Optimal Investment Problem Research Articles

Indefinite risk-sensitive control

We consider a risk-sensitive control problem with indefinite cost matrices. All solutions to such an optimal control problem are obtained in an explicit closed-form by the completion of squares and the change of measure methods. Both the finite and infinite horizon cases are considered. As an application, we find all solutions to an optimal investment problem in a market with a stochastic interest rate.

Convergence of Optimal Investment Problems in the Vanishing Fixed Cost Limit

We consider an optimal investment problem for an investor facing constant and proportional transaction costs and study the limit as the constant cost tends to zero. Combining the stochastic Perron's method with stability arguments for viscosity solutions, we show that the value function converges to the value function of the problem with purely proportional costs. Moreover, using a Komlos-type argument, we show that forward-convex combinations of the optimal strategies in the problem with constant costs converge to an optimal strategy without a constant cost.

Optimal Reinsurance and Investment Strategies Under Mean-Variance Criteria: Partial and Full Information

This paper is concerned with an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. The driving Brownian motion and the rate in return of the risky asset price dynamic equation cannot be directly observed. And the short-selling of stocks is prohibited. The problem is formulated as a stochastic linear-quadratic control problem where the control variables are constrained. Based on the separation principle and stochastic filtering theory, the partial information problem is solved. Efficient strategies and efficient frontier are presented in closed forms via solutions to two extended stochastic Riccati equations. As a comparison, the efficient strategies and efficient frontier are given by the viscosity solution to the HJB equation in the full information case. Some numerical illustrations are also provided.

Optimal Investment, Consumption and Life Insurance Problem with Stochastic Environments

Optimal investment, consumption and life insurance problem with stochastic environments for a CRRA wage-earner is solved in this study.  The wage-earner invests in the financial market with one risk-free security, one risky security, receives labor income and has a life insurance policy in the insurance market. A life insurance policy is purchased to hedge the financial wealth for the beneficiary in case of wage earner premature death. The interest rate and the volatility are stochastic. The stochastic interest rate dynamics of risk-free security follow a Ho-Lee model and the risky security follow Heston’s model with stochastic volatility parameter dynamics following a Cox-Ingersoll-Ross (CIR) model. The objective of the wage-earner is to allocate wealth between risky security and risk-free security but also buy a life insurance policy during the investment period to maximize the expected discounted utilities derived from consumption, legacy and terminal wealth over an uncertain lifetime. By applying Bellman's optimality principle, the associated HJB PDE for the value function is established. The power utility function is employed for our analysis to obtain the value function and optimal policies. Finally, numerical examples and simulations are provided.

Numerical Approximation of a System of Hamilton–Jacobi–Bellman Equations Arising in Innovation Dynamics

We consider a system of fully nonlinear partial differential equations that corresponds to the Hamilton–Jacobi–Bellman equations for the value functions of an optimal innovation investment problem of a monopoly firm facing bankruptcy risk. We compare several algorithms for the numerical solution of the considered problem: the collocation method, the finite difference method, WENO method and the adaptive finite element method. We discuss implementation issues for the considered schemes and perform numerical studies for different model parameters to assess their performance.

Inter‐temporal mutual‐fund management

AbstractTraditionally, mutual funds are mostly managed via an ad hoc approach, namely a terminal‐only optimization. Due to the intricate mathematical complexity of a continuum of constraints imposed, effects of the inter‐temporal reward for the managers are essentially neglected in the previous literature. For instance, the inter‐temporal optimal investment problem from the fund manager's viewpoint, who earns proportional management fees continuously (a golden rule in practice), has been outstanding for long. This article completely resolves this challenging question especially under generic running and terminal utilities, via the Dynamic Programming Principle which leads to a nonconventional, highly nonlinear HJB equation. We develop an original mathematical analysis to establish the unique existence of the classical solution of the primal problem. Further numerical calibrations and simulations for both the portfolio weight and the value functions illustrate the robustness of the optimal portfolio towards the manager's risk attitude, which allows different managers with various risk characteristics to sell essentially the same investment vehicle. Simulation studies also indicate that the policy of charging a substantial terminal‐only management fee can be replaced by another one with only a negligible amount over the interim period, which substantially reduces the total management fee paid by the clients without lowering the manager's satisfaction at all; this last observation echoes the magic of the alchemy of finance.

Investments with declining cost following a Lévy process

We consider an optimal investment problem in which the cost of the investment decreases over time. This decrease is modelled using the negative of a non-decreasing Lévy process. The decreasing cost is a way of modelling that innovations drive down the cost of the investment. We present general results on how to compute both the value of the investment, as well as the optimal time at which the investment should be done. Several explicit examples of how different Lévy processes influence the value of the investment are given as illustrations of the general results. The main tools used are fluctuation theory for Lévy processes and inversion of Laplace transforms. When the inversion can be done analytically, we can present analytical solutions where in some cases only numerical solution has previously been known.

