Optimal Investment Problem Research Articles

Necessary and Sufficient Optimality Conditions for Regular–Singular Stochastic Differential Games with Asymmetric Information

We consider a class of regular–singular stochastic differential games arising in the optimal investment and dividend problem of an insurer under model uncertainty. The information available to the two players is asymmetric partial information and the control variable of each player consists of two components: regular control and singular control. We establish the necessary and sufficient optimality conditions for the saddle point of the zero-sum game. Then, as an application, these conditions are applied to an optimal investment and dividend problem of an insurer under model uncertainty. Furthermore, we generalize our results to the nonzero-sum regular–singular game with asymmetric information, and then the Nash equilibrium point is characterized.

Strict local martingales and optimal investment in a Black–Scholes model with a bubble

AbstractThere are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen–Ledoit–Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between these previously disconnected approaches. While the original JLS model is never a strict local martingale, there are relaxations that can be strict local martingales and that preserve the key assumption of a log‐periodic power law for the hazard rate of the time of the crash. We then study the optimal investment problem for an investor with constant relative risk aversion in this model. We show that for positive instantaneous expected returns, investors with relative risk aversion above one always ride the bubble.

THE EFFECTS OF TAXATION ON OPTIMAL CONSUMPTION AND INVESTMENT

We investigate the optimal consumption and investment problem of working agent who faces tax system on consumption, labor income, savings and...

Optimal investment under VaR-Regulation and Minimum Insurance

We look at an optimal investment problem of a financial institution operating under a joint Value-at-Risk and a portfolio insurance constraint. This analysis is particularly relevant for an insurance company operating under the Solvency II regulation which aims to maximize the expected utility of its shareholders, while at the same time being required to provide its policyholders a minimum guaranteed amount. Using static Lagrangian method, we solve the pointwise utility optimization problem to achieve the global maximum by carefully comparing the local maximizers with the jump point or the boundary. Our theoretical and numerical results show that contrary to a pure Value-at-Risk regulation, an insurance company that operates not only under a Solvency II VaR constraint but additionally has to serve a minimal guarantee admits a comprehensive but not too costly protection, and at the same time displays prudent investment behavior. This result holds for both constant and stochastic volatility settings.

Optimal portfolio of continuous‐time mean‐variance model with futures and options

SummaryThis paper is concerned with the portfolio selection of options that have different sale and purchase prices based on the continuous‐time mean‐variance criterion in financial markets. The optimal investment problem is formulated as a continuous‐time mathematical model. These price processes follow jump‐diffusion processes (the Weiner process and the Poisson process). With the theory of stochastic linear‐quadratic control and viscosity solutions, the corresponding Hamilton‐Jacobi‐Bellman equation of the problem is represented and its solutions are obtained in different conditions. The optimal investment strategies are presented. In addition, the efficient frontier is also illustrated. Finally, an example and some discussions illustrating these results are also presented.

Optimal Investment Problem in Stochastic and Local Volatility Models

We revisit the classical Merton's optimal allocation problem and show that significant corrections to the Merton ratio arise from hard to observe behaviour of the volatility process around zero. Having regularised this behaviour, we show that the adjustment to the myopic Merton ratio can be largely deduced from observed option prices, which paves the way to a practical approach to the more efficient asset allocation.

Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Game contingent claims (GCCs) generalize American contingent claims in allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we construct Nash equilibria by rewriting the game as a non-zero-sum stopping game in which players compare payoffs in terms of their exponential utility indifference values. As a by-product, we also obtain an existence result for the optimal exercise time of an American claim under utility indifference valuation by relating it to the corresponding nonlinear Snell envelope.

The optimal investment problem in stochastic and local volatility models

The optimal investment problem in stochastic and local volatility models

Optimal Investment of DC Pension Plan Under Short-Selling Constraints and Portfolio Insurance

In this paper we investigate an optimal investment problem with short-selling constraints and portfolio insurance faced by a defined contribution pension fund manager who is loss averse under inflationary risk. The financial market consists of a cash bond, an indexed bond and a stock. The manager aims to maximize the expected S-shaped utility of the terminal wealth exceeding a minimum guarantee. We apply the dual control method to solve the problem and derive the representations of the optimal wealth process and trading strategies in terms of the dual controlled process and the dual value function. We also perform some numerical tests and show how the loss aversion, the short-selling constraints and the portfolio insurance impact the optimal terminal wealth.

Optimal Investment, Derivative Demand and Arbitrage Under Price Impact

This paper studies the optimal investment problem with random endowment in an inventory-based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading, and non-linear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, non-linear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.

