Optimal Investment Problem Research Articles

Optimal investment with intermediate consumption under no unbounded profit with bounded risk

Abstract We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions.

Optimal investment of variance-swaps in jump-diffusion market with regime-switching

We consider a general jump-diffusion market with regime-switching where the jump risk is modeled as a Markov-modulated Poisson random measure. In this incomplete market, we price the variance-swaps using a combination of the Esscher transform and change of measure on time-inhomogeneous Markov chains. We study the dynamic optimal investment problem of the variance-swaps and characterize the optimal feedback strategy. Moreover, a closed-form solution to the HJB PDE associated with the stochastic control problem is established and the verification theorem is proved. The numerical analysis based on a two-state Markov chain uncovers some robust features of the optimal investment strategy.

Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading

AbstractWe consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint. We derive the dynamic programming equations for the resulting stochastic optimal control problems and solve them numerically. Our numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. We then investigate time discretization effects and find that the loss of portfolio performance resulting from imposing a risk constraint is typically bigger than the loss resulting from infrequent trading.

An optimal consumption and investment problem with quadratic utility and negative wealth constraints

In this paper, we investigate the optimal consumption and portfolio selection problem with negative wealth constraints for an economic agent who has a quadratic utility function of consumption and receives a constant labor income. Due to the property of the quadratic utility function, we separate our problem into two cases and derive the closed-form solutions for each case. We also illustrate some numerical implications of the optimal consumption and portfolio.

Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model

In this paper, we consider the optimal investment problem for both an insurer and a reinsurer. The insurer’s wealth process is described by a jump diffusion risk model and the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset whose price process follows the constant elasticity of variance (CEV) model. Moreover, the correlation between risk model and the risky asset’s price is considered. The objective is maximizing the expected utility of the insurer’s and the reinsurer’s terminal wealth. Applying stochastic control theory, we establish the corresponding Hamilton–Jacobi–Bellman (HJB) equations and derive optimal investment–reinsurance strategies for exponential utility function. Finally, numerical examples are provided to analyze the effects of parameters on the optimal strategies.

Resource extraction with a carbon tax and regime switching prices: Exercising your options

This paper develops a model of a profit maximizing firm with the option to exploit a non-renewable resource, choosing the timing and pace of development. The resource price is modelled as a regime switching process, which is calibrated to oil futures prices. A Hamilton-Jacobi-Bellman equation is specified that describes the profit maximization decision of the firm. The model is applied to a problem of optimal investment in a typical oils sands in situ operation, and solved for critical levels of oil prices that would motivate a firm to make the large scale investment needed for oil sands extraction, as well as to operate, mothball or abandon the facility. Regime shifts can have an important effect on the optimal timing of investment and extraction. The paper examines the effect of several carbon tax schemes on optimal timing of construction, production and abandonment. A form of Green Paradox is identified.

Analytic value function for optimal regime-switching pairs trading rules

We introduce a regime-switching Ornstein–Uhlenbeck (O–U) model to address an optimal investment problem. Our study gives a closed-form expression for a regime-switching pairs trading value function consisting of probability and expectation of the double boundary stopping time of the Markov-modulated O–U process. We derive analytic solutions for the homogenous and non-homogenous ODE systems with initial value conditions for probability and expectation of the double boundary stopping time, and translate the solutions with boundary value conditions into solutions with initial value conditions. Based on the smoothness and continuity of the value function, we can obtain the optimum of the value function with thresholds and guarantee the existence of optimal thresholds in a finite closed interval. Our numerical analysis illustrates the rationality of theoretical model and the shape of transition probability and expected stopping time, as well as discusses sensitivity analysis in both one-state and two-state regime-switching models. We find that the optimal expected return per unit time in the two-state regime-switching model is higher than that of one-state regime-switching model. Likewise, the regime-switching model’s optimal thresholds are closer and more symmetric to the long-term mean.

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

In this paper, we consider the problem of optimal investment by an insurer. The wealth of the insurer is described by a Cramér–Lundberg process. The insurer invests in a market consisting of a bank account and m risky assets. The mean returns and volatilities of the risky assets depend linearly on economic factors that are formulated as the solutions of linear stochastic differential equations. Moreover, the insurer preferences are exponential. With this setting, a Hamilton–Jacobi–Bellman equation that is derived via a dynamic programming approach has an explicit solution found by solving the matrix Riccati equation. Hence, the optimal strategy can be constructed explicitly. Finally, we present some numerical results related to the value function and the ruin probability using the optimal strategy.