Valuation of compound real options for co-investment in residential battery systems

The process of evaluating investments by electricity utilities to subsidize residential battery installations in distribution networks is challenging, because finding efficient investment plans involves reasoning over various sources of uncertainty and consideration of different sizing, location and timing options. This work proposes a novel financial framework using real options valuation (ROV) to solve the optimal investment problem for subsidizing residential battery installations, considering compound options including the option to delay the investment and its subsequent option (expand). We incorporate random variations in PV generation and load behaviour, making use of Monte Carlo (MC) power flow analysis that incorporates battery scheduling profiles. Specifically, the ROV framework solves stochastic power flow analysis with battery scheduling in a forward-looking MC model, and determines the optimal plan for executing the options that maximize the investment value using backward induction over all MC paths. However, this approach to MC becomes computationally prohibitive when battery operational profiles are generated by optimization-based home energy management. To reduce the computational burden, we use policy function approximation to provide fast battery operational profiles, drawing on machine learning methods that reduce MC computation time by 95%. Compared to standard financial analysis, the proposed model allows ROV to capture all uncertainties while allowing utilities to make contingent decisions to maximize the benefits from subsidizing residential battery storage.

Robust optimal proportional reinsurance and investment problem for an insurer with delay and dependent risks

In this article, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and dependent risks, in which the insurer’s surplus process is assumed to follow a risk model with two dependent classes of insurance business. The insurer is allowed to purchase proportional reinsurance and invest his surplus in a financial market, which contains one risk-free asset and one risky asset whose price process satisfies a jump-diffusion model. Under the consideration of the performance-related capital inflow or outflow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s aim is to develop the robust optimal reinsurance and investment strategy for the insurer by maximizing the expected exponential utility of the combination of the average historical performance and terminal wealth under the worst-case scenario of the alternative measures. By adopting the stochastic dynamic programming technique, the expressions of the robust optimal strategy and value function are explicitly obtained. Finally, we present some numerical examples to illustrate the effects of some model parameters on the optimal strategy.

Asset allocation for a DC pension plan with learning about stock return predictability

<p style='text-indent:20px;'>This paper investigates an optimal investment problem for a defined contribution pension plan member who receives a stochastic salary, and considers inflation risk and stock return predictability. The member aims to maximize the expected power utility from her terminal real wealth by investing her pension account wealth in a financial market consisting of a risk-free asset, an inflation-indexed bond and a stock. The expected excess return on the stock can be predicted by both an observable predictor and an unobservable predictor, and the member has to estimate the unobservable predictor by learning the history information. By using the filtering techniques and dynamic programming approach, the closed-form optimal investment strategy and the corresponding value function are derived. Finally, with the help of numerical analysis, we explore the impact of model parameters on the optimal investment strategy, and analyze the welfare benefits from leaning and using inflation-indexed bond to hedge the stock return predictors.</p>

Acceptability maximization

<p style='text-indent:20px;'>The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of risk minimization problems. The results are applied to several important CAIs, such as the gain-to-loss ratio, the risk-adjusted return on capital and the tail-value-at-risk based CAI. In the second part of the paper we investigate the acceptability maximization in a discrete time dynamic setup. Using robust representations of CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs is recursive (i.e. strongly time consistent) and assuming some recursive structure of the market model, then the acceptability maximization problem reduces to just a one period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a time-inconsistent stochastic control problem, similar to the classical mean-variance criteria. To overcome this form of time-inconsistency, we adapt to our setup the set-valued Bellman's principle recently proposed in [<xref ref-type="bibr" rid="b23">23</xref>] applied to two particular dynamic CAIs - the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio. The obtained theoretical results are illustrated via numerical examples that include, in particular, the computation of the intermediate mean-risk efficient frontiers.</p>

Multi-agent Robust Optimal Investment Problem in Incomplete Market

Multi-agent Robust Optimal Investment Problem in Incomplete Market

Asymptotic solution of optimal reinsurance and investment problem with correlation risk for an insurer under the CEV model

In this paper, we consider an optimal proportional reinsurance and investment problem for an insurer whose objective is to maximise the expected exponential utility of terminal wealth. Suppose that the insurer's surplus process follows a Brownian motion with drift. The insurer is allowed to purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. The correlation between risk model and the risky asset's price is considered. By applying dynamic programming approach, we derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation. The asymptotic expansions of the solution to the partial differential equation (PDE) derived from the HJB equation are presented as the parameter appearing in the exponent of the diffusion coefficient tends to 0. We use perturbation theory for partial differential equations (PDEs) to obtain the asymptotic solutions of the optimal reinsurance and investment strategies. Finally, we provide numerical examples and sensitivity analyses to illustrate the effects of model parameters on the optimal reinsurance and investment strategies.

