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arXiv2013

Homogenization and asymptotics for small transaction costs: the multidimensional case

In the context of the multi-dimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. Similar to the one-dimensional derivation in our accompanying paper [42], the asymptotic expansion is expressed in terms of a singular ergodic control problem, and our arguments are based on the theory of viscosity solutions, and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available anymore. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.

math.AP↗

arXiv2015

Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation

We consider a utility maximization problem for an investment-consumption portfolio when the current utility depends also on the wealth process. Such kind of problems arise, e.g., in portfolio optimization with random horizon or with random trading times. To overcome the difficulties of the problem we use the dual approach. We define a dual problem and treat it by means of dynamic programming, showing that the viscosity solutions of the associated Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth functions. This allows to define a smooth solution of the primal Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique in a suitable class and coincides with the value function of the primal problem. Some financial applications of the results are provided.

q-fin.PM↗

arXiv2013

Optimal replication of random claims by ordinary integrals with applications in finance

By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential equation. We found that the solution of this replication problem exists and is not unique. This leads to a new optimal control problem: find a replicating process that is minimal in an integral norm. We found an explicit solution of this problem. Possible applications to portfolio selection problems and to bond pricing models are suggested.

q-fin.PM↗

arXiv2013

Gambling in contests with regret

This paper discusses the gambling contest introduced in Seel & Strack (Gambling in contests, Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 375, Mar 2012.) and considers the impact of adding a penalty associated with failure to follow a winning strategy. The Seel & Strack model consists of $n$-agents each of whom privately observes a transient diffusion process and chooses when to stop it. The player with the highest stopped value wins the contest, and each player's objective is to maximise their probability of winning the contest. We give a new derivation of the results of Seel & Strack based on a Lagrangian approach. Moreover, we consider an extension of the problem in which in the case when an agent is penalised when their strategy is suboptimal, in the sense that they do not win the contest, but there existed an alternative strategy which would have resulted in victory.

q-fin.PM↗

arXiv2013

On a dynamic adaptation of the Distribution Builder approach to investment decisions

Sharpe et al. proposed the idea of having an expected utility maximizer choose a probability distribution for future wealth as an input to her investment problem instead of a utility function. They developed a computer program, called The Distribution Builder, as one way to elicit such a distribution. In a single-period model, they then showed how this desired distribution for terminal wealth can be used to infer the investor's risk preferences. We adapt their idea, namely that a risk-averse investor can choose a desired distribution for future wealth as an alternative input attribute for investment decisions, to continuous time. In a variety of scenarios, we show how the investor's desired distribution combines with her initial wealth and market-related input to determine the feasibility of her distribution, her implied risk preferences, and her optimal policies throughout her investment horizon. We then provide several examples.

q-fin.PM↗

arXiv2013

Portfolio Management Approach in Trade Credit Decision Making

The basic financial purpose of an enterprise is maximization of its value. Trade credit management should also contribute to realization of this fundamental aim. Many of the current asset management models that are found in financial management literature assume book profit maximization as the basic financial purpose. These book profitbased models could be lacking in what relates to another aim (i.e., maximization of enterprise value). The enterprise value maximization strategy is executed with a focus on risk and uncertainty. This article presents the consequences that can result from operating risk that is related to purchasers using payment postponement for goods and/or services. The present article offers a method that uses portfolio management theory to determine the level of accounts receivable in a firm. An increase in the level of accounts receivables in a firm increases both net working capital and the costs of holding and managing accounts receivables. Both of these decrease the value of the firm, but a liberal policy in accounts receivable coupled with the portfolio management approach could increase the value. Efforts to assign ways to manage these risks were also undertaken; among them, special attention was paid to adapting assumptions from portfolio theory as well as gauging the potential effect on the firm value.

q-fin.PM↗

arXiv2013

Planning Optimal From the Firm Value Creation Perspective Levels of Operating Cash Investments

The basic financial purpose of corporation is creation of its value. Liquidity management should also contribute to realization of this fundamental aim. Many of the current asset management models that are found in financial management literature assume book profit maximization as the basic financial purpose. These book profit based models could be lacking in what relates to another aim like maximization of enterprise value. The corporate value creation strategy is executed with a focus on risk and uncertainty. Firms hold cash for a variety of reasons. Generally, cash balances held in a firm can be called considered, precautionary, speculative, transactional and intentional. The first are the result of management anxieties. Managers fear the negative part of the risk and hold cash to hedge against it. Second, cash balances are held to use chances that are created by the positive part of the risk equation. Next, cash balances are the result of the operating needs of the firm. In this article, we analyze the relation between these types of cash balances and risk. This article presents the discussion about relations between firm net working investment policy and as result operating cash balances and firm value. This article also contains propositions for marking levels of precautionary cash balances and speculative cash balances. Application of these propositions should help managers to make better decisions to maximize the value of a firm.

