Multi-Period Trading via Convex Optimization

We consider a basic model of multi-period trading, which can be used to evaluate the performance of a trading strategy. We describe a framework for single-period optimization, where the trades in each period are found by solving a convex optimization problem that trades off expected return, risk, transaction cost and holding cost such as the borrowing cost for shorting assets. We then describe a multi-period version of the trading method, where optimization is used to plan a sequence of trades, with only the first one executed, using estimates of future quantities that are unknown when the trades are chosen. The single-period method traces back to Markowitz; the multi-period methods trace back to model predictive control. Our contribution is to describe the single-period and multi-period methods in one simple framework, giving a clear description of the development and the approximations made. In this paper we do not address a critical component in a trading algorithm, the predictions or forecasts of future quantities. The methods we describe in this paper can be thought of as good ways to exploit predictions, no matter how they are made. We have also developed a companion open-source software library that implements many of the ideas and methods described in the paper

Long memory of financial time series and hidden Markov models with time-varying parameters

Hidden Markov models are often used to model daily returns and to infer the hidden state of financial markets. Previous studies have found that the estimated models change over time, but the implications of the time-varying behavior have not been thoroughly examined. This paper presents an adaptive estimation approach that allows for the parameters of the estimated models to be time varying. It is shown that a two-state Gaussian hidden Markov model with time-varying parameters is able to reproduce the long memory of squared daily returns that was previously believed to be the most difficult fact to reproduce with a hidden Markov model. Capturing the time-varying behavior of the parameters also leads to improved one-step density forecasts. Finally, it is shown that the forecasting performance of the estimated models can be further improved using local smoothing to forecast the parameter variations

Greedy Gaussian Segmentation of Multivariate Time Series

We consider the problem of breaking a multivariate (vector) time series into segments over which the data is well explained as independent samples from a Gaussian distribution. We formulate this as a covariance-regularized maximum likelihood problem, which can be reduced to a combinatorial optimization problem of searching over the possible breakpoints, or segment boundaries. This problem can be solved using dynamic programming, with complexity that grows with the square of the time series length. We propose a heuristic method that approximately solves the problem in linear time with respect to this length, and always yields a locally optimal choice, in the sense that no change of any one breakpoint improves the objective. Our method, which we call greedy Gaussian segmentation (GGS), easily scales to problems with vectors of dimension over 1000 and time series of arbitrary length. We discuss methods that can be used to validate such a model using data, and also to automatically choose appropriate values of the two hyperparameters in the method. Finally, we illustrate our GGS approach on financial time series and Wikipedia text data

Dynamic Portfolio Optimization Across Hidden Market Regimes

Regime-based asset allocation has been shown to add value over rebalancing to static weights and, in particular, reduce potential drawdowns by reacting to changes in market conditions. The predominant approach in previous studies has been to specify in advance a static decision rule for changing the allocation based on the state of financial markets or the economy. This talk proposes the use of model predictive control to dynamically optimize a portfolio based on forecasts of the mean and variance of financial returns from a hidden Markov model with time-varying parameters. There are computational advantages to using model predictive control when estimates of future returns are updated repeatedly, since the optimal control actions are reconsidered anyway every time a new observation becomes available. Results from testing the approach on market data are presented and compared with previous, rule-based approaches. Further, imposing a trading penalty that reduces the number of trades is discussed as a way to increase the robustness of the approach

Hyperparameter Optimization for Portfolio Selection

Portfolio selection involves a trade-off between maximizing expected return and minimizing risk. In practice, useful formulations also include various costs and constraints that regularize the problem and reduce the risk due to estimation errors, resulting in solutions that depend on a number of hyperparameters. As the number of hyperparameters grows, selecting their value becomes increasingly important and difficult. In this article, the authors propose a systematic approach to hyperparameter optimization by leveraging recent advances in automated machine learning and multiobjective optimization. They optimize hyperparameters on a train set to yield the best result subject to market-determined realized costs. In applications to single- and multiperiod portfolio selection, they show that sequential hyperparameter optimization finds solutions with better risk–return trade-offs than manual, grid, and random search over hyperparameters using fewer function evaluations. At the same time, the solutions found are more stable from in-sample training to out-of-sample testing, suggesting they are less likely to be extremities that randomly happened to yield good performance in training