Hedging and utility valuation of a defaultable claim driven by Hawkes processes

AbstractThis article studies the problem of hedging a defaultable claim via the maximization of the mean value of exponential utility, over a set of admissible strategies. The dynamics of the underlying asset is assumed to be governed by mutually exciting Hawkes processes, which captures the jumps clustering phenomenon observed in the market. The resulting market is incomplete and does not allow perfect replication. Hence, a dynamic programming approach is adopted to characterize the value function as the largest solution to a suitable backward stochastic differential equation (BSDE) with a non‐Lipschitz generator. The value function of the optimal investment problem and of the indifference prices are represented in terms of limits of the sequence of value functions of suitable Lipschitz BSDEs and, further, a result of uniqueness is achieved. Finally, numerical experiments are performed to demonstrate the applicability of the proposed framework and to understand the impact of the jump‐clustering on the values of the claims.

On an irreversible investment problem with two-factor uncertainty

We consider a real options model for the optimal irreversible investment problem of a profit-maximizing company. The company has the opportunity to invest in a production plant capable of producing two products, of which the prices follow two independent geometric Brownian motions. After paying a constant sunk investment cost, the company sells the products on the market and thus receives a continuous stochastic revenue flow. This investment problem is set as a two-dimensional optimal stopping problem. We find that the optimal investment decision is triggered by a convex curve, which we characterize as the unique continuous solution to a nonlinear integral equation. Furthermore, we provide analytical and numerical comparative statics results of the dependency of the project's value and investment decision with respect to the model's parameters.

On Market Completions Approach to Option Pricing

Option pricing is one of the most important problems of contemporary quantitative finance. It can be solved in complete markets with non-arbitrage option price being uniquely determined via averaging with respect to a unique risk-neutral measure. In incomplete markets, an adequate option pricing is achieved by determining an interval of non-arbitrage option prices as a region of negotiation between seller and buyer of the option. End points of this interval characterise the minimum and maximum average of discounted pay-off function over the set of equivalent risk-neutral measures. By estimating these end points, one constructs super hedging strategies providing a risk-management in such contracts. The current paper analyses an interesting approach to this pricing problem, which consists of introducing the necessary amount of auxiliary assets such that the market becomes complete with option price uniquely determined. One can estimate the interval of non-arbitrage prices by taking minimal and maximal price values from various numbers calculated with the help of different completions. It is a dual characterisation of option prices in incomplete markets, and it is described here in detail for the multivariate diffusion market model. Besides that, the paper discusses how this method can be exploited in optimal investment and partial hedging problems.

Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks

This article investigates the optimal investment-reinsurance problem for state-dependent risk aversion with delay and common shock dependence. We suppose that the insurer manages insurance risk which is characterized by a two-dimensional dependent claim process by purchasing proportional reinsurance, and the insurer can invest his wealth in a financial market that consists of a risk-free asset and a risky asset. On the basis of the consideration of the performance-related capital inflow/outflow, the wealth process of the insurer is modeled by a stochastic delay differential equation. Then we formulate the optimal investment and reinsurance mean-variance problem within a game-theoretic framework and then applying the stochastic control theory with delay, we derive the corresponding extended HJB equations with delay. Next by solving the extended HJB equations and constructing the exponential martingale process, the closed-form solution to the optimal investment-reinsurance strategy and the corresponding equilibrium value function are derived. Besides, we use a numerical example to analyze the influence of the model parameters on the optimal time-consistent strategy and give an economic explanation.

Asymptotic Portfolio Strategy Based on the CEV Model with General Utility Function

The optimal investment problem is a hot field of financial risk control. The analytical solution of investment strategy can be obtained with the power function utility and exponential function utility when the stock price obeys the constant elasticity of variance (CEV) model. However, different investors have different risk preferences; it means that different investors have different utility functions. In this paper, we propose an asymptotic analysis method to obtain the asymptotic solution of investment strategy with the general utility function. The value function is expanded in the form of series, the expressions of the zero-order term and first-order term of the series expansion are derived, respectively, and the error between the asymptotic approximation and the optimal value function is calculated. Finally, the numerical examples provide comparative analysis between the analytical solution and the asymptotic solution to verify the effectiveness of the proposed method.

Robust Optimal Investment Problem with Delay under Heston’s Model

Robust Optimal Investment Problem with Delay under Heston’s Model

Optimal regulators for a class of nonlinear stochastic systems

We consider a class of nonlinear stochastic systems with square-root nonlinearities appearing in the diffusion terms. The optimal control problems with indefinite quadratic criteria in both finite and infinite horizon are formulated and solved in an explicit closed form. It turns out that all optimal controls are of an affine state-feedback form, despite the fact that the system is nonlinear. We use the method of completion of squares and new types of Riccati differential and algebraic equations to find the solutions. An application to the problem of optimal investment in a market with a stochastic interest rate is given.