q-fin.RM↗

arXiv2013

Value-Based Inventory Management

The basic financial purpose of a firm is to maximize its value. An inventory management system should also contribute to realization of this basic aim. Many current asset management models currently found in financial management literature were constructed with the assumption of book profit maximization as basic aim. However these models could lack what relates to another aim, i.e., maximization of enterprise value. This article presents a modified value-based inventory management model.

q-fin.PM↗

arXiv2014

Diversity and no arbitrage

A stock market is called diverse if no stock can dominate the market in terms of relative capitalization. On one hand, this natural property leads to arbitrage in diffusion models under mild assumptions. On the other hand, it is also easy to construct diffusion models which are both diverse and free of arbitrage. Can one tell whether an observed diverse market admits arbitrage? In the present paper we argue that this may well be impossible by proving that the known examples of diverse markets in the literature (which do admit arbitrage) can be approximated uniformly (on the logarithmic scale) by models which are both diverse and arbitrage-free.

q-fin.PM↗

arXiv2013

Financial Portfolio Optimization: Computationally guided agents to investigate, analyse and invest!?

Financial portfolio optimization is a widely studied problem in mathematics, statistics, financial and computational literature. It adheres to determining an optimal combination of weights associated with financial assets held in a portfolio. In practice, it faces challenges by virtue of varying math. formulations, parameters, business constraints and complex financial instruments. Empirical nature of data is no longer one-sided; thereby reflecting upside and downside trends with repeated yet unidentifiable cyclic behaviours potentially caused due to high frequency volatile movements in asset trades. Portfolio optimization under such circumstances is theoretically and computationally challenging. This work presents a novel mechanism to reach an optimal solution by encoding a variety of optimal solutions in a solution bank to guide the search process for the global investment objective formulation. It conceptualizes the role of individual solver agents that contribute optimal solutions to a bank of solutions, a super-agent solver that learns from the solution bank, and, thus reflects a knowledge-based computationally guided agents approach to investigate, analyse and reach to optimal solution for informed investment decisions. Conceptual understanding of classes of solver agents that represent varying problem formulations and, mathematically oriented deterministic solvers along with stochastic-search driven evolutionary and swarm-intelligence based techniques for optimal weights are discussed. Algorithmic implementation is presented by an enhanced neighbourhood generation mechanism in Simulated Annealing algorithm. A framework for inclusion of heuristic knowledge and human expertise from financial literature related to investment decision making process is reflected via introduction of controlled perturbation strategies using a decision matrix for neighbourhood generation.

q-fin.PM↗

arXiv2013

On the optimal allocation of assets in investment portfolio with application of modern portfolio and nonlinear dynamic chaos theories in investment, commercial and central banks

The investment economy is a main characteristic of prosperous society. The investment portfolio management is a main financial problem, which has to be solved by the investment, commercial and central banks with the application of modern portfolio theory in the investment economy. We use the learning analytics together with the integrative creative imperative intelligent conceptual co-lateral adaptive thinking with the purpose to advance our scientific knowledge on the diversified investment portfolio management in the nonlinear dynamic financial system. We apply the econophysics principles and the econometrics methods with the aim to find the solution to the problem of the optimal allocation of assets in the investment portfolio, using the advanced risk management techniques with the efficient frontier modeling in agreement with the modern portfolio theory and using the stability management techniques with the dynamic regimes modeling on the bifurcation diagram in agreement with the dynamic chaos theory. We show that the bifurcation diagram, created with the use of the logistic function in Matlab, can provide some valuable information on the stability of combining risky investments in the investment portfolio, solving the problem of optimization of assets allocation in the investment portfolio. We propose the Ledenyov investment portfolio theorem, based on the Lyapunov stability criteria, with the aim to create the optimized investment portfolio with the uncorrelated diversified assets, which can deliver the increased expected returns to the institutional and private investors in the nonlinear dynamic financial system in the frames of investment economy.

