Portfolio Optimization via Risk-Sensitive Control

本研究关注投资：投资者应该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Harry Markowitz引入的“均值-方差分析”，精确地量化了风险和回报之间的关系。马科维茨的思想渗透到金融行业，每个人都熟悉投资组合管理。然而，均值-方差理论有一些众所周知的缺点。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。虽然有可能估计后者，但不可能根据历史数据以任何可用的准确度估计前者。因此，需要进一步的建模步骤，将平均回报与其他市场或经济因素联系起来。其次，这是一个单期理论，在概念上，投资者购买一个投资组合，持有一段时间，然后兑现。什么是需要的是一个动态的理论，其中投资者的最佳反应，随着时间的推移，市场价格的运动是determined. System研究的动态理论的投资组合优化是在1969年开创性的工作，由罗伯特默顿谁制定了它作为一个随机控制问题的目标是最大化的终身效用的消费。这导致了效用最大化和随机控制数学的广泛发展，但其中很少进入资产管理的实际世界，主要可能是因为它过于依赖程式化的数学模型。另一种不同但相关的方法是最大化长期增长率（有时被称为“凯利准则”）。该理论对模型的依赖性较低，但它导致的战略，可以是非常危险的，在现实的投资horizons.In这项研究中，我们开发了一种方法，在我们看来，结合了马科维茨，默顿和凯利的优点，并解决各自的不足之处。这个主题是由Jacobson在20世纪70年代在控制系统文献中介绍的。Bielecki和普利斯卡在20世纪90年代率先将其应用于资产管理。在传统的随机控制中，人们寻求最大化某些性能指标的期望值。在RSC中，这被最大化性能指标的指数所取代。在资产管理应用程序中，该指数是增长率，结果表明，目标相当于最大化预期增长率，并对方差（即风险）进行惩罚。这项研究是与新债券街资产管理公司（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，即资产价格的跳跃。目前RSC理论只能以有限的方式做到这一点，因此需要新的数学。这项工作有四个组成部分，（a）发展RSC理论，（B）产生可证明的鲁棒算法来求解RSC方程，（c）研究NBSAM交易的各类信贷风险证券的数学模型，以及进行经济计量分析，以确定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”的理论，以涵盖具有相当弱增长条件的无界区域中的积分-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分配模型，这是对现有技术的巨大改进。

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.