Economy and Management Science
Over past decades, deceptive counterfeits which cannot be recognized by ordinary consumers when purchasing, such as counterfeit cosmetics, have posed serious threats on consumers’ health and safety, and resulted in huge economic loss and inestimable brand damages to the genuine goods at the same time. Thus, how to effectively control and eliminate deceptive counterfeits in the market has become a critical problem to the local government. One of the principal challenges in combating the cheating action for the government is how to enhance the enforcement of relative quality inspection agencies like industrial administration office (IAO). In this paper,we formulate a two-stage counterfeit product model with a fixed checking rate from IAO and a penalty for holding counterfeits. Tominimize the total expected cost over two stages,the retailer adopts optimal ordering policies which are correlated with the checking rate and penalty. Under certain circumstances, we find that the optimal expected cost function for the retailer is first-order continuous and convex. The optimal ordering policy in stage two depends closely on the inventory level after the first sales period. When the checking rate in stage one falls into a certain range, the optimal ordering policy for the retailer at each stage is to order both kinds of products. Knowing the retailer’s optimal ordering policy at each stage, IAO can modify the checking rate accordingly to keep the ratio of deceptive counterfeits on the market under a certain level.
Model uncertainty is a type of inevitable financial risk. Mistakes on the choice of pricing model may cause great financial losses. In this paper we investigate financial markets with mean-volatility uncertainty. Models for stock market and option market with uncertain prior distributions are established by Peng’s G-stochastic calculus. On the hedging market, the upper price of an (exotic) option is derived following the Black–Scholes–Barenblatt equation. It is interesting that the corresponding Barenblatt equation does not depend on mean uncertainty of the underlying stocks.Appropriate definitions of arbitrage for super- and sub-hedging strategies are presented such that the super- and sub-hedging prices are reasonable. In particular, the condition of arbitrage for sub-hedging strategy fills the gap of the theory of arbitrage under model uncertainty. Finally we show that the term K of finite variance arising in the superhedging strategy is interpreted as the max Profit & Loss (P&L) of shorting a delta-hedged option. The ask-bid spread is in fact an accumulation of the superhedging P&L and the sub-hedging P&L.
This work studies the constrained optimal execution problem with a random market depth in the limit order market. Motivated from the real trading activities, our execution model considers the execution bounds and allows the random market depth to be statistically correlated in different periods. Usually, it is difficult to achieve the analytical solution for this class of constrained dynamic decision problem. Thanks to the special structure of this model, by applying the proposed state separation theorem and dynamic programming, we successfully obtain the analytical execution policy. The revealed policy is of feedback nature. Examples are provided to illustrate our solution methods. Simulation results demonstrate the advantages of our model comparing with the classical execution policy.
In this paper, we propose an inexact proximal point method to solve equilibrium problems using proximal distances and the diagonal subdifferential. Under some natural assumptions on the problem and the quasimonotonicity condition on the bifunction, we prove that the sequence generated by the method converges to a solution point of the problem.
Bond portfolio immunization is a classical issue in finance. Since Macaulay gave the concept of duration in 1938, many scholars proposed different kinds of duration immunization models. In the literature of bond portfolio immunization using multifactor model, to the best of our knowledge, researchers only use the first-order immunization, which is usually called as duration immunization, and no one has considered second-order effects in immunization, which is well known as “convexity” in the case of single-factor model. In this paper, we introduce the second-order information associated with multifactor model into bond portfolio immunization and reformulate the corresponding problems as tractable semidefinite programs. Both simulation analysis and empirical study show that the second-order immunization strategies exhibit more accurate approximation to the value change of bonds and thus result in better immunization performance.
In this paper, we construct tight lower and upper bounds for the price of an American strangle, which is a special type of strangle consisting of long positions in an American put and an American call, where the early exercise of one side of the position will knock out the remaining side. This contract was studied in Chiarella and Ziogas (J Econ Dyn Control 29:31–62, 2005) with the corresponding nonlinear integral equations derived, which are hard to be solved efficiently through numerical methods. We extend the approach in the paper of Broadie and Detemple (RevFinance Stud 9:1211–1250, 1996) from the case of American call options to the case of American strangles. We establish theoretical properties of the lower and upper bounds, and propose a sequential optimization algorithm in approximating the early exercise boundary of the American strangle. The theoretical bounds obtained can be easily evaluated, and numerical examples confirm the accuracy of the approximations compared to the literature.
Baton and Lemaire (Astin Bull 12:57–71, 1981) proved the nonemptiness of the core of a reinsurance market in which the risks of companies are independent. However, cases involving dependent risks have received increasing concerns in modern actuarial science. In this paper, we investigate the nonemptiness of the core of a reinsurance market where the risks of different companies may be dependent. When the exponential utility function is employed, we find an important property on risk premium and show that the core of the market is always nonempty.