On optimal investment with processes of long or negative memory

We consider the problem of utility maximization for investors with power utility functions. Building on the earlier work Larsen et al. (2016), we prove that the value of the problem is a Fréchet-differentiable function of the drift of the price process, provided that this drift lies in a suitable Banach space.We then study optimal investment problems with non-Markovian driving processes. In such models there is no hope to get a formula for the achievable maximal utility. Applying results of the first part of the paper we provide first order expansions for certain problems involving fractional Brownian motion either in the drift or in the volatility. We also point out how asymptotic results can be derived for models with strong mean reversion.

Research and Development Cooperatives and Market Collusion: A Global Dynamic Approach

We present a continuous-time generalization of the seminal research and development model of d’Aspremont and Jacquemin (Am Econ Rev 78(5):1133–1137, 1988) to examine the trade-off between the benefits of allowing firms to cooperate in research and the corresponding increased potential for product market collusion. We show the existence of a solution to the optimal investment problem using a combination of results from viscosity theory and the theory of planar dynamical systems. In particular, we show that there is a critical level of marginal cost at which firms are indifferent between doing nothing and starting to develop the technology. We find that colluding firms develop further a wider range of initial technologies, pursue innovations more quickly, and are less likely to abandon a technology. Product market collusion could thus yield higher total surplus.

An optimal consumption, leisure, and investment problem with an option to retire and negative wealth constraints

This paper attempts to choose the optimal consumption, leisure, investment, and voluntary retirement time under the negative wealth constraint. The Dynamic Programming method is used to derive the value function and to identify the optimal policies when the agent’s utility function of consumption and leisure is given in the form of Cobb–Douglas. Finally, the effects of negative wealth constraints were discussed by examining the optimal policies that vary depending on the degree of the negative wealth constraint.

Optimal reinsurance and investment in a jump-diffusion financial market with common shock dependence

In this paper, we study the optimal reinsurance and investment problem in a financial market with jump-diffusion risky asset. It is assumed that the insurance risk model is modulated by a compound Poisson process, and that the jumps in both the risky asset and insurance risk process are correlated through a common shock. Under the criterion of maximizing the expected exponential utility, we adopt a nonstandard approach to examine the existence and uniqueness of the optimal strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived not only for the expected value principle but also for the variance premium principle. Also, we investigate the effect of the common shock parameter as well as some other important parameters on the optimal strategies. In particular, a numerical example shows that the optimal investment strategy decreases as the degree of common shock dependence increases but the optimal insurance retention level does not behave the same.

Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer

We study an optimal investment and dividend problem of an insurer, where the aggregate insurance claims process is modeled by a pure jump Levy process. We allow the management of the dividend payment policy and the investment of surplus in a continuous-time financial market, which is composed of a risk free asset and a risky asset. The information available to the insurer is partial information. We generalize this problem as a partial information regular-singular stochastic control problem, where the control variable consists of regular control and singular control. Then maximum principles are established to give sufficient and necessary optimality conditions for the solutions of the regular-singular control problem. Finally we apply the maximum principles to solve the investment and dividend problem of an insurer.

Optimal Consumption and Investment for Exponential Utility Function

We investigate an optimal consumption and investment problem for Black-Scholes type financial market on the whole investment interval [0, T]. We formulate various utility maximization problem, which can be solved explicitly. The method of solution uses the convex dual function (Legendre transform) of the utility function. Related to this concept, we introduce and study the convex dual of the value function for our problem.

OPTIMAL INVESTMENT IN HEDGE FUNDS UNDER LOSS AVERSION

We study optimal investment problems in hedge funds for a loss averse manager under the framework of cumulative prospect theory. We obtain explicit solutions for a general utility function satisfying the Inada conditions and a piece-wise exponential utility function. Through a sensitivity analysis, we find that the manager reduces the risk of the hedge fund when her/his loss aversion, risk aversion, ownership in the fund, or management fee ratio increases. However, the increase of incentive fee ratio drives the manager to seek more risk in order to achieve higher prospect utility.