On Long Term Investment Optimality

We study the problem of optimal long term investment with a view to beat a benchmark for a diffusion model of asset prices. Two kinds of objectives are considered. One criterion concerns the probability of outperforming the benchmark and seeks either to minimise the decay rate of the probability that a portfolio exceeds the benchmark or to maximise the decay rate that the portfolio falls short. The other criterion concerns the growth rate of an expected risk-sensitised utility of wealth which has to be either minimised, for a risk-averse investor, or maximised, for a risk-seeking investor. It is assumed that the mean returns and volatilities of the securities are affected by an economic factor, possibly, in a nonlinear fashion. The economic factor and the benchmark are modelled with general Ito differential equations. The results identify asymptotically optimal portfolios and produce the decay, or growth, rates. The proportions of wealth invested in the individual securities are time-homogeneous functions of the economic factor. Furthermore, a uniform treatment is given to the out—and under—performance probability optimisation as well as to the risk-averse and risk-seeking portfolio optimisation. It is shown that there exists a portfolio that optimises the decay rates of both the outperformance probability and the underperformance probability. While earlier research on the subject has relied, for the most part, on the techniques of stochastic optimal control and dynamic programming, in this contribution the quantities of interest are studied directly by employing the methods of the large deviation theory. The key to the analysis is to recognise the setup in question as a case of coupled diffusions with time scale separation, with the economic factor representing “the fast motion”.

Dynamic derivative-based investment strategy for mean–variance asset–liability management with stochastic volatility

This paper considers the derivative-based optimal investment strategies for an asset–liability management (ALM) problem under the mean–variance criterion in the presence of stochastic volatility. Specifically, an asset–liability manager is allowed to invest not only in a risk-free bond and a stock, but also in a derivative, whose price depends on the underlying price of the stock and its volatility. By solving a system of two backward stochastic differential equations, we derive the explicit expressions of the efficient strategies and the corresponding efficient frontiers in two cases, with and without the derivative asset. Moreover, we consider the special case of an optimal investment problem with no liability commitment, which is also not studied in the literature. We also provide some numerical examples to illustrate our results and find that the efficient frontier of the case with the derivative is always better than that of the case without the derivative. Moreover, under the same variance, the expectation of the case with the derivative can reach up to as twice as that of the case without the derivative in some situations.

Optimal dividend and investment problems under Sparre Andersen model

In this paper, we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a “renewal” process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique, we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated HJB equation on a cylindrical domain on which the problem is well defined.

Multi-objective Optimization Algorithms with the Island Metaheuristic for Effective Project Management Problem Solving

Abstract Background and Purpose: In every organization, project management raises many different decision-making problems, a large proportion of which can be efficiently solved using specific decision-making support systems. Yet such kinds of problems are always a challenge since there is no time-efficient or computationally efficient algorithm to solve them as a result of their complexity. In this study, we consider the problem of optimal financial investment. In our solution, we take into account the following organizational resource and project characteristics: profits, costs and risks. Design/Methodology/Approach: The decision-making problem is reduced to a multi-criteria 0-1 knapsack problem. This implies that we need to find a non-dominated set of alternative solutions, which are a trade-off between maximizing incomes and minimizing risks. At the same time, alternatives must satisfy constraints. This leads to a constrained two-criterion optimization problem in the Boolean space. To cope with the peculiarities and high complexity of the problem, evolution-based algorithms with an island meta-heuristic are applied as an alternative to conventional techniques. Results: The problem in hand was reduced to a two-criterion unconstrained extreme problem and solved with different evolution-based multi-objective optimization heuristics. Next, we applied a proposed meta-heuristic combining the particular algorithms and causing their interaction in a cooperative and collaborative way. The obtained results showed that the island heuristic outperformed the original ones based on the values of a specific metric, thus showing the representativeness of Pareto front approximations. Having more representative approximations, decision-makers have more alternative project portfolios corresponding to different risk and profit estimations. Since these criteria are conflicting, when choosing an alternative with an estimated high profit, decision-makers follow a strategy with an estimated high risk and vice versa. Conclusion: In the present paper, the project portfolio decision-making problem was reduced to a 0-1 knapsack constrained multi-objective optimization problem. The algorithm investigation confirms that the use of the island meta-heuristic significantly improves the performance of genetic algorithms, thereby providing an efficient tool for Financial Responsibility Centres Management.

