Portfolio optimization in markets with stochastic dependence

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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.