Data envelopment analysis has been successfully used in resource allocation problems. However, to the best of our knowledge, there are no allocation models proposed in the literature that simultaneously take both the global efficiency and growing potential into account. Hence, this research aims at developing an allocation model for extra input resources, which maximizes the global technical efficiency and scale efficiency of a decision-making unit (DMU) set while maintaining the pure technical efficiency (i.e., growing potential) of each DMU. To this purpose, we first discuss the optimal resources required by each DMU. We prove that the optimal inputs for the DMU are actually the inputs of some most productive scale size (MPSS). We then propose the allocation model based on the discussion on the case of one DMU. The allocation model is illustrated using two numerical examples.
In the real-world environments, different individuals have different risk preferences. This paper investigates the optimal portfolio and consumption rule with a Cox–Ingersoll–Ross (CIR) model in a more general utility framework. After consumption, an individual invests his wealth into the financial market with one risk-free asset and multiple risky assets, where the short-term rate is driven by the CIR model and stock price dynamics are simultaneously influenced by random sources from both stochastic interest rate and stock market itself. The individual hopes to optimize their portfolios and consumption rules to maximize expected utility of terminal wealth and intermediate consumption. Risk preference of individual is assumed to satisfy hyperbolic absolute risk aversion (HARA) utility, which contains power utility, logarithm utility, and exponential utility as special cases. By using the principle of stochastic optimality and Legendre transform-dual theory, the explicit expressions of the optimal portfolio and consumption rule are obtained. The sensitivity of the optimal strategies to main parameters is analysed by a numerical example. In addition, economic implications are also presented. Our research results show that Legendre transform-dual theory is an effective methodology in dealing with the portfolio selection problems with HARA utility and interest rate risk can be completely hedged by constructing specific portfolios.
In this paper, we first construct a time consistent multi-period worst-case risk measure, which measures the dynamic investment risk period-wise from a distributionally robust perspective. Under the usually adopted uncertainty set, we derive the explicit optimal investment strategy for the multi-period robust portfolio selection problem under the multi-period worst-case risk measure. Empirical results demonstrate that the portfolio selection model under the proposed risk measure is a good complement to existing multi-period robust portfolio selection models using the adjustable robust approach.
In reality, when facing a multi-period asset-liability portfolio selection problem, the risk aversion attitude of a mean-variance investor may depend on the wealth level and liability level. Thus, in this paper, we propose a state-dependent risk aversion model for the investor, in which risk aversion is a linear function of current wealth level and current liability level. Due to the time inconsistency of the resulting multi-period asset-liability mean-variance model, we investigate its time-consistent portfolio policy by solving a nested mean-variance game formulation. We derive the analytical time-consistent portfolio policy, which takes a linear form of current wealth level and current liability level. We also analyze the influence of the risk aversion coefficients on the time-consistent portfolio policy and the investment performance via a numerical example.
Error Bounds for Generalized Mixed Vector Equilibrium Problems via a Minimax Strategy
In this paper, by using scalarization techniques and a minimax strategy, error bound results in terms of gap functions for a generalized mixed vector equilibrium problem are established, where the solutions for vector problems may be general sets under natural assumptions, but are not limited to singletons. The other essentially equivalent approach via a separation principle is analyzed. Special cases to the classical vector equilibrium problem and vector variational inequality are also discussed.
We study the Fisher model of a competitive market from the algorithmic perspective. For that, the related convex optimization problem due to Gale and Eisenberg (Ann Math Stat 30(1):165–168,1959) is used. The latter problem is known to yield a Fisher equilibrium under some structural assumptions on consumers’ utilities, e.g., homogeneity of degree 1, homotheticity. Our goal is to examine applicability of the convex optimization framework by departing from these traditional assumptions. We just assume the concavity of consumers’ utility functions. For this case, we suggest a novel concept of Fisher–Gale equilibrium by using consumers’ utility prices. The prices of utility transfer the utility of consumption bundle to a common numéraire. We develop a subgradient-type algorithm from Convex Analysis to compute a Fisher–Gale equilibrium via Gale’s approach. In order to decentralize prices, we additionally implement the auction design, i.e., consumers settle and update their individual prices and producers sell at the highest offer price. Our price adjustment is based on a tatonnement procedure, i.e., the prices change proportionally to consumers’ individual excess supplies. Historical averages of consumption are shown to clear the market of goods. Our algorithm is justified by a global rate of convergence. In the worst case, the number of price updates needed to achieve an ε-tolerance is proportional to 1/ε2.