Optimal investment with transaction costs under cumulative prospect theory in discrete time

We study optimal investment problems under the framework of cumulative prospect theory (CPT). A CPT investor makes investment decisions in a single-period financial market with transaction costs. The objective is to seek the optimal investment strategy that maximizes the prospect value of the investor’s final wealth. We obtain the optimal investment strategy explicitly in two examples. An economic analysis is conducted to investigate the impact of the transaction costs and risk aversion on the optimal investment strategy.

Optional Service Problem between Two Insurance Companies with Investment

Service has become an important factor that affects insurance holders’ purchase behaviors, competition between insurance companies, and even the survival of insurance companies. This paper first introduces the service quality into the optimal investment problem between two competing insurance companies, company 1 and 2. Company 1 provides service while company 2 does not, which results in attracting insurance business from company 2 to company 1, partly. Moreover, the service quality affects the cost, the premium and the demand of the insurance. The surplus processes of the two insurance companies are assumed to follow classical Cram´er-Lundberg (C-L) model. Both the two insurance companies are allowed to invest in a risk-free asset and two different risky assets, respectively. Dynamic mean-variance criterion is considered in this paper. Each insurance company wants to maximize the expectation of the difference of wealth between itself and the other one, and to minimize the variance. By applying stochastic control approach, we establish the corresponding extended Hamilton-Jacobi-Bellman (HJB) system of equations. Furthermore, we derive the equilibrium service and investment strategies, and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated. It is interesting to find that the equilibrium value function of company 1 who provides service is larger than that of company 2, and for company 1, the equilibrium value function with service is larger than that without service, while the situation of company 2 is opposite.

Optimal transportation and shoreline infrastructure investment planning under a stochastic climate future

This paper studies the problem of optimal long-term transportation investment planning to protect from and mitigate impacts of climate change on roadway performance. The problem of choosing the extent, specific system components, and timing of these investments over a long time horizon (e.g., 40–60 years) is modeled as a multi-stage, stochastic, bi-level, mixed-integer program wherein cost-effective investment decisions are taken in the upper level. The effects of possible episodic precipitation events on experienced travel delays are estimated from solution of a lower-level, traffic equilibrium problem. The episodic events and longer-term sea level changes exist on different time scales, making their integration a crucial element in model development. The optimal investment strategy is obtained at a Stackelberg equilibrium that is reached upon solution to the bilevel program. A recursive noisy genetic algorithm (rNGA), designed to address large-scale applications, is proposed for this purpose. The rNGA seeks the optimal combination of investment decisions to take now given only probabilistic information on the predicted sea level rise trend for a long planning horizon and associated likely extreme climatic events (in terms of their frequencies and intensities) that might arise over that planning period. The proposed solution method enables the evaluation of decisions concerning where, when and to what level to make infrastructure investments. The proposed rNGA has broad applicability to more general multi-stage, stochastic, bilevel, nonconvex, mixed integer programs that arise in many applications. The proposed solution methodology is demonstrated on an example representing a portion of the Washington, D.C. Greater Metropolitan area adjacent to the Potomac River.

Consumption with Unhedgeable Interest Rate Risk

This paper investigates the optimal investment and consumption problem in a continuous-time financial market for investors with power utility on consumption, who face partially hedgeable interest rate risk. With no analytical solution to the optimal strategies, closed-form approximate strategies are derived by solving the same optimization problem in two fictitious complete markets. The approximate solution helps verify the existence and the optimality of the solution to the original optimization problem and provides bounds of the optimal consumption strategy and the approximation error, both in closed form. As the interest rate increases, if the investor's risk aversion is greater than one, the wealth effect dominates the substitution effect, and consumption increases. If risk aversion is less than one, then the substitution effect dominates, and the investor consumes less.

A spectral method for an Optimal Investment problem with transaction costs under Potential Utility

This paper concerns the numerical solution of the finite-horizon Optimal Investment problem with transaction costs under Potential Utility. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints. In Day and Yi (2009), the problem is reformulated as a non-linear parabolic double obstacle problem posed in one spatial variable and defined in an unbounded domain where several explicit properties and formulae are obtained. The restatement of the problem in polar coordinates allows to pose the problem in one spatial variable in a finite domain, avoiding some of the technical difficulties of the numerical solution of the previous statement of the problem. If high precision is required, the spectral numerical method proposed becomes more efficient than simpler methods as finite differences for example.