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

A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.Comment: 25 pages, 5 figure

Cooperative Energy Trading in CoMP Systems Powered by Smart Grids

This paper studies the energy management in the coordinated multi-point (CoMP) systems powered by smart grids, where each base station (BS) with local renewable energy generation is allowed to implement the two-way energy trading with the grid. Due to the uneven renewable energy supply and communication energy demand over distributed BSs as well as the difference in the prices for their buying/selling energy from/to the gird, it is beneficial for the cooperative BSs to jointly manage their energy trading with the grid and energy consumption in CoMP based communication for reducing the total energy cost. Specifically, we consider the downlink transmission in one CoMP cluster by jointly optimizing the BSs' purchased/sold energy units from/to the grid and their cooperative transmit precoding, so as to minimize the total energy cost subject to the given quality of service (QoS) constraints for the users. First, we obtain the optimal solution to this problem by developing an algorithm based on techniques from convex optimization and the uplink-downlink duality. Next, we propose a sub-optimal solution of lower complexity than the optimal solution, where zero-forcing (ZF) based precoding is implemented at the BSs. Finally, through extensive simulations, we show the performance gain achieved by our proposed joint energy trading and communication cooperation schemes in terms of energy cost reduction, as compared to conventional schemes that separately design communication cooperation and energy trading