Super-Exponential RE Bubble Model with Efficient Crashes

We propose a dynamic Rational Expectations (RE) bubble model of prices with the intention to exploit it for and evaluate it on optimal investment strategies. Our bubble model is defined as a geometric Brownian motion combined with separate crash (and rally) discrete jump distributions associated with positive (and negative) bubbles. We assume that crashes tend to efficiently bring back excess bubble prices close to a “normal” or fundamental value (“efficient crashes”). Then, the RE condition implies that the excess risk premium of the risky asset exposed to crashes is an increasing function of the amplitude of the expected crash, which itself grows with the bubble mispricing: hence, the larger the bubble price, the larger its subsequent growth rate. This positive feedback of price on return is the archetype of super-exponential price dynamics, which has been previously proposed as a general definition of bubbles. Our bubble model also allows for a sequence of small jumps or long-term corrections. We use the RE condition to estimate the realtime crash probability dynamically through an accelerating probability function depending on the increasing expected return. After showing how to estimate the model parameters, we examine the optimal investment problem in the context of the bubble model by obtaining an analytic expression for maximizing the expected log of wealth (Kelly criterion) for the risky asset and a risk-free asset. We also obtain a closed-form approximation for the optimal investment. We demonstrate, on seven historical crashes, the promising outperformance of the method compared to a 60/40 portfolio, the classic Kelly allocation, and the risky asset, and how it mitigates jumps, both positive and negative.

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity, as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem. We provide the optimal control as the solution of a reflected diffusion at a suitable boundary surface. Such boundary arises from the analysis of a family of two-dimensional parameter-dependent optimal stopping problems, and it is characterized in terms of the family of unique continuous solutions to parameter-dependent, nonlinear integral equations of Fredholm type.

About two-criteria optimization of the stock portfolio

This scientific research is devoted to developing the new and known application of mathematical modeling for solving the problem of optimal investment in risky securities. The objectives for new problems are formulated and trajectory modeling methods are constructed for the market value dynamics of a single share and the whole stock portfolio. While solving the problem of modeling the optimal trajectory of the portfolio of shares, methods of optimal system control were utilized in which the fractions of different kinds of shares in the portfolio were used as the control parameters. The problems for the optimal control dynamics of the investment portfolio are formulated for quality criteria, one of which uses "a soft path" and the second - a deviation from the estimated trajectory of the terminal value. Thus, the first part of the work is devoted to solving the problem of constructing the optimal expected market value of the portfolio shares. The second part deals with the problem of optimizing the portfolio based on risk. To solve it, the admissible and effective sets of investment portfolios are applied. The algorithm for solving the problem allows to dynamically take into account the instrumental market constraints that are formulated in the mathematical formulation of the problem.

Optimal reinsurance and investment problem in a defaultable market

ABSTRACTThis article investigates the optimal reinsurance and investment problem involving a defaultable security. The insurer can purchase reinsurance and allocate his wealth among three financial securities: a money account, a stock, and a defaultable corporate bond. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. Using techniques of stochastic control theory, we derive the corresponding Hamilton–Jacobi–Bellman equation and decompose the original optimization problem into a predefault case and a postdefault case. Explicit expressions for optimal strategies and the corresponding value functions are derived, and the verification theorem is given. Finally, we present numerical examples to illustrate our results.

A Class of Solvable Multiple Entry Problems with Forced Exits

We study an optimal investment problem with multiple entries and forced exits. A closed form solution of the optimisation problem is presented for general underlying diffusion dynamics and a general running payoff function in the case when forced exits occur on the jump times of a Poisson process. Furthermore, we allow the investment opportunity to be subject to the risk of a catastrophe that can occur at the jumps of the Poisson process. More precisely, we attach IID Bernoulli trials to the jump times and if the trial fails, no further re-entries are allowed. Interestingly, we find in the general case that the optimal investment threshold is independent of the success probability is the Bernoulli trials. The results are illustrated with explicit examples.

Optimal Trading in Correlated Anomalies

The empirical asset pricing literature documents a myriad of anomalies. Accounting for the correlated and mean-reverting nature of these anomalies, I provide an explicit solution to the optimal dynamic investment problem of a risk-averse investor who trades in an arbitrary number of potentially mispriced assets. The solution can be characterized in terms of a passive and an active portfolio, where the active positions respond to market-timing and stock-selection considerations. Mispricing risk can have a large impact on the investor's optimal portfolio, which the investor hedges by increasing the size of the position in the mispriced asset. When the market portfolio is subject to mispricing, market-timing and stock-selection considerations interact to determined the optimal position in each asset. I show that diversification considerations can lead the investor to optimally take a positive active position in an overpriced asset, and even to increase the allocation in this asset as overpricing worsens. These results have important positive and normative implications.