q-fin.GN↗

arXiv2014

Suitability of Capital Allocations for Performance Measurement

Capital allocation principles are used in various contexts in which a risk capital or a cost of an aggregate position has to be allocated among its constituent parts. We study capital allocation principles in a performance measurement framework. We introduce the notation of suitability of allocations for performance measurement and show under different assumptions on the involved reward and risk measures that there exist suitable allocation methods. The existence of certain suitable allocation principles generally is given under rather strict assumptions on the underlying risk measure. Therefore we show, with a reformulated definition of suitability and in a slightly modified setting, that there is a known suitable allocation principle that does not require any properties of the underlying risk measure. Additionally we extend a previous characterization result from the literature from a mean-risk to a reward-risk setting. Formulations of this theory are also possible in a game theoretic setting.

q-fin.RM↗

arXiv2013

Switching Portfolios

A constant rebalanced portfolio is an asset allocation algorithm which keeps the same distribution of wealth among a set of assets along a period of time. Recently, there has been work on on-line portfolio selection algorithms which are competitive with the best constant rebalanced portfolio determined in hindsight. By their nature, these algorithms employ the assumption that high returns can be achieved using a fixed asset allocation strategy. However, stock markets are far from being stationary and in many cases the wealth achieved by a constant rebalanced portfolio is much smaller than the wealth achieved by an ad-hoc investment strategy that adapts to changes in the market. In this paper we present an efficient Bayesian portfolio selection algorithm that is able to track a changing market. We also describe a simple extension of the algorithm for the case of a general transaction cost, including the transactions cost models recently investigated by Blum and kalai. We provide a simple analysis of the competitiveness of the algorithm and check its performance on real stock data from the New York Stock Exchange accumulated during a 22-year period.

q-fin.PM↗

arXiv2014

Maximization of Non-Concave Utility Functions in Discrete-Time Financial Market Models

This paper investigates the problem of maximizing expected terminal utility in a (generically incomplete) discrete-time financial market model with finite time horizon. In contrast to the standard setting, a possibly non-concave utility function $U$ is considered, with domain of definition $\mathbb{R}$. Simple conditions are presented which guarantee the existence of an optimal strategy for the problem. In particular, the asymptotic elasticity of $U$ plays a decisive role: existence can be shown when it is strictly greater at $-\infty$ than at $+\infty$.

q-fin.PM↗

arXiv2013

Robust Hedging with Proportional Transaction Costs

Duality for robust hedging with proportional transaction costs of path dependent European options is obtained in a discrete time financial market with one risky asset. Investor's portfolio consists of a dynamically traded stock and a static position in vanilla options which can be exercised at maturity. Both the stock and the option trading is subject to proportional transaction costs. The main theorem is duality between hedging and a Monge-Kantorovich type optimization problem. In this dual transport problem the optimization is over all the probability measures which satisfy an approximate martingale condition related to consistent price systems in addition to the usual marginal constraints.

q-fin.PM↗

arXiv2013

Risks of Large Portfolios

Estimating and assessing the risk of a large portfolio is an important topic in financial econometrics and risk management. The risk is often estimated by a substitution of a good estimator of the volatility matrix. However, the accuracy of such a risk estimator for large portfolios is largely unknown, and a simple inequality in the previous literature gives an infeasible upper bound for the estimation error. In addition, numerical studies illustrate that this upper bound is very crude. In this paper, we propose factor-based risk estimators under a large amount of assets, and introduce a high-confidence level upper bound (H-CLUB) to assess the accuracy of the risk estimation. The H-CLUB is constructed based on three different estimates of the volatility matrix: sample covariance, approximate factor model with known factors, and unknown factors (POET, Fan, Liao and Mincheva, 2013). For the first time in the literature, we derive the limiting distribution of the estimated risks in high dimensionality. Our numerical results demonstrate that the proposed upper bounds significantly outperform the traditional crude bounds, and provide insightful assessment of the estimation of the portfolio risks. In addition, our simulated results quantify the relative error in the risk estimation, which is usually negligible using 3-month daily data. Finally, the proposed methods are applied to an empirical study.

stat.AP↗

arXiv2014

On the optimal dividend problem for a spectrally positive Levy process

In this paper we study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Levy process. This model including the dual model of the classical risk model and the dual model with diffusion as special cases. We assume that dividends are paid to the shareholders according to admissible strategy whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that the optimal dividend strategy is formed by a threshold strategy.

q-fin.PM↗

arXiv2014

Rationalizing Investors Choice

Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer with an explicitly derived concave non-decreasing utility function. This result enables us to infer the utility and risk aversion of agents from their investment choice in a non-parametric way. We relate the property of decreasing absolute risk aversion (DARA) to distributional properties of the terminal wealth and of the financial market. Specifically, we show that DARA is equivalent to a demand for a terminal wealth that has more spread than the opposite of the log pricing kernel at the investment horizon.

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