Portfolio optimization in markets with stochastic dependence

自从诺贝尔奖得主马科维茨和夏普的工作以来，数学一直是金融部门投资组合构建考虑的核心。简而言之，投资组合理论遵循三个步骤：首先，它们对未来资产价格分布做出分配假设；其次，它们提出目标函数，通常涉及风险和回报参数；最后，他们解决了相关的优化问题。这一领域的许多研究都是建立在基于资产收益正态性的分布假设之上的。**本提案旨在将分布假设扩展到高斯分布之外，考虑风险收益目标，并解决相关的优化问题。高斯分布通常与没有危机的市场概念或具有恒定统计特性的市场联系在一起。该提议最新颖的方面将集中在分布假设上，这些假设表现出依赖结构的分解或过渡，无论是相关性、联结还是其他任何东西。**在过去的几年里，我的大部分研究都集中在随机相关随机过程的推导和估计上，这些过去的研究将构成提案的基石之一。我们的资产价格分布模型的一部分特征将包括随机相关性，但也包括马尔可夫制度转换过程，条件异方差时间序列和允许非对称依赖的概率分布，如α稳定分布。**传统高斯情况下的投资组合优化会引起二次规划问题。在完全非高斯的情况下，它们经常导致不适定优化问题。我们的建议将侧重于仔细选择风险-收益目标函数，允许可解决的优化问题，然后解决问题的投资组合优化方面。换句话说，研究的目标是接近这三个步骤，而不是孤立的，而是为资产价格的概率假设、目标函数和优化问题寻找一个全局一致的定义环境。我们期望从这个提议中产生几项应用。首先，我们的目标是获得与不断变化的市场条件一致的金融时间序列的风险因素特征；在投资组合优化框架中，当我们将风险视为时间序列行为的来源而不仅仅是目标函数中的一个元素时，它就会出现。我们期望这将导致投资组合构建技术，该技术将适用于使用可交易工具的风险管理叠加。接下来，我们期望能够生成实时变化的合成时间序列，并将作为宏观经济变量的替代品，而宏观经济变量通常仅限于每三个月发生一次的观察。总的来说，我们希望为一个广阔的领域或研究提供另一种视角，这些领域或研究在资产管理行业中发挥着核心作用，但其中包含需要更好理解的数学元素，将其置于严格的背景下，并提供一套更强大的工具，使理论适应市场的现实。

Since the work of Nobel laureates Markowitz and Sharpe, Mathematics has been at the core of portfolio construction considerations in the financial sector. In a nutshell, portfolio theories follow three steps: first, they make distributional assumptions about future asset price distributions; next, they pose objective functions, typically involving risk and return parameters; and last, they tackle the associated optimization problem. Much of the research in this area is built on distributional assumptions based on normality of asset returns.**This proposal will aim to expand the distributional assumptions beyond gaussian distributions, consider risk-return objectives, and solve the associated optimization problems. Gaussian distributions are typically associated with the concept of markets without crises, or markets with constant statistical properties. The most novel aspects of the proposal will focus on distributional assumptions which exhibit a breakdown, or a transition, of dependence structures, be it correlations, copulas or anything else.**Much of my research over the last few years has focused on the derivation and estimation of stochastic processes with stochastic correlation, and this past research will constitute one of the building blocks for the proposal. Some of the characteristics that will be part of our model for asset price distributions will include stochastic correlation but also Markov regime switching processes, conditional heteroskedasticity time series and probability distributions that allow for asymmetric dependence, such as alpha-stable distributions.**Portfolio optimization in the traditional gaussian case gives rise to quadratic programming problems. In the fully non-gaussian case they often lead to ill-posed optimization problems. Our proposal will focus on a careful selection of risk-return objective functions that allow for solvable optimization problems, and then address the portfolio optimization aspects of the problem. In other words, the research will aim to approach the three steps, not isolation, but looking for a globally consistent definitional environment for the probabilistic assumptions of asset prices, the objective function and the optimization problem.**We expect several applications to emerge from this proposal. First, we will aim to obtain a risk factor characterization of financial time series which is consistent with changing market conditions; in the portfolio optimization framework, it will arise when we view risk as a source of time series behavior and not just as an element in the objective function. We expect this to lead to portfolio construction techniques that will be amenable of risk management overlays using marketable instruments. Next, we expect to be able to produce synthetic time series, which change in real time, and which will act as surrogates of macroeconomic variables, which are usually limited to observations that occur once every three months. And in general, we expect to provide an alternative perspective on a vast field or research, which enjoys a central role in the asset management industry, but has mathematical elements that need to be understood better, put into a rigorous context and provide a stronger set of tools to adapt theories to the realities of markets.