion. We assume that there exist both proportional and fixed transaction costs when issuing new equity. Our objective is to maximize the expected cumulative present value of the dividend payout minus the equity issuance until the time of bankruptcy,which is defined as the first time when the company’s capital reserve falls below zero. The solution to the mixed impulse-singular control problem relies on two auxiliary subproblems: one is the classical dividend problem without equity issuance, and the other one assumes that the company never goes bankrupt by equity issuance.We first provide closed-form expressions of the value functions and the optimal strategies for both auxiliary subproblems. We then identify the solution to the original problem with either of the auxiliary problems. Our results show that the optimal strategy should either allow for bankruptcy or keep the company’s reserve above zero by issuing new equity, depending on the model’s parameters. We also present some economic interpretations and sensitivity analysis for our results by theoretical analysis and numerical examples.     

Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs

We consider the optimal financing and dividend control problem of the insurance company with fixed and proportional transaction costs. The management of the company controls the reinsurance rate, dividends payout as well as the equity issuance process to maximize the expected present value of the dividends payout minus the equity issuance until the time of bankruptcy. This is the first time that the financing process in an insurance model with two kinds of transaction costs, which come from real financial market has been considered. We solve the mixed classical-impulse control problem by constructing two categories of suboptimal models, one is the classical model without equity issuance, the other never goes bankrupt by equity issuance.     

Additive habit formation: consumption in incomplete markets with random endowments

We provide a detailed characterization of the optimal consumption stream for the additive habit-forming utility maximization problem, in a framework of general discrete-time incomplete markets and random endowments. This characterization allows us to derive the monotonicity and concavity of the optimal consumption as a function of wealth, for several important classes of incomplete markets and preferences. These results yield a deeper understanding of the fine structure of the optimal consumption and provide a further theoretical support for the classical conjectures of Keynes (The general theory of employment, interest and money. Cambridge University Press, Cambridge, 1936).     

Perturbation solution of optimal portfolio theory with transaction costs for any utility function

The solution to the optimal portfolio selection and consumptionrule with small transaction costs is derived via the use ofperturbation analysis for the case when one risky and one risklessasset are available for investment. This methodology allowsus to apply a broader specification for the utility function.     

Multi-asset investment-consumption model with transaction costs

In this paper, we consider the multi-asset optimal investment-consumption model: a riskless asset and d risky assets. when the initial time is t?0, for a proportional transaction costs and discount factors, we proof that the value function of the model is a unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equations.     

VaR optimal portfolio with transaction costs

We consider the problem of portfolio optimization under VaR risk measure taking into account transaction costs. Fixed costs as well as impact costs as a nonlinear function of trading activity are incorporated in the optimal portfolio model. Thus the obtained model is a nonlinear optimization problem with nonsmooth objective function. The model is solved by an iterative method based on a smoothing VaR technique. We prove the convergence of the considered iterative procedure and demonstrate the nontrivial influence of transaction costs on the optimal portfolio weights.     

Generalized average shadow prices and bottlenecks

Usually some of the constraints of a 0-1-Mixed Integer Linear Programming problem correspond to resources and in this paper we suppose that they may be redefined. For the availability of the resources the average shadow price is the maximum price that the decision maker is willing to pay for an additional unit of the package (i.e. a combination) of resources defined by some direction. In this paper we present a generalization of the average shadow price and its relation with bottlenecks including the analysis relative to the coefficients matrix of resource constraints. The generalization presented does not have some limitations of the usual average shadow price. A mathematical programming approach to find the strategy for investment in resources is presented.     

Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs

In this paper we consider the dividend payments and capital injections control problem in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. The objective is to maximize the expected present value of the dividends minus the discounted costs of capital injections. This paper can be considered as an extension of Yao et al. (2010), we include fixed transaction costs incurred by capital injections in this paper. This leads to an impulse control problem. Using the techniques of quasi-variational inequalities (QVI), this optimal control problem is solved. Numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.     

Index tracking with fixed and variable transaction costs

Index tracking is a form of passive portfolio (fund) management that attempts to mirror the performance of a specific index and generate returns that are equal to those of the index, but without purchasing all of the stocks that make up the index. We present two mixed-integer linear programming formulations of this problem. In particular we explicitly consider both fixed and variable transaction costs. Computational results are presented for data sets drawn from major world markets.     

Asset allocation with distorted beliefs and transaction costs

In this paper we study the problem of the optimal portfolio selection with transaction costs for a decision-maker who is faced with Knightian uncertainty. The decision-maker’s portfolio consists of one risky and one risk-free asset, and we assume that the transaction costs are proportional to the traded volume of the risky asset. The attitude to uncertainty is modeled by the Choquet expected utility. We derive optimal strategies and bounds of the no-transaction region for both optimistic and pessimistic decision-makers. The no-transaction region of a pessimistic investor is narrower and its bounds lie closer to the origin than that of an optimistic trader. Moreover, under the Choquet expected utility the structure of the no-transaction region is not necessarily a closed interval as it is under the standard expected utility model.