Portfolio Optimization via Risk-Sensitive Control

本研究涉及投资：投资者应如何构建和管理风险资产的投资组合或基金，以实现收益最大化？至少在过去的50年里，投资组合优化的数学理论一直是工业界和学术界深入研究的对象。最初的突破是由哈里·马科维茨（Harry Markowitz）提出的“均值方差分析”（mean-variance analysis），精确地量化了风险与回报之间的关系。马科维茨的观点在金融行业广为流传，每个关心投资组合管理的人都很熟悉。尽管如此，均值-方差理论仍存在某些众所周知的缺陷。首先，优化需要资产收益的均值向量和协方差矩阵作为输入。虽然有可能估计出后者，但不可能从历史数据中以任何可用的精度估计出前者。因此，需要进一步的建模步骤来将平均回报与其他市场或经济因素联系起来。其次，这是一个单周期理论，从概念上讲，投资者购买一个投资组合，持有一段时间，然后套现。我们需要的是一个动态理论，在这个理论中，投资者对市场价格随时间变化的最佳反应是确定的。1969年，罗伯特·默顿（Robert Merton）开创性地对投资组合优化的动态理论进行了系统研究，他将其表述为一个随机控制问题，其目标是使消费的终身效用最大化。这导致了效用最大化和随机控制数学的广泛发展，但这些成果几乎没有进入资产管理的实际世界，主要可能是因为它过于依赖于程式化的数学模型。另一种不同但相关的方法是最大化长期增长率（这有时被称为“凯利准则”）。这方面的理论不太依赖模型，但它导致的策略在现实的投资范围内可能非常冒险。在这项研究中，我们开发了一种方法，在我们看来，它结合了马科维茨、默顿和凯利的优点，并解决了各自的不足之处。该方法基于风险敏感控制（RSC）理论。这个主题是由雅各布森在20世纪70年代在控制系统文献中介绍的。它在资产管理中的应用是由Bielecki和Pliska在20世纪90年代首创的。在传统的随机控制中，人们寻求使某些性能指标的期望值最大化。在RSC中，这被性能指标的指数最大化所取代。在资产管理应用程序中，该指标是增长率，结果证明，目标相当于在方差（即风险）的惩罚下最大化预期增长率。这项研究是与新邦德街资产管理公司（NBSAM）的联合项目，该公司的业务是管理涉及信用风险的投资组合，因此我们的数学模型必须包括信用事件，即资产价格的跃升。目前，RSC理论只在有限的方法中做到这一点，因此需要新的数学方法。这项工作有四个组成部分，(a)发展RSC理论，(b)为求解RSC方程提供可证明的鲁棒算法，(c)研究NBSAM交易的各类信用风险证券的数学模型，(d)进行计量经济学分析，以建立价格对经济因素的依赖性。部分(a)涉及扩展非线性偏微分方程的所谓“粘性解”理论，以涵盖具有相当弱增长条件的无界域的积分微分算子和解，而部分(b)将涵盖与马尔可夫链控制问题相关的有限差分格式。完成后，这项工作将为投资组合经理提供一个完全动态的资产配置模型，这代表了对当前技术的巨大改进。

This research concerns investment: how should an investor construct and manage a portfolio or fund of risky assets so as to maximize returns? Mathematical theories of portfolio optimization have been the object of intensive study, both in industry and academia, for at least the last 50 years. The initial breakthrough was made by Harry Markowitz with the introduction of 'mean-variance analysis', quantifying precisely the relationship between risk and return. Markowitz' ideas pervade the finance industry and are familiar to everyone concerned with portfolio management. Nonetheless, the mean-variance theory suffers from certain well-known drawbacks. Firstly, the optimization requires as inputs the mean vector and covariance matrix of asset returns. While it may be possible to estimate the latter, it is impossible to estimate the former from historical data with any usable degree of accuracy. Thus a further modelling step is required to relate the mean returns to other market or economic factors. Secondly, this is a one-period theory in which, conceptually, investors buy a portfolio, hold it for some period and then cash out. What is needed is a dynamic theory in which the investors' optimal response to the movement of market prices over time is determined.Systematic study of a dynamic theory of portfolio optimization was initiated in 1969 in ground-breaking work by Robert Merton who formulated it as a stochastic control problem where the objective was to maximize the lifetime utility of consumption. This led to wide-ranging developments in the mathematics of utility maximization and stochastic control, but little of this has found its way into the practical world of asset management, mainly perhaps because it is too dependent on a stylized mathematical model. A different but related approach is to maximize the long-run growth rate (this is sometimes known as the 'Kelly criterion'). The theory for this is less model-dependent, but it leads to strategies that can be very risky over realistic investment horizons.In this research we develop an approach which, in our view, combines the virtues of Markowitz, Merton and Kelly and addresses the deficiencies of each.The approach based on the theory of risk-sensitive control (RSC). This subject was introduced in the control systems literature by Jacobson in the 1970s. Its application to asset management was pioneered by Bielecki and Pliska in the 1990s. In traditional stochastic control one seeks to maximize the expected value of some performance index. In RSC this is replaced by maximizing the exponential of a performance index. In the asset management application this index is the growth rate, and it turns out that the objective is equivalent to maximizing the expected growth rate with a penalty for variance (i.e. risk). This research is a joint project with New Bond Street Asset Management (NBSAM), whose business is managing portfolios involving credit risk, so it is essential that our mathematical models include credit events, i.e. jumps in asset prices. At present RSC theory only does this in limited ways, so new mathematics is required.There are four components to the work, (a) developing the theory of RSC, (b) producing demonstrably robust algorithms for solving the equations of RSC, (c) studying the mathematical modelling of the various classes of credit-risky securities traded by NBSAM and (d) undertaking an econometric analysis to establish the dependence of prices on economic factors. Part (a) involves extending the theory of so-called 'viscosity solutions' of nonlinear partial differential equations to cover integro-differential operators and solutions in unbounded domains with quite weak growth conditions, while part (b) will cover finite-difference schemes which are related to control problems for Markov chains.When completed, this work will give portfolio managers a fully dynamic asset allocation model that represents a huge improvement over current techniques.